https://en.formulasearchengine.com/api.php?action=feedcontributions&user=Admin&feedformat=atomformulasearchengine - User contributions [en]2020-04-06T07:51:44ZUser contributionsMediaWiki 1.35.0-alphahttps://en.formulasearchengine.com/index.php?title=Template:Dump1911&diff=329759Template:Dump19112019-12-04T08:16:21Z<p>Admin: </p>
<hr />
<div>*[https://en.formulasearchengine.com/images/datasets/2019/raw/{{{1}}}-latest-pages-articles-multistream.xml.bz2 {{{1}}}],{{{2}}},{{{3}}}</div>Adminhttps://en.formulasearchengine.com/index.php?title=Template:Dump1911&diff=329758Template:Dump19112019-12-04T08:14:22Z<p>Admin: </p>
<hr />
<div>*[https://en.formulasearchengine.com/images/datasets/2019/raw/{{{1}}}wiki-latest-pages-articles-multistream.xml.bz2 {{{1}}}],{{{2}}},{{{3}}}</div>Adminhttps://en.formulasearchengine.com/index.php?title=Template:Dump1911&diff=329757Template:Dump19112019-12-04T08:13:27Z<p>Admin: </p>
<hr />
<div>*[{{{1}}} https://en.formulasearchengine.com/images/datasets/2019/{{{1}}}wiki-latest-pages-articles-multistream.xml.bz2],{{{2}}},{{{3}}}</div>Adminhttps://en.formulasearchengine.com/index.php?title=Template:Dump1911&diff=329756Template:Dump19112019-12-02T23:19:13Z<p>Admin: Created page with "*{{{1}}},{{{2}}},{{{3}}}"</p>
<hr />
<div>*{{{1}}},{{{2}}},{{{3}}}</div>Adminhttps://en.formulasearchengine.com/index.php?title=Dumps&diff=329755Dumps2019-12-02T23:18:01Z<p>Admin: Created page with "This page lists recent database dumps from 2019-11-20 {{dump1911|aawiki|898d76c401441dbc317fe4445ff600a0|35K}} {{dump1911|abwiki|a0e86d7cbcd62583ef0904315abc07c8|1.6M}} {{dum..."</p>
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<div>This page lists recent database dumps from 2019-11-20<br />
<br />
{{dump1911|aawiki|898d76c401441dbc317fe4445ff600a0|35K}}<br />
{{dump1911|abwiki|a0e86d7cbcd62583ef0904315abc07c8|1.6M}}<br />
{{dump1911|acewiki|9bb76f32950e2bf270adaf9c29b3cce2|2.9M}}<br />
{{dump1911|adywiki|7b6bde8793011e24b7e22e6730610f9d|368K}}<br />
{{dump1911|afwiki|33f9b7c93e991674589cc4e59c5bcaa0|96M}}<br />
{{dump1911|akwiki|c39cd8bd53a5f1547a427345b5e1484c|402K}}<br />
{{dump1911|alswiki|7a1deb47bcced0b4d782a6c3d71d5c5e|51M}}<br />
{{dump1911|amwiki|d3a0de4ce927ec397f1963da7a2b0a56|6.8M}}<br />
{{dump1911|angwiki|60fbd38816498d673c8bd4923e619ebe|4.1M}}<br />
{{dump1911|anwiki|6ef2a479617f5d0825893183d9c4125f|33M}}<br />
{{dump1911|arcwiki|ab0341ea8540542e0036abc7eb0625b6|1.1M}}<br />
{{dump1911|arwiki|71401925f31e3c1f7a23fc9731a89fa9|1.1G}}<br />
{{dump1911|arzwiki|3d61bf40c4635bebe54491ef5b0315d8|24M}}<br />
{{dump1911|astwiki|a97b7dc22a3242ab6ce3555a809b6093|217M}}<br />
{{dump1911|aswiki|cf30fdcf25ae98dada6bd3687a14f2e4|21M}}<br />
{{dump1911|atjwiki|1f8a525f363e11c26ba6a939d2f78bd3|493K}}<br />
{{dump1911|avwiki|3858a917c06a07d397b5a503ea01142c|4.1M}}<br />
{{dump1911|aywiki|1e99800c55890fd2fcbccc0b8182d5e0|2.2M}}<br />
{{dump1911|azbwiki|57464a1a562f9fbae03cac3436027067|68M}}<br />
{{dump1911|azwiki|66162c76998c533342a6c736cce5bb90|176M}}<br />
{{dump1911|barwiki|c63817d3c226199fdee85db23b8ad477|32M}}<br />
{{dump1911|bat_smgwiki|e855ec6baaff24a9a2c19c8db964c5ee|4.8M}}<br />
{{dump1911|bawiki|7f8679033aadfd8ef9b1a2aa6cb83000|63M}}<br />
{{dump1911|bclwiki|e516e6b820781d31a75364a4fc72412d|7.3M}}<br />
{{dump1911|bewiki|6664bf756459005dad9da13ef3fc1d8b|205M}}<br />
{{dump1911|be_x_oldwiki|6a319701b6fad31873502ff643c31f87|79M}}<br />
{{dump1911|bgwiki|0f0d0a9ae9b6cde8bd70c24c874360fd|341M}}<br />
{{dump1911|bhwiki|3b1570b20d46ea5d83dd0d48dbdd94af|14M}}<br />
{{dump1911|biwiki|bd2b7fe2e8e11faa86dee9e4cc7338ba|433K}}<br />
{{dump1911|bjnwiki|788c768779f7c5479aa02d5393e46623|2.6M}}<br />
{{dump1911|bmwiki|78300c846724081464a0a56bc383da5b|450K}}<br />
{{dump1911|bnwiki|61e075efe01edc6fb1b936f3a7573cd5|174M}}<br />
{{dump1911|bowiki|d3977868cd2a0d82abd8afed038a9cb4|14M}}<br />
{{dump1911|bpywiki|58b879fa88dd1068233d83faca468d5d|5.1M}}<br />
{{dump1911|brwiki|a628ce7bb12064e3c0669babbd1580ec|50M}}<br />
{{dump1911|bswiki|5b669c92f8ac407bd1b1674cc17f5de2|109M}}<br />
{{dump1911|bugwiki|f9d86c383d31f73c9643a65fcc240acf|1.8M}}<br />
{{dump1911|bxrwiki|13dde056516f3e48e24d6ece1eaeb32a|3.3M}}<br />
{{dump1911|cawiki|d40569a47a262c6328e7aa2e382e38ec|879M}}<br />
{{dump1911|cbk_zamwiki|b60787945294596c93836ae75e67ce91|1.9M}}<br />
{{dump1911|cdowiki|d2e2f167558d386319e4321830c60fce|4.3M}}<br />
{{dump1911|cebwiki|fcf2be9f5e0e35353693d4ae9ee2e8d9|1.9G}}<br />
{{dump1911|cewiki|a4e82f4efb2b61448e292ee0d8bf9c22|50M}}<br />
{{dump1911|chowiki|0d57d49578e9a7ec6ad803ed4ab9b799|17K}}<br />
{{dump1911|chrwiki|0464d726ea9fe4ae178512d73eb278dd|628K}}<br />
{{dump1911|chwiki|45f7cd6747f5a661e1a91de6fef1eaa2|690K}}<br />
{{dump1911|chywiki|e21d64325634c741604c967bce96a310|329K}}<br />
{{dump1911|ckbwiki|4a483c0329e276e4242760688febf571|26M}}<br />
{{dump1911|cowiki|be80f3fa70ddd010a7a7aa19c7ca531e|3.6M}}<br />
{{dump1911|crhwiki|4220dd1bb05db2546a00af7f72b76ab7|4.3M}}<br />
{{dump1911|crwiki|cb4898db3be0de6bad21c02f1162ea1f|260K}}<br />
{{dump1911|csbwiki|a8c9c7793d065f4d74f945b968c6b5a3|2.2M}}<br />
{{dump1911|cswiki|ecab691eb226e8b032e5a6a35595e27d|805M}}<br />
{{dump1911|cuwiki|54b4d4ceb75d82aadf0ccfef377480c6|649K}}<br />
{{dump1911|cvwiki|cf475be23c3d7477b7a3006c72ac02f9|23M}}<br />
{{dump1911|cywiki|38399ce262851608a0650a5ed295d766|69M}}<br />
{{dump1911|dawiki|5d20a04e4cea326859df577ec42f7728|337M}}<br />
{{dump1911|dewiki|cba54bf2fc1bf428d480b2219229dda3|5.3G}}<br />
{{dump1911|dinwiki|73ffc49835f1eb7741769235b5b2e663|474K}}<br />
{{dump1911|diqwiki|8e066a903133433d18ab96b1a5ad3221|8.2M}}<br />
{{dump1911|dsbwiki|d721386d4244d852aa4ff27593c62f5d|3.8M}}<br />
{{dump1911|dtywiki|78d81663076b13781bec3f652c53c6c4|6.2M}}<br />
{{dump1911|dvwiki|ec5a6fc7b6515b09488b36de6e21decc|4.4M}}<br />
{{dump1911|dzwiki|fdaf16ce9dc481434193d0c88f77f8b6|366K}}<br />
{{dump1911|eewiki|2115ffa12e08b71fbed90734205d3c4d|445K}}<br />
{{dump1911|elwiki|b17b8aa97261c41143f273e5262bc298|350M}}<br />
{{dump1911|emlwiki|859315448798aa60df97276402653f0a|8.0M}}<br />
{{dump1911|enwiki|9ff4f7dfca3add074de12363ae5e6638|17G}}<br />
{{dump1911|eowiki|25a888e2eba7da9f0f5dcffb5e8a861b|260M}}<br />
{{dump1911|eswiki|6ac6582ba87c4191fee1f14f256545fa|3.1G}}<br />
{{dump1911|etwiki|9b2d1f488c3d9b69f0bf016602357bdb|208M}}<br />
{{dump1911|euwiki|96f994edd39264791241953441610e8b|192M}}<br />
{{dump1911|extwiki|3fa68dcbbd37951ec8f4710a0d95aaec|2.5M}}<br />
{{dump1911|fawiki|7b79aaaabbcfa8d73905680b03488dfc|750M}}<br />
{{dump1911|ffwiki|5dfddef4714ffae5d61901363fc5e7bc|396K}}<br />
{{dump1911|fiu_vrowiki|cb99e9d09caa0964bd459e032c24ab5d|2.1M}}<br />
{{dump1911|fiwiki|1592f3613fd9c6ffa119d5f7f745f21a|692M}}<br />
{{dump1911|fjwiki|42ee4297a63038b81e1e1fe548e79010|266K}}<br />
{{dump1911|fowiki|976d8c03d8703214aa6c76c3badd8e44|14M}}<br />
{{dump1911|frpwiki|43e6d67a9176008fa3a0b585605d1c15|2.2M}}<br />
{{dump1911|frrwiki|6d72f1875abc92fd7638010b04c74443|8.3M}}<br />
{{dump1911|frwiki|a0c267ada13dfc4da745ad12ceecf8c4|4.4G}}<br />
{{dump1911|furwiki|ec7707174d8335dc7280269a1f6f8e83|2.4M}}<br />
{{dump1911|fywiki|3623b17fb2eb9f72b167c9e269841b94|49M}}<br />
{{dump1911|gagwiki|099d2fad868cbe9eb5b73ef8cdfe142a|2.1M}}<br />
{{dump1911|ganwiki|807ffbc6fc960aacf3c17e8da49dd118|3.9M}}<br />
{{dump1911|gawiki|42baa9139c14d7d40a2ce3c5dab56b9b|27M}}<br />
{{dump1911|gdwiki|67227e80e09d42988f51b40467124a65|8.6M}}<br />
{{dump1911|glkwiki|d5f9871c8b6f1f565b53730c4fe8d187|2.0M}}<br />
{{dump1911|glwiki|64aae2e653ed2c2ce646cd847390c3a9|250M}}<br />
{{dump1911|gnwiki|4446919c1e52e1c8b40ed9cbf89f16ca|3.5M}}<br />
{{dump1911|gomwiki|1a0b17fa10a41283d08fda6ce61b6dc8|6.3M}}<br />
{{dump1911|gorwiki|ec18eeea322fd4d70482d0c4551c3bbb|1.5M}}<br />
{{dump1911|gotwiki|a5c06e6edf3f61e1a98dd5682790d5be|634K}}<br />
{{dump1911|guwiki|0f6f07223ffa2f8879783da9256b56c1|29M}}<br />
{{dump1911|gvwiki|510f41c9f4be929cec5b79402a8e7dd6|5.4M}}<br />
{{dump1911|hakwiki|d3f0f8e6bf9588657a8d42043f30cfe3|3.7M}}<br />
{{dump1911|hawiki|7ad0895c919a320cc69cc7a58d8153a3|2.1M}}<br />
{{dump1911|hawwiki|88cca4f830c0013116a3b109847c81ba|1.4M}}<br />
{{dump1911|hewiki|9c2d5eef4643a3050269ba767af696e8|612M}}<br />
{{dump1911|hifwiki|5f9df4034ac216d4b706cf79f5922a07|4.6M}}<br />
{{dump1911|hiwiki|90a74ef0a886e5c3b0465e96082ae86a|148M}}<br />
{{dump1911|howiki|91c5b91294df4fd2e71e78c501c4a3e4|8.7K}}<br />
{{dump1911|hrwiki|6374a4f6d48ff5965c5d79e0f7a9dd96|257M}}<br />
{{dump1911|hsbwiki|d5b261605bdd913b41de4b54d7f1948d|11M}}<br />
{{dump1911|htwiki|5e6c17f467228683df5cb423ee48fa21|13M}}<br />
{{dump1911|huwiki|697c88c3d0e893d3817c5aa58b69b16b|849M}}<br />
{{dump1911|hywiki|b5b3bf60c0e0b39a406f80e05962dca6|296M}}<br />
{{dump1911|hywwiki|f0f9337c84777fd7e15745f099854451|13M}}<br />
{{dump1911|hzwiki|e0928ac36c72bb2523586cfa633d99b9|6.7K}}<br />
{{dump1911|iawiki|60a1ce632886635498d42ceb0ab3f008|8.5M}}<br />
{{dump1911|idwiki|fcf67b1c20fb4af75fce5305baa80cea|573M}}<br />
{{dump1911|iewiki|94f57adf8bcf2dc80362dc2077147a9e|1.8M}}<br />
{{dump1911|igwiki|b0cec3f566129140e0d5b95d3431faa4|1.1M}}<br />
{{dump1911|iiwiki|c147163269d3b6ca173a866533ffd9f0|22K}}<br />
{{dump1911|ikwiki|3745eef735c2cbad8362761f7c82fcc3|240K}}<br />
{{dump1911|ilowiki|853baccc4dd60d2d3fe30e6c644f91d6|17M}}<br />
{{dump1911|inhwiki|3260ba6cd0d92c12a7e1b97c214c0321|1.9M}}<br />
{{dump1911|iowiki|34d2321c006db5b2ccb638a6307d4736|14M}}<br />
{{dump1911|iswiki|487547f943c3a5088192505cc2bd7786|45M}}<br />
{{dump1911|itwiki|19d51e4531135a11165b0494cba0e3a8|2.8G}}<br />
{{dump1911|iuwiki|45d3dd0cf409fa1ad1b677b9f098702a|281K}}<br />
{{dump1911|jamwiki|f4399ab8493997336ad358e5301a46f5|895K}}<br />
{{dump1911|jawiki|280d712a6d3871426d724a65ae086a03|2.9G}}<br />
{{dump1911|jbowiki|baac96b4f0032d7c57bbf4a36a6461d4|1.1M}}<br />
{{dump1911|jvwiki|763bce273f371c2273eca663278e6132|42M}}<br />
{{dump1911|kaawiki|2e65b0207d2486c8ddda5f0a006fc98a|1.4M}}<br />
{{dump1911|kabwiki|47cf299c24fdc2bd8c61ae67f457fa56|2.9M}}<br />
{{dump1911|kawiki|4b6d90084aed499e4a17785eb3bdb31b|140M}}<br />
{{dump1911|kbdwiki|9c10f1450a3732d1fb11b71d862019ca|1.7M}}<br />
{{dump1911|kbpwiki|bd2dcfe5efe6516dfe7c7dbf4d5aedd7|1.4M}}<br />
{{dump1911|kgwiki|1453870b5bd13cf37edb8be127384a14|442K}}<br />
{{dump1911|kiwiki|8a32ddb8d596160fe9692a7a95a389a7|365K}}<br />
{{dump1911|kjwiki|9c3fb82a6798ef2b1a52c9f2930fdede|6.9K}}<br />
{{dump1911|kkwiki|9e025458b47147d01a4095c43438d679|116M}}<br />
{{dump1911|klwiki|51094bdf38bcbe3e1ded7d272593755a|866K}}<br />
{{dump1911|kmwiki|a8f7c29e085762c0531ab9bae8995aed|23M}}<br />
{{dump1911|knwiki|a2e3b0dc222faae423d0f14927fbe35f|72M}}<br />
{{dump1911|koiwiki|0e92d1b0902defc4feac5cd650a8741a|2.2M}}<br />
{{dump1911|kowiki|f2adf170c0340e309c35362f0fdf82bb|668M}}<br />
{{dump1911|krcwiki|28eaf0e7cc73f1dfe5863cd32f5fbb5f|3.2M}}<br />
{{dump1911|krwiki|de8b78ed46b67d3a5442910aaeebdc84|4.3K}}<br />
{{dump1911|kshwiki|1c3d32039f18b48c25ed70e936c1eed2|3.1M}}<br />
{{dump1911|kswiki|e362b67e5f8aa72aad21c869c2c34347|312K}}<br />
{{dump1911|kuwiki|27b403c8fa854598c5a57a3c0d198397|18M}}<br />
{{dump1911|kvwiki|b1e21dd7d025ca6c4da63043571e325b|3.5M}}<br />
{{dump1911|kwwiki|1dd1601616adba00c3ced750e4ee1856|1.9M}}<br />
{{dump1911|kywiki|20614893ec185a3e3f56c4411bb0f10f|34M}}<br />
{{dump1911|ladwiki|8825bebcfa3b773b6251cc471f0900ea|3.4M}}<br />
{{dump1911|lawiki|32ca061a7c46c1c46950872b0a527b4d|86M}}<br />
{{dump1911|lbewiki|da8d1be25ffd27845b805ad875813396|1.3M}}<br />
{{dump1911|lbwiki|cec70bf74fc588550a3a1be788b941cf|47M}}<br />
{{dump1911|lezwiki|b060b29978c5a83591af5bd417007f71|4.4M}}<br />
{{dump1911|lfnwiki|8ee6c55e337c0d064e3dd94e709b6a95|3.5M}}<br />
{{dump1911|lgwiki|485f03480b8115d4cf692d5d63ec4320|1.6M}}<br />
{{dump1911|lijwiki|d6b96ff1929d03c4c950250fc4bd2ed9|2.9M}}<br />
{{dump1911|liwiki|7c012d0eff304cc3e410fdfdf4f4ae09|15M}}<br />
{{dump1911|lmowiki|efdb1ba54fe9ef3173c68a349944d1b1|22M}}<br />
{{dump1911|lnwiki|12995794c9cc4f503ab2b70c213ffe97|1.9M}}<br />
{{dump1911|lowiki|6d396d44e34ea8ab9f7058edcf1c91fb|4.2M}}<br />
{{dump1911|lrcwiki|7903700947a7c106b66e75cf4f7f352e|5.4M}}<br />
{{dump1911|ltgwiki|13a0a706093161e73780ef156ba3ce00|856K}}<br />
{{dump1911|ltwiki|7b36d4aa488248454f486e0888708644|180M}}<br />
{{dump1911|lvwiki|efe9d5f18b97e2103a38f9f24353d1e0|135M}}<br />
{{dump1911|maiwiki|05e3e29656534de86a424b1ae09d1c47|12M}}<br />
{{dump1911|map_bmswiki|c59bbcba05cb9a61f8236565f6706329|4.6M}}<br />
{{dump1911|mdfwiki|6e815899dc52102cd669727c1caf2a1b|1.2M}}<br />
{{dump1911|mgwiki|69d0394c5c672e57021d34f1c5d4d99d|27M}}<br />
{{dump1911|mhrwiki|a4c9650012394b191b3ba32e9c994d00|5.9M}}<br />
{{dump1911|mhwiki|5781c543e18349e2556cf47d667f2788|19K}}<br />
{{dump1911|minwiki|cf209f5b606073102db741918b841c27|27M}}<br />
{{dump1911|miwiki|a9eba40c389b98d51d760e05c2817e80|2.0M}}<br />
{{dump1911|mkwiki|04b60c140bc9383c6453898b55817141|150M}}<br />
{{dump1911|mlwiki|58a2055d32d6abcaddc67906601e0cc1|128M}}<br />
{{dump1911|mnwiki|6043ee3ea75fc7df5aaf786835566c2e|30M}}<br />
{{dump1911|mrjwiki|cd8556538eaf7c76e7ff6d3f8f9be4d4|3.1M}}<br />
{{dump1911|mrwiki|9d960f6ef6567dd9059198e8ff5502a6|53M}}<br />
{{dump1911|mswiki|c27c8fdcb92bff72568fd86af4e72a88|222M}}<br />
{{dump1911|mtwiki|abb838b473ecf6d86ed78ac472b2b46b|8.5M}}<br />
{{dump1911|muswiki|b0e3285701bcf0d0a998ecab884c6174|4.5K}}<br />
{{dump1911|mwlwiki|e9b5dbc191011edd39a6db6df97bf4de|9.1M}}<br />
{{dump1911|myvwiki|4b561496c0f427470bf365468af2a678|8.7M}}<br />
{{dump1911|mywiki|01e22936904688a752c16afb4d7e2c5f|37M}}<br />
{{dump1911|mznwiki|5f1606cae3167943a941d4d0fb000fea|6.7M}}<br />
{{dump1911|nahwiki|f3a3227407ba969d8bd64bedeb412cdf|4.4M}}<br />
{{dump1911|napwiki|c329b5222115a0b590dda6021e9150fb|5.2M}}<br />
{{dump1911|nawiki|8ead0924ce321b29e303434c58172347|480K}}<br />
{{dump1911|nds_nlwiki|a7bbe57132a70a12f783a8dc6ed02701|6.8M}}<br />
{{dump1911|ndswiki|303a0c3d672d13ee878735dca21d960d|37M}}<br />
{{dump1911|newiki|9e4a5d2e0faad9253b56ee7db04e870b|33M}}<br />
{{dump1911|newwiki|1d8c0ce67ed8e88290066e53c9feaa84|17M}}<br />
{{dump1911|ngwiki|3272fdf764606bff9b5c8ebe72f7ba0f|82K}}<br />
{{dump1911|nlwiki|21e0dd91f90bc765301ac04b057ed0d8|1.5G}}<br />
{{dump1911|nnwiki|9dd8aa3737f373fddcdea06956de2b5a|132M}}<br />
{{dump1911|novwiki|8ce7657718f481e6ec61dccf828a30ce|1.2M}}<br />
{{dump1911|nowiki|d57264a880a18672f30f7c55bbff6783|610M}}<br />
{{dump1911|nrmwiki|77216c216cd4a71c449b0f6ff6d1f2bd|1.7M}}<br />
{{dump1911|nsowiki|ddbd9e774f823e31ab689689f8b882d0|2.3M}}<br />
{{dump1911|nvwiki|50bbb721a13b1438b615e5b5886e0b73|3.1M}}<br />
{{dump1911|nywiki|06e8737ba55e721feda5b237b44e512b|1.3M}}<br />
{{dump1911|ocwiki|f4f11cfe2f7f77a16eeaba48e85d9453|73M}}<br />
{{dump1911|olowiki|a59f77a904bda95c70098929b1483157|1.8M}}<br />
{{dump1911|omwiki|f9d890f8c934960690db881b279d077d|1.1M}}<br />
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{{dump1911|oswiki|35c03b54b17af34b9e27465c8032d1d5|7.7M}}<br />
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{{dump1911|pihwiki|15f5f5d52ae52acd5409ad185aaac5a3|709K}}<br />
{{dump1911|piwiki|b8054a1bb3fe942aae1aed9c2c45227f|591K}}<br />
{{dump1911|plwiki|06c9e0861c1e4a55fcd3b0546f9885f9|1.9G}}<br />
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{{dump1911|sahwiki|6591af04339d4eae235c5f34604dc481|13M}}<br />
{{dump1911|satwiki|ce0fb6f3e4677529a81359d36eb36e92|5.1M}}<br />
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{{dump1911|sowiki|00aff0aa5f602d8f2e89fc911d8ecbe9|8.4M}}<br />
{{dump1911|sqwiki|9b8e9c4def8060932da49e8ef93cd7e8|83M}}<br />
{{dump1911|srnwiki|b401cef4a648be3c9ccf98adb0d7d5f8|634K}}<br />
{{dump1911|srwiki|c0c024afc49902b872a7887b50d309cf|765M}}<br />
{{dump1911|sswiki|d4b5449b9ffbb9d75f7da5be4e8a311f|748K}}<br />
{{dump1911|stqwiki|f40201f798fc22800cbe1561b5f1f79f|3.4M}}<br />
{{dump1911|stwiki|c14c957ab387a8490f9769c6d556c5e6|530K}}<br />
{{dump1911|suwiki|75d0518b6673fd550088cdfe460fa999|24M}}<br />
{{dump1911|svwiki|8cecb08dd4fc764df5c042fdc9141f54|1.7G}}<br />
{{dump1911|swwiki|513721a587324b0b0c13d193edd3a460|30M}}<br />
{{dump1911|szlwiki|e2b5e87d1927f55bff7b6c6dcb66f4de|12M}}<br />
{{dump1911|tawiki|94de6245fd3801bfa7ee4111d37a6ae7|150M}}<br />
{{dump1911|tcywiki|b7ec020120b7105553d42be3963e95ec|2.7M}}<br />
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{{dump1911|tewiki|d19472dedb47e33812c08558ef2a283b|119M}}<br />
{{dump1911|tgwiki|ce7e7c64f265f56fd583fab42a610df3|39M}}<br />
{{dump1911|thwiki|842e4ba2cee443a015a40c37009b42df|267M}}<br />
{{dump1911|tiwiki|f8951cdf5f3f27e4206e0592126cd5f9|312K}}<br />
{{dump1911|tkwiki|207e63661636229844b826193a46cf86|4.6M}}<br />
{{dump1911|tlwiki|90c92d743e177343b45d787be54e627a|54M}}<br />
{{dump1911|tnwiki|c1ce695b062680e0a48c8c44474d2a1b|1.4M}}<br />
{{dump1911|towiki|e724df76042c82c1020723b5cdc80feb|775K}}<br />
{{dump1911|tpiwiki|d82dd69bc72bad6fda1661a5f75dc837|1.5M}}<br />
{{dump1911|trwiki|61b065669effd2413f3db038bb49e7ec|519M}}<br />
{{dump1911|tswiki|ba03a5da28de1bfaee53234a9a49c6ee|1.5M}}<br />
{{dump1911|ttwiki|7c5f89569141421e53f55be9960e5e48|57M}}<br />
{{dump1911|tumwiki|cf5fcdb96b361e432d5e63e64913cb2b|323K}}<br />
{{dump1911|twwiki|420f6022496a1ef21a8616a5ed52684e|379K}}<br />
{{dump1911|tyvwiki|52f0c3f4d84db1c773bf9705a8920546|2.8M}}<br />
{{dump1911|tywiki|bcbac7e1667bd88744676737ec42b800|485K}}<br />
{{dump1911|udmwiki|edd4a33eddaba5e32bf6d4190ba0818d|3.2M}}<br />
{{dump1911|ugwiki|86297681bfa0f3b23539de8f2b8da7a5|5.8M}}<br />
{{dump1911|ukwiki|3a0399caf881c9cafc3ca6d040c54812|1.4G}}<br />
{{dump1911|urwiki|fc8f34ad56977cbf6aa3a7c452087f22|139M}}<br />
{{dump1911|uzwiki|03deef2ec3fe2d99859f7e6dfc96a225|63M}}<br />
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{{dump1911|vepwiki|c637a7ad4faadaf5f905a3a6fb7de1d6|5.3M}}<br />
{{dump1911|vewiki|ae2bf72ad27d4528ee5f4552eb769c1a|262K}}<br />
{{dump1911|viwiki|fcee1b252eb13621b57f7830088748c5|694M}}<br />
{{dump1911|vlswiki|04c617d10290b091eae8de5d580699f8|6.8M}}<br />
{{dump1911|vowiki|a48fb6146c06c3dfad8b3ab86e74101f|25M}}<br />
{{dump1911|warwiki|5842c758f598d1a73f2b20242184721b|258M}}<br />
{{dump1911|wawiki|7a7cdfeab227d7f9bca5628802121654|9.1M}}<br />
{{dump1911|wowiki|41347c11f4c8862ebaa7d89187ecebd7|1.8M}}<br />
{{dump1911|wuuwiki|325be95b68b4cce1d2bd55149a99b046|11M}}<br />
{{dump1911|xalwiki|78ee34c0172d6dd13f632f6bf977a40d|1.7M}}<br />
{{dump1911|xhwiki|2c382c4c5948c6d41fc9791afe5b04cb|1.4M}}<br />
{{dump1911|xmfwiki|26aa1ee61940575324b108fed7927d0c|11M}}<br />
{{dump1911|yiwiki|8c1d141f69a08f426a105fb5120af90c|12M}}<br />
{{dump1911|yowiki|f122df0dd882a4f744e6306a5746fc66|12M}}<br />
{{dump1911|zawiki|94dd2bc16f4223045afe44bef71a99a0|760K}}<br />
{{dump1911|zeawiki|98ffc72b1565c0198ff34c723a80b9a2|2.5M}}<br />
{{dump1911|zh_classicalwiki|a7849dc5626ebfbd647f5214dcd98098|15M}}<br />
{{dump1911|zh_min_nanwiki|899fa9ba15dd96648d309780362cadb9|51M}}<br />
{{dump1911|zhwiki|5db1a24da4d7164cc4ae9dff95e8b56c|1.9G}}<br />
{{dump1911|zh_yuewiki|84641c0781458424b83bc80c606ae5cd|56M}}<br />
{{dump1911|zuwiki|1c4b63f74a3ed7802bbcb9c8d91362df|1.8M}}</div>Adminhttps://en.formulasearchengine.com/index.php?title=Main_Page&diff=329754Main Page2019-09-15T21:52:09Z<p>Admin: </p>
<hr />
<div>This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.<br />
<br />
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]<br />
* Only registered users will be able to execute this rendering mode.<br />
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.<br />
<br />
Registered users will be able to choose between the following three rendering modes: <br />
<br />
'''MathML'''<br />
:<math forcemathmode="mathml">E=mc^2</math><br />
<br />
<!--'''PNG''' (currently default in production)<br />
:<math forcemathmode="png">E=mc^2</math><br />
<br />
'''source'''<br />
:<math forcemathmode="source">E=mc^2</math> --><br />
<br />
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].<br />
<br />
==Demos==<br />
<br />
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:<br />
<br />
<br />
* accessibility:<br />
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]<br />
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].<br />
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].<br />
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.<br />
<br />
==Test pages ==<br />
<br />
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:<br />
*[[Displaystyle]]<br />
*[[MathAxisAlignment]]<br />
*[[Styling]]<br />
*[[Linebreaking]]<br />
*[[Unique Ids]]<br />
*[[Help:Formula]]<br />
<br />
*[[Inputtypes|Inputtypes (private Wikis only)]]<br />
*[[Url2Image|Url2Image (private Wikis only)]]<br />
==Bug reporting==<br />
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .</div>Adminhttps://en.formulasearchengine.com/index.php?title=Bug/T108388&diff=218403Bug/T1083882015-08-09T10:13:36Z<p>Admin: Created page with "<math>\Alpha\Beta\Delta\Gamma</math>"</p>
<hr />
<div><math>\Alpha\Beta\Delta\Gamma</math></div>Adminhttps://en.formulasearchengine.com/index.php?title=User:Admin&diff=218402User:Admin2015-08-05T17:21:26Z<p>Admin: </p>
<hr />
<div>The formula, <math forcemathmode=mathml>1=1</math>, is valid.<br />
<br />
The formula, <math forcemathmode=mathml>E=mc^2,</math> is valid.</div>Adminhttps://en.formulasearchengine.com/index.php?title=User:Admin&diff=218401User:Admin2015-08-05T17:20:55Z<p>Admin: Created page with "The formula, <math forcemathmode=mathml>E=mc^2</math>, is valid. The formula, <math forcemathmode=mathml>E=mc^2,</math> is valid."</p>
<hr />
<div>The formula, <math forcemathmode=mathml>E=mc^2</math>, is valid.<br />
<br />
The formula, <math forcemathmode=mathml>E=mc^2,</math> is valid.</div>Adminhttps://en.formulasearchengine.com/index.php?title=Main_Page&diff=218400Main Page2015-08-05T13:09:29Z<p>Admin: </p>
<hr />
<div>This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.<br />
<br />
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]<br />
* Only registered users will be able to execute this rendering mode.<br />
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.<br />
<br />
Registered users will be able to choose between the following three rendering modes: <br />
<br />
'''MathML'''<br />
:<math forcemathmode="mathml">E=mc^2</math><br />
<br />
'''PNG''' (currently default in production)<br />
:<math forcemathmode="png">E=mc^2</math><br />
<br />
'''source'''<br />
:<math forcemathmode="source">E=mc^2</math> <br />
<br />
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].<br />
<br />
==Demos==<br />
<br />
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:<br />
<br />
<br />
* accessibility:<br />
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]<br />
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].<br />
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].<br />
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.<br />
<br />
==Test pages ==<br />
<br />
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:<br />
*[[Displaystyle]]<br />
*[[MathAxisAlignment]]<br />
*[[Styling]]<br />
*[[Linebreaking]]<br />
*[[Unique Ids]]<br />
*[[Help:Formula]]<br />
<br />
*[[Inputtypes|Inputtypes (private Wikis only)]]<br />
*[[Url2Image|Url2Image (private Wikis only)]]<br />
==Bug reporting==<br />
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .</div>Adminhttps://en.formulasearchengine.com/index.php?title=Main_Page&diff=218399Main Page2015-08-05T13:09:02Z<p>Admin: </p>
<hr />
<div>This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.<br />
<br />
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]<br />
* Only registered users will be able to execute this rendering mode.<br />
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.<br />
<br />
Registered users will be able to choose between the following three rendering modes: <br />
<br />
'''MathML'''<br />
:<math forcemathmode="mathml">E=mc^2</math><br />
<br />
'''PNG''' (currently default in production)<br />
:<math forcemathmode="png">E=mc^2</math><br />
<br />
'''source'''<br />
:<math forcemathmode="source">E=mc^2</math> <!-- forcemathmode="3" temporary disabled --><br />
<br />
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].<br />
<br />
==Demos==<br />
<br />
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:<br />
<br />
<br />
* accessibility:<br />
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]<br />
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].<br />
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].<br />
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.<br />
<br />
==Test pages ==<br />
<br />
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:<br />
*[[Displaystyle]]<br />
*[[MathAxisAlignment]]<br />
*[[Styling]]<br />
*[[Linebreaking]]<br />
*[[Unique Ids]]<br />
*[[Help:Formula]]<br />
<br />
*[[Inputtypes|Inputtypes (private Wikis only)]]<br />
*[[Url2Image|Url2Image (private Wikis only)]]<br />
==Bug reporting==<br />
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .</div>Adminhttps://en.formulasearchengine.com/index.php?title=Help:Color&diff=218398Help:Color2015-07-31T15:04:45Z<p>Admin: Created page with "== Color == Equations can use color with the <code>\color</code> command. The default Texvc renderer and the MathJax renderers have different syntaxes..."</p>
<hr />
<div>== Color ==<br />
Equations can use color with the <code>\color</code> command. The default [[Wikipedia:Texvc|Texvc]] renderer and the [[MathJax]] renderers have different syntaxes to support both use <code>{\color{Blue}{text}}</code>. For example<br />
* <source lang="text" enclose="none">{\color{Blue}{x^2}}+{\color{Orange}{2x}}-{\color{LimeGreen}{1}}</source><br />
*: <math>{\color{Blue}{x^2}}+{\color{Orange}{2x}}-{\color{LimeGreen}{1}}</math><br />
* <source lang="text" enclose="none">x_{1,2}=\frac{{\color{Blue}{-b}}\pm\sqrt{\color{Red}{b^2-4ac}}}{\color{Green}{2a}}</source><br />
*: <math>x_{1,2}=\frac{{\color{Blue}{-b}}\pm\sqrt{\color{Red}{b^2-4ac}}}{\color{Green}{2a}}</math><br />
There are several alternate notations styles<br />
* <source lang="text" enclose="none">{\color{Blue}x^2}+{\color{Orange}2x}-{\color{LimeGreen}1}</source> works with both texvc and MathJax<br />
*: <math>{\color{Blue}x^2}+{\color{Orange}2x}-{\color{LimeGreen}1}</math><br />
* <source lang="text" enclose="none">\color{Blue}x^2\color{Black}+\color{Orange}2x\color{Black}-\color{LimeGreen}1</source> works with both texvc and MathJax<br />
*: <math>\color{Blue}x^2\color{Black}+\color{Orange}2x\color{Black}-\color{LimeGreen}1</math><br />
* <source lang="text" enclose="none">\color{Blue}{x^2}+\color{Orange}{2x}-\color{LimeGreen}{1}</source> only works with MathJax<br />
*: <math>\color{Blue}{x^2}+\color{Orange}{2x}-\color{LimeGreen}{1}</math><br />
<br />
Some color names are predeclared according to the following table, you can use them directly for the rendering of formulas (or for declaring the intended color of the page background).<br />
<br />
{| class="wikitable"<br />
|+ Colors supported<br />
|-<br />
| <math>\color{Apricot}{\text{Apricot}}</math> ||style="background-color: gray;"| <math>\pagecolor{Gray}\color{Aquamarine}{\text{Aquamarine}}</math> || <math>\color{Bittersweet}{\text{Bittersweet}}</math> || <math>\color{Black}{\text{Black}}</math><br />
|-<br />
| <math>\color{Blue}{\text{Blue}}</math> || <math>\color{BlueGreen}{\text{BlueGreen}}</math> || <math>\color{BlueViolet}{\text{BlueViolet}}</math> || <math>\color{BrickRed}{\text{BrickRed}}</math><br />
|-<br />
| <math>\color{Brown}{\text{Brown}}</math> || <math>\color{BurntOrange}{\text{BurntOrange}}</math> || <math>\color{CadetBlue}{\text{CadetBlue}}</math> || <math>\color{CarnationPink}{\text{CarnationPink}}</math><br />
|-<br />
| <math>\color{Cerulean}{\text{Cerulean}}</math> || <math>\color{CornflowerBlue}{\text{CornflowerBlue}}</math> ||style="background-color: gray;"| <math>\pagecolor{Gray}\color{Cyan}{\text{Cyan}}</math> || <math>\color{Dandelion}{\text{Dandelion}}</math><br />
|-<br />
| <math>\color{DarkOrchid}{\text{DarkOrchid}}</math> || <math>\color{Emerald}{\text{Emerald}}</math> || <math>\color{ForestGreen}{\text{ForestGreen}}</math> || <math>\color{Fuchsia}{\text{Fuchsia}}</math><br />
|-<br />
| <math>\color{Goldenrod}{\text{Goldenrod}}</math> || <math>\color{Gray}{\text{Gray}}</math> || <math>\color{Green}{\text{Green}}</math> ||style="background-color: gray;"| <math>\pagecolor{Gray}\color{GreenYellow}{\text{GreenYellow}}</math><br />
|-<br />
| <math>\color{JungleGreen}{\text{JungleGreen}}</math> ||style="background-color: gray;"| <math>\pagecolor{Gray}\color{Lavender}{\text{Lavender}}</math> || <math>\color{LimeGreen}{\text{LimeGreen}}</math> || <math>\color{Magenta}{\text{Magenta}}</math><br />
|-<br />
| <math>\color{Mahogany}{\text{Mahogany}}</math> || <math>\color{Maroon}{\text{Maroon}}</math> || <math>\color{Melon}{\text{Melon}}</math> || <math>\color{MidnightBlue}{\text{MidnightBlue}}</math><br />
|-<br />
| <math>\color{Mulberry}{\text{Mulberry}}</math> || <math>\color{NavyBlue}{\text{NavyBlue}}</math> || <math>\color{OliveGreen}{\text{OliveGreen}}</math> || <math>\color{Orange}{\text{Orange}}</math><br />
|-<br />
| <math>\color{OrangeRed}{\text{OrangeRed}}</math> || <math>\color{Orchid}{\text{Orchid}}</math> || <math>\color{Peach}{\text{Peach}}</math> || <math>\color{Periwinkle}{\text{Periwinkle}}</math><br />
|-<br />
| <math>\color{PineGreen}{\text{PineGreen}}</math> || <math>\color{Plum}{\text{Plum}}</math> || <math>\color{ProcessBlue}{\text{ProcessBlue}}</math> || <math>\color{Purple}{\text{Purple}}</math><br />
|-<br />
| <math>\color{RawSienna}{\text{RawSienna}}</math> || <math>\color{Red}{\text{Red}}</math> || <math>\color{RedOrange}{\text{RedOrange}}</math> || <math>\color{RedViolet}{\text{RedViolet}}</math><br />
|-<br />
| <math>\color{Rhodamine}{\text{Rhodamine}}</math> || <math>\color{RoyalBlue}{\text{RoyalBlue}}</math> || <math>\color{RoyalPurple}{\text{RoyalPurple}}</math> || <math>\color{RubineRed}{\text{RubineRed}}</math><br />
|-<br />
| <math>\color{Salmon}{\text{Salmon}}</math> || <math>\color{SeaGreen}{\text{SeaGreen}}</math> || <math>\color{Sepia}{\text{Sepia}}</math> || <math>\color{SkyBlue}{\text{SkyBlue}}</math><br />
|-<br />
| <math>\color{SpringGreen}{\text{SpringGreen}}</math> || <math>\color{Tan}{\text{Tan}}</math> || <math>\color{TealBlue}{\text{TealBlue}}</math> ||style="background-color: gray;"| <math>\pagecolor{Gray}\color{Thistle}{\text{Thistle}}</math><br />
|-<br />
| <math>\color{Turquoise}{\text{Turquoise}}</math> || <math>\color{Violet}{\text{Violet}}</math> || <math>\color{VioletRed}{\text{VioletRed}}</math> ||style="background-color: gray;"| <math>\pagecolor{Gray}{\color{White}{\text{White}}}</math><br />
|-<br />
| <math>\color{WildStrawberry}{\text{WildStrawberry}}</math> ||style="background-color: gray;"| <math>\pagecolor{Gray}\color{Yellow}{\text{Yellow}}</math> || <math>\color{YellowGreen}{\text{YellowGreen}}</math> || <math>\color{YellowOrange}{\text{YellowOrange}}</math><br />
|}<br />
<br />
Note that color should not be used as the ''only'' way to identify something, because it will become meaningless on black-and-white media or for color-blind people. See [[Wikipedia:Manual of Style (accessibility)#Color]].<br />
<br />
Latex does not have a command for setting the background color. The most effective of setting a background color is by setting a CSS styling rules for a table cell<br />
<pre style="display: inline-block;"><br />
{| class="wikitable"<br />
|-<br />
| style="background: gray" | <math>\pagecolor{Gray}x^2</math> || style="background: Goldenrod" | <math>\pagecolor{Goldenrod}y^3</math><br />
|}<br />
</pre><br />
<br />
Rendered as<br />
{| class="wikitable"<br />
|-<br />
| style="background: grey" | <math>\pagecolor{Gray}x^2</math> || style="background: Goldenrod" | <math>\pagecolor{Goldenrod}y^3</math><br />
|}<br />
<br />
The <code>\pagecolor{Goldenrod}</code> command is necessary for the Texvc renderer to use the correct anti-aliasing around the edges of the semi-transparent images.<br />
<br />
Custom colours can be defined using<br />
<syntaxhighlight lang="latex">\definecolor{myorange}{RGB}{255,165,100}\color{myorange}e^{i \pi}\color{Black} + 1 = 0</syntaxhighlight><br />
<br />
:<math>\definecolor{myorange}{RGB}{255,165,100}\color{myorange}e^{i \pi}\color{Black} + 1 = 0</math></div>Adminhttps://en.formulasearchengine.com/index.php?title=Main_Page&diff=218397Main Page2015-07-30T11:08:23Z<p>Admin: </p>
<hr />
<div>This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.<br />
<br />
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]<br />
* Only registered users will be able to execute this rendering mode.<br />
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.<br />
<br />
Registered users will be able to choose between the following three rendering modes: <br />
<br />
'''MathML'''<br />
:<math forcemathmode="5">E=mc^2</math><br />
<br />
'''PNG''' (currently default in production)<br />
:<math forcemathmode="0">E=mc^2</math><br />
<br />
'''source'''<br />
:<math>E=mc^2</math> <!-- forcemathmode="3" temporary disabled --><br />
<br />
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].<br />
<br />
==Demos==<br />
<br />
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:<br />
<br />
<br />
* accessibility:<br />
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]<br />
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].<br />
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].<br />
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.<br />
<br />
==Test pages ==<br />
<br />
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:<br />
*[[Displaystyle]]<br />
*[[MathAxisAlignment]]<br />
*[[Styling]]<br />
*[[Linebreaking]]<br />
*[[Unique Ids]]<br />
*[[Help:Formula]]<br />
<br />
*[[Inputtypes|Inputtypes (private Wikis only)]]<br />
*[[Url2Image|Url2Image (private Wikis only)]]<br />
==Bug reporting==<br />
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .</div>Adminhttps://en.formulasearchengine.com/index.php?title=MediaWiki:Common.js&diff=218396MediaWiki:Common.js2015-07-26T00:14:21Z<p>Admin: imported from https://en.wikipedia.org/w/index.php?title=MediaWiki:Common.css&oldid=672449626</p>
<hr />
<div>/**<br />
* Keep code in MediaWiki:Common.js to a minimum as it is unconditionally<br />
* loaded for all users on every wiki page. If possible create a gadget that is<br />
* enabled by default instead of adding it here (since gadgets are fully<br />
* optimized ResourceLoader modules with possibility to add dependencies etc.)<br />
*<br />
* Since Common.js isn't a gadget, there is no place to declare its<br />
* dependencies, so we have to lazy load them with mw.loader.using on demand and<br />
* then execute the rest in the callback. In most cases these dependencies will<br />
* be loaded (or loading) already and the callback will not be delayed. In case a<br />
* dependency hasn't arrived yet it'll make sure those are loaded before this.<br />
*/<br />
<br />
/* global mw, $, importStylesheet, importScript */<br />
/* jshint strict:false, browser:true */<br />
<br />
mw.loader.using( ['mediawiki.util', 'mediawiki.notify', 'jquery.client'] ).done( function () {<br />
/* Begin of mw.loader.using callback */<br />
<br />
/**<br />
* Main Page layout fixes<br />
*<br />
* Description: Adds an additional link to the complete list of languages available.<br />
* Maintainers: [[User:AzaToth]], [[User:R. Koot]], [[User:Alex Smotrov]]<br />
*/<br />
if ( mw.config.get( 'wgPageName' ) === 'Main_Page' || mw.config.get( 'wgPageName' ) === 'Talk:Main_Page' ) {<br />
$( function () {<br />
mw.util.addPortletLink( 'p-lang', '//meta.wikimedia.org/wiki/List_of_Wikipedias',<br />
'Complete list', 'interwiki-completelist', 'Complete list of Wikipedias' );<br />
} );<br />
}<br />
<br />
/**<br />
* Redirect User:Name/skin.js and skin.css to the current skin's pages<br />
* (unless the 'skin' page really exists)<br />
* @source: http://www.mediawiki.org/wiki/Snippets/Redirect_skin.js<br />
* @rev: 2<br />
*/<br />
if ( mw.config.get( 'wgArticleId' ) === 0 && mw.config.get( 'wgNamespaceNumber' ) === 2 ) {<br />
var titleParts = mw.config.get( 'wgPageName' ).split( '/' );<br />
/* Make sure there was a part before and after the slash<br />
and that the latter is 'skin.js' or 'skin.css' */<br />
if ( titleParts.length == 2 ) {<br />
var userSkinPage = titleParts.shift() + '/' + mw.config.get( 'skin' );<br />
if ( titleParts.slice( -1 ) == 'skin.js' ) {<br />
window.location.href = mw.util.getUrl( userSkinPage + '.js' );<br />
} else if ( titleParts.slice( -1 ) == 'skin.css' ) {<br />
window.location.href = mw.util.getUrl( userSkinPage + '.css' );<br />
}<br />
}<br />
}<br />
<br />
/**<br />
* Map addPortletLink to mw.util<br />
* @deprecated: Use mw.util.addPortletLink instead.<br />
*/<br />
mw.log.deprecate( window, 'addPortletLink', mw.util.addPortletLink, 'Use mw.util.addPortletLink instead' );<br />
<br />
/**<br />
* Extract a URL parameter from the current URL<br />
* @deprecated: Use mw.util.getParamValue with proper escaping<br />
*/<br />
mw.log.deprecate( window, 'getURLParamValue', mw.util.getParamValue, 'Use mw.util.getParamValue instead' );<br />
<br />
/**<br />
* Test if an element has a certain class<br />
* @deprecated: Use $(element).hasClass() instead.<br />
*/<br />
mw.log.deprecate( window, 'hasClass', function ( element, className ) {<br />
return $( element ).hasClass( className );<br />
}, 'Use jQuery.hasClass() instead' );<br />
<br />
/**<br />
* @source www.mediawiki.org/wiki/Snippets/Load_JS_and_CSS_by_URL<br />
* @rev 6<br />
*/<br />
var extraCSS = mw.util.getParamValue( 'withCSS' ),<br />
extraJS = mw.util.getParamValue( 'withJS' );<br />
<br />
if ( extraCSS ) {<br />
if ( extraCSS.match( /^MediaWiki:[^&<>=%#]*\.css$/ ) ) {<br />
importStylesheet( extraCSS );<br />
} else {<br />
mw.notify( 'Only pages from the MediaWiki namespace are allowed.', { title: 'Invalid withCSS value' } );<br />
}<br />
}<br />
<br />
if ( extraJS ) {<br />
if ( extraJS.match( /^MediaWiki:[^&<>=%#]*\.js$/ ) ) {<br />
importScript( extraJS );<br />
} else {<br />
mw.notify( 'Only pages from the MediaWiki namespace are allowed.', { title: 'Invalid withJS value' } );<br />
}<br />
}<br />
<br />
/**<br />
* Import more specific scripts if necessary<br />
*/<br />
if ( mw.config.get( 'wgAction' ) === 'edit' || mw.config.get( 'wgAction' ) === 'submit' || mw.config.get( 'wgCanonicalSpecialPageName' ) === 'Upload' ) {<br />
/* scripts specific to editing pages */<br />
importScript( 'MediaWiki:Common.js/edit.js' );<br />
} else if ( mw.config.get( 'wgCanonicalSpecialPageName' ) === 'Watchlist' ) {<br />
/* watchlist scripts */<br />
importScript( 'MediaWiki:Common.js/watchlist.js' );<br />
}<br />
<br />
/**<br />
* Helper script for .hlist class in Common.css<br />
* Add pseudo-selector class to last-child list items in IE8<br />
* @source mediawiki.org/wiki/Snippets/Horizontal_lists<br />
* @revision 6 (2014-08-23)<br />
* @author [[User:Edokter]]<br />
*/<br />
( function ( mw, $ ) {<br />
var profile = $.client.profile();<br />
if ( profile.name === 'msie' && profile.versionNumber === 8 ) {<br />
mw.hook( 'wikipage.content' ).add( function ( $content ) {<br />
$content.find( '.hlist' ).find( 'dd:last-child, dt:last-child, li:last-child' )<br />
.addClass( 'hlist-last-child' );<br />
} );<br />
}<br />
}( mediaWiki, jQuery ) );<br />
<br />
/**<br />
* Fix for Windows XP Unicode font rendering<br />
*/<br />
if ( navigator.appVersion.search(/windows nt 5/i) !== -1 ) {<br />
mw.util.addCSS( '.IPA { font-family: "Lucida Sans Unicode", "Arial Unicode MS"; } ' +<br />
'.Unicode { font-family: "Arial Unicode MS", "Lucida Sans Unicode"; } ' );<br />
}<br />
<br />
/**<br />
* WikiMiniAtlas<br />
*<br />
* Description: WikiMiniAtlas is a popup click and drag world map.<br />
* This script causes all of our coordinate links to display the WikiMiniAtlas popup button.<br />
* The script itself is located on meta because it is used by many projects.<br />
* See [[Meta:WikiMiniAtlas]] for more information.<br />
* Maintainers: [[User:Dschwen]]<br />
*/<br />
( function () {<br />
var require_wikiminiatlas = false;<br />
var coord_filter = /geohack/;<br />
$( function () {<br />
$( 'a.external.text' ).each( function( key, link ) {<br />
if ( link.href && coord_filter.exec( link.href ) ) {<br />
require_wikiminiatlas = true;<br />
// break from loop<br />
return false;<br />
}<br />
} );<br />
if ( $( 'div.kmldata' ).length ) {<br />
require_wikiminiatlas = true;<br />
}<br />
if ( require_wikiminiatlas ) {<br />
mw.loader.load( '//meta.wikimedia.org/w/index.php?title=MediaWiki:Wikiminiatlas.js&action=raw&ctype=text/javascript' );<br />
}<br />
} );<br />
} )();<br />
<br />
/**<br />
* Collapsible tables<br />
*<br />
* Allows tables to be collapsed, showing only the header. See [[Wikipedia:NavFrame]].<br />
*<br />
* @version 2.0.3 (2014-03-14)<br />
* @source https://www.mediawiki.org/wiki/MediaWiki:Gadget-collapsibleTables.js<br />
* @author [[User:R. Koot]]<br />
* @author [[User:Krinkle]]<br />
* @deprecated Since MediaWiki 1.20: Use class="mw-collapsible" instead which<br />
* is supported in MediaWiki core.<br />
*/<br />
<br />
var autoCollapse = 2;<br />
var collapseCaption = 'hide';<br />
var expandCaption = 'show';<br />
var tableIndex = 0;<br />
<br />
function collapseTable( tableIndex ) {<br />
var Button = document.getElementById( 'collapseButton' + tableIndex );<br />
var Table = document.getElementById( 'collapsibleTable' + tableIndex );<br />
<br />
if ( !Table || !Button ) {<br />
return false;<br />
}<br />
<br />
var Rows = Table.rows;<br />
var i;<br />
<br />
if ( Button.firstChild.data === collapseCaption ) {<br />
for ( i = 1; i < Rows.length; i++ ) {<br />
Rows[i].style.display = 'none';<br />
}<br />
Button.firstChild.data = expandCaption;<br />
} else {<br />
for ( i = 1; i < Rows.length; i++ ) {<br />
Rows[i].style.display = Rows[0].style.display;<br />
}<br />
Button.firstChild.data = collapseCaption;<br />
}<br />
}<br />
<br />
function createClickHandler( tableIndex ) {<br />
return function ( e ) {<br />
e.preventDefault();<br />
collapseTable( tableIndex );<br />
};<br />
}<br />
<br />
function createCollapseButtons( $content ) {<br />
var NavigationBoxes = {};<br />
var $Tables = $content.find( 'table' );<br />
var i;<br />
<br />
$Tables.each( function( i, table ) {<br />
if ( $(table).hasClass( 'collapsible' ) ) {<br />
<br />
/* only add button and increment count if there is a header row to work with */<br />
var HeaderRow = table.getElementsByTagName( 'tr' )[0];<br />
if ( !HeaderRow ) {<br />
return;<br />
}<br />
var Header = table.getElementsByTagName( 'th' )[0];<br />
if ( !Header ) {<br />
return;<br />
}<br />
<br />
NavigationBoxes[ tableIndex ] = table;<br />
table.setAttribute( 'id', 'collapsibleTable' + tableIndex );<br />
<br />
var Button = document.createElement( 'span' );<br />
var ButtonLink = document.createElement( 'a' );<br />
var ButtonText = document.createTextNode( collapseCaption );<br />
// Styles are declared in [[MediaWiki:Common.css]]<br />
Button.className = 'collapseButton';<br />
<br />
ButtonLink.style.color = Header.style.color;<br />
ButtonLink.setAttribute( 'id', 'collapseButton' + tableIndex );<br />
ButtonLink.setAttribute( 'href', '#' );<br />
$( ButtonLink ).on( 'click', createClickHandler( tableIndex ) );<br />
ButtonLink.appendChild( ButtonText );<br />
<br />
Button.appendChild( document.createTextNode( '[' ) );<br />
Button.appendChild( ButtonLink );<br />
Button.appendChild( document.createTextNode( ']' ) );<br />
<br />
Header.insertBefore( Button, Header.firstChild );<br />
tableIndex++;<br />
}<br />
} );<br />
<br />
for ( i = 0; i < tableIndex; i++ ) {<br />
if ( $( NavigationBoxes[i] ).hasClass( 'collapsed' ) ||<br />
( tableIndex >= autoCollapse && $( NavigationBoxes[i] ).hasClass( 'autocollapse' ) )<br />
) {<br />
collapseTable( i );<br />
}<br />
else if ( $( NavigationBoxes[i] ).hasClass ( 'innercollapse' ) ) {<br />
var element = NavigationBoxes[i];<br />
while ((element = element.parentNode)) {<br />
if ( $( element ).hasClass( 'outercollapse' ) ) {<br />
collapseTable ( i );<br />
break;<br />
}<br />
}<br />
}<br />
}<br />
}<br />
<br />
mw.hook( 'wikipage.content' ).add( createCollapseButtons );<br />
<br />
/**<br />
* Dynamic Navigation Bars (experimental)<br />
*<br />
* Description: See [[Wikipedia:NavFrame]].<br />
* Maintainers: UNMAINTAINED<br />
*/<br />
<br />
/* set up the words in your language */<br />
var NavigationBarHide = '[' + collapseCaption + ']';<br />
var NavigationBarShow = '[' + expandCaption + ']';<br />
var indexNavigationBar = 0;<br />
<br />
/**<br />
* Shows and hides content and picture (if available) of navigation bars<br />
* Parameters:<br />
* indexNavigationBar: the index of navigation bar to be toggled<br />
**/<br />
window.toggleNavigationBar = function ( indexNavigationBar, event ) {<br />
var NavToggle = document.getElementById( 'NavToggle' + indexNavigationBar );<br />
var NavFrame = document.getElementById( 'NavFrame' + indexNavigationBar );<br />
var NavChild;<br />
<br />
if ( !NavFrame || !NavToggle ) {<br />
return false;<br />
}<br />
<br />
/* if shown now */<br />
if ( NavToggle.firstChild.data === NavigationBarHide ) {<br />
for ( NavChild = NavFrame.firstChild; NavChild != null; NavChild = NavChild.nextSibling ) {<br />
if ( $( NavChild ).hasClass( 'NavContent' ) || $( NavChild ).hasClass( 'NavPic' ) ) {<br />
NavChild.style.display = 'none';<br />
}<br />
}<br />
NavToggle.firstChild.data = NavigationBarShow;<br />
<br />
/* if hidden now */<br />
} else if ( NavToggle.firstChild.data === NavigationBarShow ) {<br />
for ( NavChild = NavFrame.firstChild; NavChild != null; NavChild = NavChild.nextSibling ) {<br />
if ( $( NavChild ).hasClass( 'NavContent' ) || $( NavChild ).hasClass( 'NavPic' ) ) {<br />
NavChild.style.display = 'block';<br />
}<br />
}<br />
NavToggle.firstChild.data = NavigationBarHide;<br />
}<br />
<br />
event.preventDefault();<br />
};<br />
<br />
/* adds show/hide-button to navigation bars */<br />
function createNavigationBarToggleButton( $content ) {<br />
var NavChild;<br />
/* iterate over all < div >-elements */<br />
var $divs = $content.find( 'div' );<br />
$divs.each( function ( i, NavFrame ) {<br />
/* if found a navigation bar */<br />
if ( $( NavFrame ).hasClass( 'NavFrame' ) ) {<br />
<br />
indexNavigationBar++;<br />
var NavToggle = document.createElement( 'a' );<br />
NavToggle.className = 'NavToggle';<br />
NavToggle.setAttribute( 'id', 'NavToggle' + indexNavigationBar );<br />
NavToggle.setAttribute( 'href', '#' );<br />
$( NavToggle ).on( 'click', $.proxy( window.toggleNavigationBar, window, indexNavigationBar ) );<br />
<br />
var isCollapsed = $( NavFrame ).hasClass( 'collapsed' );<br />
/**<br />
* Check if any children are already hidden. This loop is here for backwards compatibility:<br />
* the old way of making NavFrames start out collapsed was to manually add style="display:none"<br />
* to all the NavPic/NavContent elements. Since this was bad for accessibility (no way to make<br />
* the content visible without JavaScript support), the new recommended way is to add the class<br />
* "collapsed" to the NavFrame itself, just like with collapsible tables.<br />
*/<br />
for ( NavChild = NavFrame.firstChild; NavChild != null && !isCollapsed; NavChild = NavChild.nextSibling ) {<br />
if ( $( NavChild ).hasClass( 'NavPic' ) || $( NavChild ).hasClass( 'NavContent' ) ) {<br />
if ( NavChild.style.display === 'none' ) {<br />
isCollapsed = true;<br />
}<br />
}<br />
}<br />
if ( isCollapsed ) {<br />
for ( NavChild = NavFrame.firstChild; NavChild != null; NavChild = NavChild.nextSibling ) {<br />
if ( $( NavChild ).hasClass( 'NavPic' ) || $( NavChild ).hasClass( 'NavContent' ) ) {<br />
NavChild.style.display = 'none';<br />
}<br />
}<br />
}<br />
var NavToggleText = document.createTextNode( isCollapsed ? NavigationBarShow : NavigationBarHide );<br />
NavToggle.appendChild( NavToggleText );<br />
<br />
/* Find the NavHead and attach the toggle link (Must be this complicated because Moz's firstChild handling is borked) */<br />
for( var j = 0; j < NavFrame.childNodes.length; j++ ) {<br />
if ( $( NavFrame.childNodes[j] ).hasClass( 'NavHead' ) ) {<br />
NavToggle.style.color = NavFrame.childNodes[j].style.color;<br />
NavFrame.childNodes[j].appendChild( NavToggle );<br />
}<br />
}<br />
NavFrame.setAttribute( 'id', 'NavFrame' + indexNavigationBar );<br />
}<br />
} );<br />
}<br />
<br />
mw.hook( 'wikipage.content' ).add( createNavigationBarToggleButton );<br />
<br />
/**<br />
* Uploadwizard_newusers<br />
* Switches in a message for non-autoconfirmed users at [[Wikipedia:Upload]]<br />
*<br />
* Maintainers: [[User:Krimpet]]<br />
*/<br />
function uploadwizard_newusers() {<br />
if ( mw.config.get( 'wgNamespaceNumber' ) === 4 && mw.config.get( 'wgTitle' ) === 'Upload' && mw.config.get( 'wgAction' ) === 'view' ) {<br />
var oldDiv = document.getElementById( 'autoconfirmedusers' ),<br />
newDiv = document.getElementById( 'newusers' );<br />
if ( oldDiv && newDiv ) {<br />
var userGroups = mw.config.get( 'wgUserGroups' );<br />
if ( userGroups ) {<br />
for ( var i = 0; i < userGroups.length; i++ ) {<br />
if ( userGroups[i] === 'autoconfirmed' ) {<br />
oldDiv.style.display = 'block';<br />
newDiv.style.display = 'none';<br />
return;<br />
}<br />
}<br />
}<br />
oldDiv.style.display = 'none';<br />
newDiv.style.display = 'block';<br />
return;<br />
}<br />
}<br />
}<br />
<br />
$(uploadwizard_newusers);<br />
<br />
/**<br />
* Magic editintros ****************************************************<br />
*<br />
* Description: Adds editintros on disambiguation pages and BLP pages.<br />
* Maintainers: [[User:RockMFR]]<br />
*/<br />
function addEditIntro( name ) {<br />
$( '.mw-editsection, #ca-edit' ).find( 'a' ).each( function ( i, el ) {<br />
el.href = $( this ).attr( 'href' ) + '&editintro=' + name;<br />
} );<br />
}<br />
<br />
if ( mw.config.get( 'wgNamespaceNumber' ) === 0 ) {<br />
$( function () {<br />
if ( document.getElementById( 'disambigbox' ) ) {<br />
addEditIntro( 'Template:Disambig_editintro' );<br />
}<br />
} );<br />
<br />
$( function () {<br />
var cats = mw.config.get('wgCategories');<br />
if ( !cats ) {<br />
return;<br />
}<br />
if ( $.inArray( 'Living people', cats ) !== -1 || $.inArray( 'Possibly living people', cats ) !== -1 ) {<br />
addEditIntro( 'Template:BLP_editintro' );<br />
}<br />
} );<br />
}<br />
<br />
/* End of mw.loader.using callback */<br />
} );<br />
/* DO NOT ADD CODE BELOW THIS LINE */</div>Adminhttps://en.formulasearchengine.com/index.php?title=MediaWiki:Common.css&diff=218395MediaWiki:Common.css2015-07-26T00:13:35Z<p>Admin: Import from https://en.wikipedia.org/w/index.php?title=MediaWiki:Common.css&oldid=672449626</p>
<hr />
<div>/* Default styling for HTML elements */<br />
dfn {<br />
font-style: inherit; /* Reset default styling for <dfn> */<br />
}<br />
q {<br />
quotes: '"' '"' "'" "'"; /* Straight quote marks for <q> */<br />
}<br />
blockquote {<br />
overflow: hidden; /* Avoid collision of background with floating elements */<br />
}<br />
strong.selflink {<br />
font-weight: 700; /* Prevent the 'double bold' bug in Firefox when using DirectWrite */<br />
}<br />
<br />
/* Consistent size for <sub> and <sup> */<br />
.mw-body sub,<br />
.mw-body sup,<br />
span.reference /* for Parsoid */ {<br />
font-size: 80%;<br />
}<br />
<br />
/* Main page fixes */<br />
#interwiki-completelist {<br />
font-weight: bold;<br />
}<br />
body.page-Main_Page #ca-delete {<br />
display: none !important;<br />
}<br />
body.page-Main_Page #mp-topbanner {<br />
clear: both;<br />
}<br />
<br />
/* Edit window toolbar */<br />
#toolbar {<br />
height: 22px;<br />
margin-bottom: 6px;<br />
}<br />
<br />
/* Hide charinsert base for those not using the gadget */<br />
#editpage-specialchars {<br />
display: none;<br />
}<br />
<br />
/* Highlight data points in the info action if specified in the URL */<br />
body.action-info :target {<br />
background: #DEF;<br />
}<br />
<br />
/* Make the list of references smaller */<br />
ol.references,<br />
div.reflist,<br />
div.refbegin {<br />
font-size: 90%; /* Default font-size */<br />
margin-bottom: 0.5em;<br />
}<br />
div.refbegin-100 {<br />
font-size: 100%; /* Option for normal fontsize in {{refbegin}} */<br />
}<br />
div.reflist ol.references {<br />
font-size: 100%; /* Reset font-size when nested in div.reflist */<br />
list-style-type: inherit; /* Enable custom list style types */<br />
}<br />
<br />
/* Highlight clicked reference in blue to help navigation */<br />
span.citation:target {<br />
background-color: #DEF;<br />
}<br />
<br />
/* Ensure refs in table headers and the like aren't bold or italic */<br />
sup.reference {<br />
font-weight: normal;<br />
font-style: normal;<br />
}<br />
<br />
/* Allow hidden ref errors to be shown by user CSS */<br />
span.brokenref {<br />
display: none;<br />
}<br />
<br />
/* Styling for citations (CSS3). Breaks long urls, etc., rather than overflowing box */<br />
.citation {<br />
word-wrap: break-word;<br />
}<br />
<br />
/* For linked citation numbers and document IDs, where<br />
the number need not be shown on a screen or a handheld,<br />
but should be included in the printed version */<br />
@media screen, handheld {<br />
.citation .printonly {<br />
display: none;<br />
}<br />
}<br />
<br />
/* Reset top margin for lists embedded in columns */<br />
div.columns {<br />
margin-top: 0.3em;<br />
}<br />
div.columns dl,<br />
div.columns ol,<br />
div.columns ul {<br />
margin-top: 0;<br />
}<br />
<br />
/* Avoid elements from breaking between columns */<br />
.nocolbreak,<br />
div.columns li,<br />
div.columns dd dd {<br />
-webkit-column-break-inside: avoid;<br />
page-break-inside: avoid;<br />
break-inside: avoid-column;<br />
}<br />
<br />
/* Style for [[Template:Flowlist]] that Lets lists flow around floating objecs */<br />
.flowlist ul {<br />
overflow-x: hidden;<br />
margin-left: 0;<br />
padding-left: 1.6em;<br />
}<br />
.flowlist ol {<br />
overflow-x: hidden;<br />
margin-left: 0;<br />
padding-left: 3.2em;<br />
}<br />
.flowlist dl {<br />
overflow-x: hidden;<br />
}<br />
<br />
/* Style for horizontal lists (separator following item).<br />
IE8-specific classes are assigned in [[MediaWiki:Common.js]].<br />
@source mediawiki.org/wiki/Snippets/Horizontal_lists<br />
@revision 6 (2014-05-09)<br />
@author [[User:Edokter]]<br />
*/<br />
.hlist dl,<br />
.hlist ol,<br />
.hlist ul {<br />
margin: 0;<br />
padding: 0;<br />
}<br />
/* Display list items inline */<br />
.hlist dd,<br />
.hlist dt,<br />
.hlist li {<br />
margin: 0;<br />
display: inline;<br />
}<br />
/* Display nested lists inline */<br />
.hlist.inline,<br />
.hlist.inline dl,<br />
.hlist.inline ol,<br />
.hlist.inline ul,<br />
.hlist dl dl, .hlist dl ol, .hlist dl ul,<br />
.hlist ol dl, .hlist ol ol, .hlist ol ul,<br />
.hlist ul dl, .hlist ul ol, .hlist ul ul {<br />
display: inline;<br />
}<br />
/* Generate interpuncts */<br />
.hlist dt:after {<br />
content: ": ";<br />
}<br />
.hlist dd:after,<br />
.hlist li:after {<br />
content: " · ";<br />
font-weight: bold;<br />
}<br />
.hlist dd:last-child:after,<br />
.hlist dt:last-child:after,<br />
.hlist li:last-child:after {<br />
content: none;<br />
}<br />
/* For IE8 */<br />
.hlist dd.hlist-last-child:after,<br />
.hlist dt.hlist-last-child:after,<br />
.hlist li.hlist-last-child:after {<br />
content: none;<br />
}<br />
/* Add parentheses around nested lists */<br />
.hlist dd dd:first-child:before, .hlist dd dt:first-child:before, .hlist dd li:first-child:before,<br />
.hlist dt dd:first-child:before, .hlist dt dt:first-child:before, .hlist dt li:first-child:before,<br />
.hlist li dd:first-child:before, .hlist li dt:first-child:before, .hlist li li:first-child:before {<br />
content: " (";<br />
font-weight: normal;<br />
}<br />
.hlist dd dd:last-child:after, .hlist dd dt:last-child:after, .hlist dd li:last-child:after,<br />
.hlist dt dd:last-child:after, .hlist dt dt:last-child:after, .hlist dt li:last-child:after,<br />
.hlist li dd:last-child:after, .hlist li dt:last-child:after, .hlist li li:last-child:after {<br />
content: ") ";<br />
font-weight: normal;<br />
}<br />
/* For IE8 */<br />
.hlist dd dd.hlist-last-child:after, .hlist dd dt.hlist-last-child:after, .hlist dd li.hlist-last-child:after,<br />
.hlist dt dd.hlist-last-child:after, .hlist dt dt.hlist-last-child:after, .hlist dt li.hlist-last-child:after,<br />
.hlist li dd.hlist-last-child:after, .hlist li dt.hlist-last-child:after, .hlist li li.hlist-last-child:after {<br />
content: ") ";<br />
font-weight: normal;<br />
}<br />
/* Put ordinals in front of ordered list items */<br />
.hlist ol {<br />
counter-reset: listitem;<br />
}<br />
.hlist ol > li {<br />
counter-increment: listitem;<br />
}<br />
.hlist ol > li:before {<br />
content: " " counter(listitem) " ";<br />
white-space: nowrap;<br />
}<br />
.hlist dd ol > li:first-child:before,<br />
.hlist dt ol > li:first-child:before,<br />
.hlist li ol > li:first-child:before {<br />
content: " (" counter(listitem) " ";<br />
}<br />
<br />
/* Unbulleted lists */<br />
.plainlist ol,<br />
.plainlist ul {<br />
line-height: inherit;<br />
list-style: none none;<br />
margin: 0;<br />
}<br />
.plainlist ol li,<br />
.plainlist ul li {<br />
margin-bottom: 0;<br />
}<br />
<br />
/* Default style for navigation boxes */<br />
.navbox { /* Navbox container style */<br />
border: 1px solid #aaa;<br />
width: 100%;<br />
margin: auto;<br />
clear: both;<br />
font-size: 88%;<br />
text-align: center;<br />
padding: 1px;<br />
}<br />
.navbox-inner,<br />
.navbox-subgroup {<br />
width: 100%;<br />
}<br />
.navbox-group,<br />
.navbox-title,<br />
.navbox-abovebelow {<br />
padding: 0.25em 1em; /* Title, group and above/below styles */<br />
line-height: 1.5em;<br />
text-align: center;<br />
}<br />
th.navbox-group { /* Group style */<br />
white-space: nowrap;<br />
/* @noflip */<br />
text-align: right;<br />
}<br />
.navbox,<br />
.navbox-subgroup {<br />
background: #fdfdfd; /* Background color */<br />
}<br />
.navbox-list {<br />
line-height: 1.5em;<br />
border-color: #fdfdfd; /* Must match background color */<br />
}<br />
.navbox th,<br />
.navbox-title {<br />
background: #ccccff; /* Level 1 color */<br />
}<br />
.navbox-abovebelow,<br />
th.navbox-group,<br />
.navbox-subgroup .navbox-title {<br />
background: #ddddff; /* Level 2 color */<br />
}<br />
.navbox-subgroup .navbox-group,<br />
.navbox-subgroup .navbox-abovebelow {<br />
background: #e6e6ff; /* Level 3 color */<br />
}<br />
.navbox-even {<br />
background: #f7f7f7; /* Even row striping */<br />
}<br />
.navbox-odd {<br />
background: transparent; /* Odd row striping */<br />
}<br />
table.navbox {<br />
margin-top: 1em; /* Prevent preceding content from clinging to navboxes */<br />
}<br />
table.navbox table.navbox {<br />
margin-top: 0; /* No top margin for nested navboxes */<br />
}<br />
table.navbox + table.navbox {<br />
margin-top: -1px; /* Single pixel border between adjacent navboxes */<br />
}<br />
.navbox .hlist td dl,<br />
.navbox .hlist td ol,<br />
.navbox .hlist td ul,<br />
.navbox td.hlist dl,<br />
.navbox td.hlist ol,<br />
.navbox td.hlist ul {<br />
padding: 0.125em 0; /* Adjust hlist padding in navboxes */<br />
}<br />
<br />
/* Default styling for Navbar template */<br />
.navbar {<br />
display: inline;<br />
font-size: 88%;<br />
font-weight: normal;<br />
}<br />
.navbar ul {<br />
display: inline;<br />
white-space: nowrap;<br />
}<br />
.mw-body-content .navbar ul {<br />
line-height: inherit;<br />
}<br />
.navbar li {<br />
word-spacing: -0.125em;<br />
}<br />
.navbar.mini li span {<br />
font-variant: small-caps;<br />
}<br />
/* Navbar styling when nested in infobox and navbox */<br />
.infobox .navbar {<br />
font-size: 100%;<br />
}<br />
.navbox .navbar {<br />
display: block;<br />
font-size: 100%;<br />
}<br />
.navbox-title .navbar {<br />
/* @noflip */<br />
float: left;<br />
/* @noflip */<br />
text-align: left;<br />
/* @noflip */<br />
margin-right: 0.5em;<br />
width: 6em;<br />
}<br />
<br />
/* 'show'/'hide' buttons created dynamically by the CollapsibleTables javascript<br />
in [[MediaWiki:Common.js]] are styled here so they can be customised. */<br />
.collapseButton {<br />
/* @noflip */<br />
float: right;<br />
font-weight: normal;<br />
/* @noflip */<br />
margin-left: 0.5em;<br />
/* @noflip */<br />
text-align: right;<br />
width: auto;<br />
}<br />
/* In navboxes, the show/hide button balances the v·d·e links<br />
from [[Template:Navbar]], so they need to be the same width. */<br />
.navbox .collapseButton {<br />
width: 6em;<br />
}<br />
<br />
/* Styling for JQuery makeCollapsible, matching that of collapseButton */<br />
.mw-collapsible-toggle {<br />
font-weight: normal;<br />
/* @noflip */<br />
text-align: right;<br />
}<br />
.navbox .mw-collapsible-toggle {<br />
width: 6em;<br />
}<br />
<br />
/* Infobox template style */<br />
.infobox {<br />
border: 1px solid #aaa;<br />
border-spacing: 3px;<br />
background-color: #f9f9f9;<br />
color: black;<br />
/* @noflip */<br />
margin: 0.5em 0 0.5em 1em;<br />
padding: 0.2em;<br />
/* @noflip */<br />
float: right;<br />
/* @noflip */<br />
clear: right;<br />
font-size: 88%;<br />
line-height: 1.5em;<br />
}<br />
.infobox caption {<br />
font-size: 125%;<br />
font-weight: bold;<br />
padding: 0.2em;<br />
}<br />
.infobox td,<br />
.infobox th {<br />
vertical-align: top;<br />
/* @noflip */<br />
text-align: left;<br />
}<br />
.infobox.bordered {<br />
border-collapse: collapse;<br />
}<br />
.infobox.bordered td,<br />
.infobox.bordered th {<br />
border: 1px solid #aaa;<br />
}<br />
.infobox.bordered .borderless td,<br />
.infobox.bordered .borderless th {<br />
border: 0;<br />
}<br />
<br />
.infobox.sisterproject {<br />
width: 20em;<br />
font-size: 90%;<br />
}<br />
<br />
.infobox.standard-talk {<br />
border: 1px solid #c0c090;<br />
background-color: #f8eaba;<br />
}<br />
.infobox.standard-talk.bordered td,<br />
.infobox.standard-talk.bordered th {<br />
border: 1px solid #c0c090;<br />
}<br />
<br />
/* styles for bordered infobox with merged rows */<br />
.infobox.bordered .mergedtoprow td,<br />
.infobox.bordered .mergedtoprow th {<br />
border: 0;<br />
border-top: 1px solid #aaa;<br />
/* @noflip */<br />
border-right: 1px solid #aaa;<br />
}<br />
<br />
.infobox.bordered .mergedrow td,<br />
.infobox.bordered .mergedrow th {<br />
border: 0;<br />
/* @noflip */<br />
border-right: 1px solid #aaa;<br />
}<br />
<br />
/* Styles for geography infoboxes, eg countries,<br />
country subdivisions, cities, etc. */<br />
.infobox.geography {<br />
border-collapse: collapse;<br />
line-height: 1.2em;<br />
font-size: 90%;<br />
}<br />
<br />
.infobox.geography td,<br />
.infobox.geography th {<br />
border-top: 1px solid #aaa;<br />
padding: 0.4em 0.6em 0.4em 0.6em;<br />
}<br />
.infobox.geography .mergedtoprow td,<br />
.infobox.geography .mergedtoprow th {<br />
border-top: 1px solid #aaa;<br />
padding: 0.4em 0.6em 0.2em 0.6em;<br />
}<br />
<br />
.infobox.geography .mergedrow td,<br />
.infobox.geography .mergedrow th {<br />
border: 0;<br />
padding: 0 0.6em 0.2em 0.6em;<br />
}<br />
<br />
.infobox.geography .mergedbottomrow td,<br />
.infobox.geography .mergedbottomrow th {<br />
border-top: 0;<br />
border-bottom: 1px solid #aaa;<br />
padding: 0 0.6em 0.4em 0.6em;<br />
}<br />
<br />
.infobox.geography .maptable td,<br />
.infobox.geography .maptable th {<br />
border: 0;<br />
padding: 0;<br />
}<br />
<br />
/* Normal font styling for table row headers with scope="row" tag */<br />
.wikitable.plainrowheaders th[scope=row] {<br />
font-weight: normal;<br />
/* @noflip */<br />
text-align: left;<br />
}<br />
<br />
/* Lists in data cells are always left-aligned */<br />
.wikitable td ul,<br />
.wikitable td ol,<br />
.wikitable td dl {<br />
/* @noflip */<br />
text-align: left;<br />
}<br />
/* ...unless they also use the hlist class */<br />
.toc.hlist ul,<br />
#toc.hlist ul,<br />
.wikitable.hlist td ul,<br />
.wikitable.hlist td ol,<br />
.wikitable.hlist td dl {<br />
text-align: inherit;<br />
}<br />
<br />
/* Icons for medialist templates [[Template:Listen]],<br />
[[Template:Multi-listen_start]], [[Template:Video]],<br />
[[Template:Multi-video_start]] */<br />
div.listenlist {<br />
background: url("//upload.wikimedia.org/wikipedia/commons/4/47/Sound-icon.svg") no-repeat scroll 0 0 transparent;<br />
background-size: 30px;<br />
padding-left: 40px;<br />
}<br />
<br />
/* Fix for hieroglyphs specificality issue in infoboxes ([[Phabricator:43869]]) */<br />
table.mw-hiero-table td {<br />
vertical-align: middle;<br />
}<br />
<br />
/* Style rules for media list templates */<br />
div.medialist {<br />
min-height: 50px;<br />
margin: 1em;<br />
/* @noflip */<br />
background-position: top left;<br />
background-repeat: no-repeat;<br />
}<br />
div.medialist ul {<br />
list-style-type: none;<br />
list-style-image: none;<br />
margin: 0;<br />
}<br />
div.medialist ul li {<br />
padding-bottom: 0.5em;<br />
}<br />
div.medialist ul li li {<br />
font-size: 91%;<br />
padding-bottom: 0;<br />
}<br />
<br />
/* Change the external link icon to an Adobe icon for all PDF files<br />
in browsers that support these CSS selectors, like Mozilla and Opera */<br />
div#content a[href$=".pdf"].external,<br />
div#content a[href*=".pdf?"].external,<br />
div#content a[href*=".pdf#"].external,<br />
div#content a[href$=".PDF"].external,<br />
div#content a[href*=".PDF?"].external,<br />
div#content a[href*=".PDF#"].external,<br />
div#mw_content a[href$=".pdf"].external,<br />
div#mw_content a[href*=".pdf?"].external,<br />
div#mw_content a[href*=".pdf#"].external,<br />
div#mw_content a[href$=".PDF"].external,<br />
div#mw_content a[href*=".PDF?"].external,<br />
div#mw_content a[href*=".PDF#"].external {<br />
background: url("//upload.wikimedia.org/wikipedia/commons/2/23/Icons-mini-file_acrobat.gif") no-repeat right;<br />
/* @noflip */<br />
padding-right: 18px;<br />
}<br />
<br />
/* Change the external link icon to an Adobe icon anywhere the PDFlink class<br />
is used (notably Template:PDFlink). This works in IE, unlike the above. */<br />
div#content span.PDFlink a,<br />
div#mw_content span.PDFlink a {<br />
background: url("//upload.wikimedia.org/wikipedia/commons/2/23/Icons-mini-file_acrobat.gif") no-repeat right;<br />
/* @noflip */<br />
padding-right: 18px;<br />
}<br />
<br />
/* Content in columns with CSS instead of tables ([[Template:Columns]]) */<br />
div.columns-2 div.column {<br />
/* @noflip */<br />
float: left;<br />
width: 50%;<br />
min-width: 300px;<br />
}<br />
div.columns-3 div.column {<br />
/* @noflip */<br />
float: left;<br />
width: 33.3%;<br />
min-width: 200px;<br />
}<br />
div.columns-4 div.column {<br />
/* @noflip */<br />
float: left;<br />
width: 25%;<br />
min-width: 150px;<br />
}<br />
div.columns-5 div.column {<br />
/* @noflip */<br />
float: left;<br />
width: 20%;<br />
min-width: 120px;<br />
}<br />
<br />
/* Messagebox templates */<br />
.messagebox {<br />
border: 1px solid #aaa;<br />
background-color: #f9f9f9;<br />
width: 80%;<br />
margin: 0 auto 1em auto;<br />
padding: .2em;<br />
}<br />
.messagebox.merge {<br />
border: 1px solid #c0b8cc;<br />
background-color: #f0e5ff;<br />
text-align: center;<br />
}<br />
.messagebox.cleanup {<br />
border: 1px solid #9f9fff;<br />
background-color: #efefff;<br />
text-align: center;<br />
}<br />
.messagebox.standard-talk {<br />
border: 1px solid #c0c090;<br />
background-color: #f8eaba;<br />
margin: 4px auto;<br />
}<br />
/* For old WikiProject banners inside banner shells. */<br />
.mbox-inside .standard-talk,<br />
.messagebox.nested-talk {<br />
border: 1px solid #c0c090;<br />
background-color: #f8eaba;<br />
width: 100%;<br />
margin: 2px 0;<br />
padding: 2px;<br />
}<br />
.messagebox.small {<br />
width: 238px;<br />
font-size: 85%;<br />
/* @noflip */<br />
float: right;<br />
clear: both;<br />
/* @noflip */<br />
margin: 0 0 1em 1em;<br />
line-height: 1.25em;<br />
}<br />
.messagebox.small-talk {<br />
width: 238px;<br />
font-size: 85%;<br />
/* @noflip */<br />
float: right;<br />
clear: both;<br />
/* @noflip */<br />
margin: 0 0 1em 1em;<br />
line-height: 1.25em;<br />
background: #F8EABA;<br />
}<br />
<br />
/* Cell sizes for ambox/tmbox/imbox/cmbox/ombox/fmbox/dmbox message boxes */<br />
th.mbox-text, td.mbox-text { /* The message body cell(s) */<br />
border: none;<br />
/* @noflip */<br />
padding: 0.25em 0.9em; /* 0.9em left/right */<br />
width: 100%; /* Make all mboxes the same width regardless of text length */<br />
}<br />
td.mbox-image { /* The left image cell */<br />
border: none;<br />
/* @noflip */<br />
padding: 2px 0 2px 0.9em; /* 0.9em left, 0px right */<br />
text-align: center;<br />
}<br />
td.mbox-imageright { /* The right image cell */<br />
border: none;<br />
/* @noflip */<br />
padding: 2px 0.9em 2px 0; /* 0px left, 0.9em right */<br />
text-align: center;<br />
}<br />
td.mbox-empty-cell { /* An empty narrow cell */<br />
border: none;<br />
padding: 0;<br />
width: 1px;<br />
}<br />
<br />
/* Article message box styles */<br />
table.ambox {<br />
margin: 0 10%; /* 10% = Will not overlap with other elements */<br />
border: 1px solid #aaa;<br />
/* @noflip */<br />
border-left: 10px solid #1e90ff; /* Default "notice" blue */<br />
background: #fbfbfb;<br />
}<br />
table.ambox + table.ambox { /* Single border between stacked boxes. */<br />
margin-top: -1px;<br />
}<br />
.ambox th.mbox-text,<br />
.ambox td.mbox-text { /* The message body cell(s) */<br />
padding: 0.25em 0.5em; /* 0.5em left/right */<br />
}<br />
.ambox td.mbox-image { /* The left image cell */<br />
/* @noflip */<br />
padding: 2px 0 2px 0.5em; /* 0.5em left, 0px right */<br />
}<br />
.ambox td.mbox-imageright { /* The right image cell */<br />
/* @noflip */<br />
padding: 2px 0.5em 2px 0; /* 0px left, 0.5em right */<br />
}<br />
<br />
table.ambox-notice {<br />
/* @noflip */<br />
border-left: 10px solid #1e90ff; /* Blue */<br />
}<br />
table.ambox-speedy {<br />
/* @noflip */<br />
border-left: 10px solid #b22222; /* Red */<br />
background: #fee; /* Pink */<br />
}<br />
table.ambox-delete {<br />
/* @noflip */<br />
border-left: 10px solid #b22222; /* Red */<br />
}<br />
table.ambox-content {<br />
/* @noflip */<br />
border-left: 10px solid #f28500; /* Orange */<br />
}<br />
table.ambox-style {<br />
/* @noflip */<br />
border-left: 10px solid #f4c430; /* Yellow */<br />
}<br />
table.ambox-move {<br />
/* @noflip */<br />
border-left: 10px solid #9932cc; /* Purple */<br />
}<br />
table.ambox-protection {<br />
/* @noflip */<br />
border-left: 10px solid #bba; /* Gray-gold */<br />
}<br />
<br />
/* Image message box styles */<br />
table.imbox {<br />
margin: 4px 10%;<br />
border-collapse: collapse;<br />
border: 3px solid #1e90ff; /* Default "notice" blue */<br />
background: #fbfbfb;<br />
}<br />
.imbox .mbox-text .imbox { /* For imboxes inside imbox-text cells. */<br />
margin: 0 -0.5em; /* 0.9 - 0.5 = 0.4em left/right. */<br />
display: block; /* Fix for webkit to force 100% width. */<br />
}<br />
.mbox-inside .imbox { /* For imboxes inside other templates. */<br />
margin: 4px;<br />
}<br />
<br />
table.imbox-notice {<br />
border: 3px solid #1e90ff; /* Blue */<br />
}<br />
table.imbox-speedy {<br />
border: 3px solid #b22222; /* Red */<br />
background: #fee; /* Pink */<br />
}<br />
table.imbox-delete {<br />
border: 3px solid #b22222; /* Red */<br />
}<br />
table.imbox-content {<br />
border: 3px solid #f28500; /* Orange */<br />
}<br />
table.imbox-style {<br />
border: 3px solid #f4c430; /* Yellow */<br />
}<br />
table.imbox-move {<br />
border: 3px solid #9932cc; /* Purple */<br />
}<br />
table.imbox-protection {<br />
border: 3px solid #bba; /* Gray-gold */<br />
}<br />
table.imbox-license {<br />
border: 3px solid #88a; /* Dark gray */<br />
background: #f7f8ff; /* Light gray */<br />
}<br />
table.imbox-featured {<br />
border: 3px solid #cba135; /* Brown-gold */<br />
}<br />
<br />
/* Category message box styles */<br />
table.cmbox {<br />
margin: 3px 10%;<br />
border-collapse: collapse;<br />
border: 1px solid #aaa;<br />
background: #DFE8FF; /* Default "notice" blue */<br />
}<br />
<br />
table.cmbox-notice {<br />
background: #D8E8FF; /* Blue */<br />
}<br />
table.cmbox-speedy {<br />
margin-top: 4px;<br />
margin-bottom: 4px;<br />
border: 4px solid #b22222; /* Red */<br />
background: #FFDBDB; /* Pink */<br />
}<br />
table.cmbox-delete {<br />
background: #FFDBDB; /* Red */<br />
}<br />
table.cmbox-content {<br />
background: #FFE7CE; /* Orange */<br />
}<br />
table.cmbox-style {<br />
background: #FFF9DB; /* Yellow */<br />
}<br />
table.cmbox-move {<br />
background: #E4D8FF; /* Purple */<br />
}<br />
table.cmbox-protection {<br />
background: #EFEFE1; /* Gray-gold */<br />
}<br />
<br />
/* Other pages message box styles */<br />
table.ombox {<br />
margin: 4px 10%;<br />
border-collapse: collapse;<br />
border: 1px solid #aaa; /* Default "notice" gray */<br />
background: #f9f9f9;<br />
}<br />
<br />
table.ombox-notice {<br />
border: 1px solid #aaa; /* Gray */<br />
}<br />
table.ombox-speedy {<br />
border: 2px solid #b22222; /* Red */<br />
background: #fee; /* Pink */<br />
}<br />
table.ombox-delete {<br />
border: 2px solid #b22222; /* Red */<br />
}<br />
table.ombox-content {<br />
border: 1px solid #f28500; /* Orange */<br />
}<br />
table.ombox-style {<br />
border: 1px solid #f4c430; /* Yellow */<br />
}<br />
table.ombox-move {<br />
border: 1px solid #9932cc; /* Purple */<br />
}<br />
table.ombox-protection {<br />
border: 2px solid #bba; /* Gray-gold */<br />
}<br />
<br />
/* Talk page message box styles */<br />
table.tmbox {<br />
margin: 4px 10%;<br />
border-collapse: collapse;<br />
border: 1px solid #c0c090; /* Default "notice" gray-brown */<br />
background: #f8eaba;<br />
}<br />
.mediawiki .mbox-inside .tmbox { /* For tmboxes inside other templates. The "mediawiki" class ensures that */<br />
margin: 2px 0; /* this declaration overrides other styles (including mbox-small above) */<br />
width: 100%; /* For Safari and Opera */<br />
}<br />
.mbox-inside .tmbox.mbox-small { /* "small" tmboxes should not be small when */<br />
line-height: 1.5em; /* also "nested", so reset styles that are */<br />
font-size: 100%; /* set in "mbox-small" above. */<br />
}<br />
<br />
table.tmbox-speedy {<br />
border: 2px solid #b22222; /* Red */<br />
background: #fee; /* Pink */<br />
}<br />
table.tmbox-delete {<br />
border: 2px solid #b22222; /* Red */<br />
}<br />
table.tmbox-content {<br />
border: 2px solid #f28500; /* Orange */<br />
}<br />
table.tmbox-style {<br />
border: 2px solid #f4c430; /* Yellow */<br />
}<br />
table.tmbox-move {<br />
border: 2px solid #9932cc; /* Purple */<br />
}<br />
table.tmbox-protection,<br />
table.tmbox-notice {<br />
border: 1px solid #c0c090; /* Gray-brown */<br />
}<br />
<br />
/* Disambig and set index box styles */<br />
table.dmbox {<br />
clear: both;<br />
margin: 0.9em 1em;<br />
border-top: 1px solid #ccc;<br />
border-bottom: 1px solid #ccc;<br />
background: transparent;<br />
}<br />
<br />
/* Footer and header message box styles */<br />
table.fmbox {<br />
clear: both;<br />
margin: 0.2em 0;<br />
width: 100%;<br />
border: 1px solid #aaa;<br />
background: #f9f9f9; /* Default "system" gray */<br />
}<br />
table.fmbox-system {<br />
background: #f9f9f9;<br />
}<br />
table.fmbox-warning {<br />
border: 1px solid #bb7070; /* Dark pink */<br />
background: #ffdbdb; /* Pink */<br />
}<br />
table.fmbox-editnotice {<br />
background: transparent;<br />
}<br />
/* Div based "warning" style fmbox messages. */<br />
div.mw-warning-with-logexcerpt,<br />
div.mw-lag-warn-high,<br />
div.mw-cascadeprotectedwarning,<br />
div#mw-protect-cascadeon,<br />
div.titleblacklist-warning,<br />
div.locked-warning {<br />
clear: both;<br />
margin: 0.2em 0;<br />
border: 1px solid #bb7070;<br />
background: #ffdbdb;<br />
padding: 0.25em 0.9em;<br />
}<br />
/* Div based "system" style fmbox messages.<br />
Used in [[MediaWiki:Readonly lag]]. */<br />
div.mw-lag-warn-normal,<br />
div.fmbox-system {<br />
clear: both;<br />
margin: 0.2em 0;<br />
border: 1px solid #aaa;<br />
background: #f9f9f9;<br />
padding: 0.25em 0.9em;<br />
}<br />
<br />
/* These mbox-small classes must be placed after all other<br />
ambox/tmbox/ombox etc classes. "body.mediawiki" is so<br />
they override "table.ambox + table.ambox" above. */<br />
body.mediawiki table.mbox-small { /* For the "small=yes" option. */<br />
/* @noflip */<br />
clear: right;<br />
/* @noflip */<br />
float: right;<br />
/* @noflip */<br />
margin: 4px 0 4px 1em;<br />
width: 238px;<br />
font-size: 88%;<br />
line-height: 1.25em;<br />
}<br />
body.mediawiki table.mbox-small-left { /* For the "small=left" option. */<br />
/* @noflip */<br />
margin: 4px 1em 4px 0;<br />
width: 238px;<br />
border-collapse: collapse;<br />
font-size: 88%;<br />
line-height: 1.25em;<br />
}<br />
<br />
/* Style for compact ambox */<br />
/* Hide the images */<br />
.compact-ambox table .mbox-image,<br />
.compact-ambox table .mbox-imageright,<br />
.compact-ambox table .mbox-empty-cell {<br />
display: none;<br />
}<br />
/* Remove borders, backgrounds, padding, etc. */<br />
.compact-ambox table.ambox {<br />
border: none;<br />
border-collapse: collapse;<br />
background: transparent;<br />
margin: 0 0 0 1.6em !important;<br />
padding: 0 !important;<br />
width: auto;<br />
display: block;<br />
}<br />
body.mediawiki .compact-ambox table.mbox-small-left {<br />
font-size: 100%;<br />
width: auto;<br />
margin: 0;<br />
}<br />
/* Style the text cell as a list item and remove its padding */<br />
.compact-ambox table .mbox-text {<br />
padding: 0 !important;<br />
margin: 0 !important;<br />
}<br />
.compact-ambox table .mbox-text-span {<br />
display: list-item;<br />
line-height: 1.5em;<br />
list-style-type: square;<br />
list-style-image: url(/w/skins/MonoBook/bullet.gif);<br />
}<br />
.skin-vector .compact-ambox table .mbox-text-span {<br />
list-style-type: disc;<br />
list-style-image: url(/w/skins/Vector/images/bullet-icon.png)<br />
}<br />
/* Allow for hiding text in compact form */<br />
.compact-ambox .hide-when-compact {<br />
display: none;<br />
}<br />
<br />
/* Remove default styles for [[MediaWiki:Noarticletext]]. */<br />
div.noarticletext {<br />
border: none;<br />
background: transparent;<br />
padding: 0;<br />
}<br />
<br />
/* Hide (formatting) elements from screen, but not from screenreaders */<br />
.visualhide {<br />
position: absolute;<br />
left: -10000px;<br />
top: auto;<br />
width: 1px;<br />
height: 1px;<br />
overflow: hidden;<br />
}<br />
<br />
/* Bold save button */<br />
#wpSave {<br />
font-weight: bold;<br />
}<br />
<br />
/* class hiddenStructure is defunct. See [[Wikipedia:hiddenStructure]] */<br />
.hiddenStructure {<br />
display: inline !important;<br />
color: #f00;<br />
background-color: #0f0;<br />
}<br />
<br />
/* suppress missing interwiki image links where #ifexist cannot<br />
be used due to high number of requests see .hidden-redlink on<br />
[[m:MediaWiki:Common.css]] */<br />
.check-icon a.new {<br />
display: none;<br />
speak: none;<br />
}<br />
<br />
/* Removes underlines from certain links */<br />
.nounderlines a,<br />
.IPA a:link, .IPA a:visited {<br />
text-decoration: none !important;<br />
}<br />
<br />
/* Standard Navigationsleisten, aka box hiding thingy<br />
from .de. Documentation at [[Wikipedia:NavFrame]]. */<br />
div.NavFrame {<br />
margin: 0;<br />
padding: 4px;<br />
border: 1px solid #aaa;<br />
text-align: center;<br />
border-collapse: collapse;<br />
font-size: 95%;<br />
}<br />
div.NavFrame + div.NavFrame {<br />
border-top-style: none;<br />
border-top-style: hidden;<br />
}<br />
div.NavPic {<br />
background-color: #fff;<br />
margin: 0;<br />
padding: 2px;<br />
/* @noflip */<br />
float: left;<br />
}<br />
div.NavFrame div.NavHead {<br />
line-height: 1.6em;<br />
font-weight: bold;<br />
background-color: #ccf;<br />
position: relative;<br />
}<br />
div.NavFrame p,<br />
div.NavFrame div.NavContent,<br />
div.NavFrame div.NavContent p {<br />
font-size: 100%;<br />
}<br />
div.NavEnd {<br />
margin: 0;<br />
padding: 0;<br />
line-height: 1px;<br />
clear: both;<br />
}<br />
a.NavToggle {<br />
position: absolute;<br />
top: 0;<br />
/* @noflip */<br />
right: 3px;<br />
font-weight: normal;<br />
font-size: 90%;<br />
}<br />
<br />
/* Hatnotes and disambiguation notices */<br />
.hatnote {<br />
font-style: italic;<br />
}<br />
.hatnote i {<br />
font-style: normal;<br />
}<br />
div.hatnote {<br />
/* @noflip */<br />
padding-left: 1.6em;<br />
margin-bottom: 0.5em;<br />
}<br />
div.hatnote + div.hatnote {<br />
margin-top: -0.5em;<br />
}<br />
<br />
/* Allow transcluded pages to display in lists rather than a table.<br />
Compatible in Firefox; incompatible in IE6. */<br />
.listify td { display: list-item; }<br />
.listify tr { display: block; }<br />
.listify table { display: block; }<br />
<br />
/* Geographical coordinates defaults. See [[Template:Coord/link]]<br />
for how these are used. The classes "geo", "longitude", and<br />
"latitude" are used by the [[Geo microformat]]. */<br />
.geo-default, .geo-dms, .geo-dec { display: inline; }<br />
.geo-nondefault, .geo-multi-punct { display: none; }<br />
.longitude, .latitude { white-space: nowrap; }<br />
<br />
/* When <div class="nonumtoc"> is used on the table of contents,<br />
the ToC will display without numbers */<br />
.nonumtoc .tocnumber {<br />
display: none;<br />
}<br />
.nonumtoc #toc ul,<br />
.nonumtoc .toc ul {<br />
line-height: 1.5em;<br />
list-style: none none;<br />
margin: .3em 0 0;<br />
padding: 0;<br />
}<br />
.hlist.nonumtoc #toc ul ul,<br />
.hlist.nonumtoc .toc ul ul {<br />
/* @noflip */<br />
margin: 0;<br />
}<br />
<br />
/* Allow limiting of which header levels are shown in a TOC;<br />
<div class="toclimit-3">, for instance, will limit to<br />
showing ==headings== and ===headings=== but no further<br />
(as long as there are no =headings= on the page, which<br />
there shouldn't be according to the MoS). */<br />
.toclimit-2 .toclevel-1 ul,<br />
.toclimit-3 .toclevel-2 ul,<br />
.toclimit-4 .toclevel-3 ul,<br />
.toclimit-5 .toclevel-4 ul,<br />
.toclimit-6 .toclevel-5 ul,<br />
.toclimit-7 .toclevel-6 ul {<br />
display: none;<br />
}<br />
<br />
/* Styling for Template:Quote */<br />
blockquote.templatequote {<br />
margin-top: 0;<br />
}<br />
blockquote.templatequote div.templatequotecite {<br />
line-height: 1.5em;<br />
/* @noflip */<br />
text-align: left;<br />
/* @noflip */<br />
padding-left: 1.6em;<br />
margin-top: 0;<br />
}<br />
<br />
/* User block messages */<br />
div.user-block {<br />
padding: 5px;<br />
margin-bottom: 0.5em;<br />
border: 1px solid #A9A9A9;<br />
background-color: #FFEFD5;<br />
}<br />
<br />
/* Prevent line breaks in silly places:<br />
1) Where desired<br />
2) Links when we don't want them to<br />
3) Bold "links" to the page itself<br />
4) Ref tags with group names <ref group="Note"> --> "[Note 1]" */<br />
.nowrap,<br />
.nowraplinks a,<br />
.nowraplinks .selflink,<br />
sup.reference a {<br />
white-space: nowrap;<br />
}<br />
/* But allow wrapping where desired: */<br />
.wrap,<br />
.wraplinks a {<br />
white-space: normal;<br />
}<br />
<br />
/* For template documentation */<br />
.template-documentation {<br />
clear: both;<br />
margin: 1em 0 0 0;<br />
border: 1px solid #aaa;<br />
background-color: #ecfcf4;<br />
padding: 1em;<br />
}<br />
<br />
/* Inline divs in ImageMaps (code borrowed from de.wiki) */<br />
.imagemap-inline div {<br />
display: inline;<br />
}<br />
<br />
/* Increase the height of the image upload box */<br />
#wpUploadDescription {<br />
height: 13em;<br />
}<br />
<br />
/* Minimum thumb width */<br />
.thumbinner {<br />
min-width: 100px;<br />
}<br />
<br />
/* Makes the background of a framed image white instead of gray.<br />
Only visible with transparent images. */<br />
div.thumb .thumbimage {<br />
background-color: #fff;<br />
}<br />
<br />
/* The backgrounds for galleries. */<br />
div#content .gallerybox div.thumb {<br />
/* Light gray padding */<br />
background-color: #F9F9F9;<br />
}<br />
/* Put a chequered background behind images, only visible if they have transparency.<br />
'.filehistory a img' and '#file img:hover' are handled by MediaWiki core (as of 1.19) */<br />
.gallerybox .thumb img {<br />
background: #fff url(//upload.wikimedia.org/wikipedia/commons/5/5d/Checker-16x16.png) repeat;<br />
}<br />
/* But not on articles, user pages, portals or with opt-out. */<br />
.ns-0 .gallerybox .thumb img,<br />
.ns-2 .gallerybox .thumb img,<br />
.ns-100 .gallerybox .thumb img,<br />
.nochecker .gallerybox .thumb img {<br />
background: #fff;<br />
}<br />
<br />
/* Prevent floating boxes from overlapping any category listings,<br />
file histories, edit previews, and edit [Show changes] views. */<br />
#mw-subcategories, #mw-pages, #mw-category-media,<br />
#filehistory, #wikiPreview, #wikiDiff {<br />
clear: both;<br />
}<br />
<br />
body.rtl #mw-articlefeedbackv5, body.rtl #mw-articlefeedback {<br />
display: block; /* Override inline block mode */<br />
margin-bottom: 1em;<br />
/* @noflip */<br />
clear: right; /* Clear any info boxes that stick out */<br />
/* @noflip */<br />
float: right; /* Prevents margin collapsing */<br />
}<br />
<br />
/* Selectively hide headers in WikiProject banners */<br />
.wpb .wpb-header { display: none; }<br />
.wpbs-inner .wpb .wpb-header { display: block; } /* for IE */<br />
.wpbs-inner .wpb .wpb-header { display: table-row; } /* for real browsers */<br />
.wpbs-inner .wpb-outside { display: none; } /* hide things that should only display outside shells */<br />
<br />
/* Styling for Abuse Filter tags */<br />
.mw-tag-markers {<br />
font-family:sans-serif;<br />
font-style:italic;<br />
font-size:90%;<br />
}<br />
<br />
/* Hide stuff meant for accounts with special permissions. Made visible again in<br />
[[MediaWiki:Group-sysop.css]], [[MediaWiki:Group-accountcreator.css]],<br />
[[MediaWiki:Group-templateeditor.css]] and [[Mediawiki:Group-autoconfirmed.css]]. */<br />
.sysop-show,<br />
.accountcreator-show,<br />
.templateeditor-show,<br />
.autoconfirmed-show {<br />
display: none;<br />
}<br />
<br />
/**<br />
* Hide the redlink generated by {{Editnotice}},<br />
* this overrides the ".sysop-show { display: none; }" above that applies<br />
* to the same link as well.<br />
*<br />
* See [[Phabricator:45013]].<br />
*/<br />
.ve-ui-mwNoticesPopupTool-item .editnotice-redlink, .mw-ve-editNotice .editnotice-redlink {<br />
display: none !important;<br />
}<br />
<br />
/* Remove bullets when there are multiple edit page warnings */<br />
ul.permissions-errors > li {<br />
list-style: none none;<br />
}<br />
ul.permissions-errors {<br />
margin: 0;<br />
}<br />
<br />
/* No linewrap on the labels of the login/signup page */<br />
body.page-Special_UserLogin .mw-label label,<br />
body.page-Special_UserLogin_signup .mw-label label {<br />
white-space: nowrap;<br />
}<br />
<br />
/* Pie chart test: Transparent borders */<br />
.transborder {<br />
border: solid transparent;<br />
}<br />
* html .transborder { /* IE6 */<br />
border: solid #000001;<br />
filter: chroma(color=#000001);<br />
}<br />
<br />
/* Styling for updated markers on watchlist, history and recent/related changes */<br />
#mw-wlheader-showupdated,<br />
#mw-wlheader-bold,<br />
#mw-wlheader-green,<br />
#mw-watchlist-resetbutton {<br />
display: none;<br />
}<br />
.updatedmarker {<br />
background-color: transparent;<br />
color: #006400;<br />
}<br />
.mw-changeslist-line-watched .mw-title,<br />
.mw-enhanced-watched .mw-enhanced-rc-time {<br />
font-weight: inherit;<br />
}<br />
<br />
/* Generic class for Times-based serif, texhtml class for inline math */<br />
.times-serif,<br />
span.texhtml {<br />
font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif;<br />
font-size: 118%;<br />
line-height: 1;<br />
}<br />
span.texhtml {<br />
white-space: nowrap;<br />
}<br />
span.texhtml span.texhtml {<br />
font-size: 100%;<br />
}<br />
<br />
/* Force tabular and lining display for digits and texhtml */<br />
.digits,<br />
.texhtml {<br />
-moz-font-feature-settings: "lnum", "tnum", "kern" 0;<br />
-webkit-font-feature-settings: "lnum", "tnum", "kern" 0;<br />
font-feature-settings: "lnum", "tnum", "kern" 0;<br />
font-variant-numeric: lining-nums tabular-nums;<br />
font-kerning: none;<br />
}<br />
<br />
/* Fix styling of transcluded prefindex tables */<br />
table#mw-prefixindex-list-table,<br />
table#mw-prefixindex-nav-table {<br />
width: 98%;<br />
}<br />
<br />
/* For portals, added 2011-12-07 -bv<br />
On wide screens, show these as two columns<br />
On narrow and mobile screens, let them collapse into a single column */<br />
.portal-column-left {<br />
float: left;<br />
width: 50%;<br />
}<br />
.portal-column-right {<br />
float: right;<br />
width: 49%;<br />
}<br />
.portal-column-left-wide {<br />
float: left;<br />
width: 60%;<br />
}<br />
.portal-column-right-narrow {<br />
float: right;<br />
width: 39%;<br />
}<br />
.portal-column-left-extra-wide {<br />
float: left;<br />
width: 70%;<br />
}<br />
.portal-column-right-extra-narrow {<br />
float: right;<br />
width: 29%;<br />
}<br />
@media only screen and (max-width: 800px) {<br />
/* Decouple the columns on narrow screens */<br />
.portal-column-left,<br />
.portal-column-right,<br />
.portal-column-left-wide,<br />
.portal-column-right-narrow,<br />
.portal-column-left-extra-wide,<br />
.portal-column-right-extra-narrow {<br />
float: inherit;<br />
width: inherit;<br />
}<br />
}<br />
<br />
/* Formerly for announcements, now used intermittently */<br />
#bodyContent .letterhead {<br />
background-image:url('//upload.wikimedia.org/wikipedia/commons/e/e0/Tan-page-corner.png');<br />
background-repeat:no-repeat;<br />
padding: 2em;<br />
background-color: #faf9f2;<br />
}<br />
<br />
/* Tree style lists */<br />
.treeview ul {<br />
padding: 0;<br />
margin: 0;<br />
}<br />
.treeview li {<br />
padding: 0;<br />
margin: 0;<br />
list-style-type: none;<br />
list-style-image: none;<br />
zoom: 1; /* BE KIND TO IE6 */;<br />
}<br />
.treeview li li {<br />
background: url("//upload.wikimedia.org/wikipedia/commons/f/f2/Treeview-grey-line.png") no-repeat 0 -2981px;<br />
/* @noflip */<br />
padding-left: 20px;<br />
text-indent: 0.3em;<br />
}<br />
.treeview li li.lastline {<br />
background-position: 0 -5971px<br />
}<br />
.treeview li.emptyline > ul {<br />
/* @noflip */<br />
margin-left: -1px;<br />
}<br />
.treeview li.emptyline > ul > li:first-child {<br />
background-position: 0 9px<br />
}<br />
<br />
/* hidden sortkey for tablesorter */<br />
td .sortkey,<br />
th .sortkey {<br />
display: none;<br />
speak: none;<br />
}<br />
<br />
/* Make it possible to hide checkboxes in <inputbox> */<br />
.inputbox-hidecheckboxes form .inputbox-element {<br />
display: none !important;<br />
}<br />
<br />
/* Work-around for [[Phabricator:25965]] (Kaltura advertisement) */<br />
.k-player .k-attribution {<br />
visibility: hidden;<br />
}<br />
<br />
/* Move 'play' button of video player to bottom left corner */<br />
.PopUpMediaTransform a .play-btn-large {<br />
margin: 0;<br />
top: auto;<br />
right: auto;<br />
bottom: 0;<br />
left: 0;<br />
}<br />
<br />
/* Workaround to keep editnotices readable in VE view.<br />
Long term, editnotices should become a core feature so that they can be designed responsive. */<br />
.mw-ve-editNotice .mbox-image {<br />
display: none;<br />
}</div>Adminhttps://en.formulasearchengine.com/index.php?title=Main_Page&diff=218394Main Page2015-07-02T09:13:52Z<p>Admin: </p>
<hr />
<div>This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.<br />
<br />
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]<br />
* Only registered users will be able to execute this rendering mode.<br />
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.<br />
<br />
Registered users will be able to choose between the following three rendering modes: <br />
<br />
'''MathML'''<br />
:<math forcemathmode="5">E=mc^2</math><br />
<br />
'''PNG''' (currently default in production)<br />
:<math forcemathmode="0">E=mc^2</math><br />
<br />
'''source'''<br />
:<math forcemathmode="3">E=mc^2</math> <br />
<br />
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].<br />
<br />
==Demos==<br />
<br />
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:<br />
<br />
<br />
* accessibility:<br />
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]<br />
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].<br />
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].<br />
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.<br />
<br />
==Test pages ==<br />
<br />
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:<br />
*[[Displaystyle]]<br />
*[[MathAxisAlignment]]<br />
*[[Styling]]<br />
*[[Linebreaking]]<br />
*[[Unique Ids]]<br />
*[[Help:Formula]]<br />
<br />
*[[Inputtypes|Inputtypes (private Wikis only)]]<br />
*[[Url2Image|Url2Image (private Wikis only)]]<br />
==Bug reporting==<br />
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .</div>Adminhttps://en.formulasearchengine.com/index.php?title=Main_Page&diff=218393Main Page2015-07-02T07:34:00Z<p>Admin: Reverted edits by Maintenance script (talk) to last revision by Admin</p>
<hr />
<div>This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is currently active on [http://en.wikipedia.beta.wmflabs.org/wiki/Main_Page BETA Wikipedia] and will be availble in production soon.<br />
<br />
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]<br />
* Only registered users will be able to execute this rendering mode.<br />
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.<br />
<br />
Registered users will be able to choose between the following three rendering modes: <br />
<br />
'''MathML'''<br />
:<math forcemathmode="5">E=mc^2</math><br />
<br />
'''PNG''' (currently default in production)<br />
:<math forcemathmode="0">E=mc^2</math><br />
<br />
'''source'''<br />
:<math forcemathmode="3">E=mc^2</math> <br />
<br />
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].<br />
<br />
==Demos==<br />
<br />
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:<br />
<br />
<br />
* accessibility:<br />
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]<br />
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].<br />
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].<br />
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.<br />
<br />
==Test pages ==<br />
<br />
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:<br />
*[[Displaystyle]]<br />
*[[MathAxisAlignment]]<br />
*[[Styling]]<br />
*[[Linebreaking]]<br />
*[[Unique Ids]]<br />
*[[Help:Formula]]<br />
<br />
*[[Inputtypes|Inputtypes (private Wikis only)]]<br />
*[[Url2Image|Url2Image (private Wikis only)]]<br />
==Bug reporting==<br />
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .</div>Adminhttps://en.formulasearchengine.com/index.php?title=User:Admin&diff=270841User:Admin2015-05-27T17:42:07Z<p>Admin: </p>
<hr />
<div>===Inline===<br />
The sum <math>\textstyle \sum_{i=0}^\infty 2^{-i}</math> converges to<br />
<br />
something so that the line width is fine.<br />
This example should be written as<br />
<br />
:<code><nowiki><math display="inline">\sum_{i=0}^\infty 2^{-i}</math></nowiki></code><br />
<br />
The convention https://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style_(mathematics)#Using_HTML is really annoying.<br />
<br />
===display===<br />
called the <br />
:<math>\text{geometric series:}\quad \begin{align} \sum_{i=0}^\infty 2^{-i}=2 \end{align}</math><br />
should be able to be written as<br />
:<code><nowiki>:<math display="block">\text{geometric series:}\quad \sum_{i=0}^\infty 2^{-i}=2 </math></nowiki></code><br />
===tidy-test===<br />
<br />
<br />
{|<br />
|| foo<br />
<tr><td>bar</td></tr><br />
|}<br />
<br />
{| class="mw-collapsible mw-collapsed wikitable"<br />
! The header || remains visible<br />
|-<br />
| This content || is hidden<br />
|-<br />
| at first || load time<br />
|}<br />
<br />
<div class="NavFrame"><br />
<div class="NavHead">'''Title 1'''</div><br />
<div class="NavContent"><br />
Hello world.<br />
</div><br />
</div><br />
<br />
<br />
<div class="NavFrame"><br />
<div class="NavHead">'''Title 2'''</div><br />
<div class="NavContent" style="display: none;"><br />
Hello world.<br />
</div><br />
</div><br />
<br />
<br />
<math id=prka>a+b+c</math></div>Adminhttps://en.formulasearchengine.com/index.php?title=Main_Page&diff=218327Main Page2015-05-14T15:11:33Z<p>Admin: </p>
<hr />
<div>This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is currently active on [http://en.wikipedia.beta.wmflabs.org/wiki/Main_Page BETA Wikipedia] and will be availble in production soon.<br />
<br />
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]<br />
* Only registered users will be able to execute this rendering mode.<br />
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.<br />
<br />
Registered users will be able to choose between the following three rendering modes: <br />
<br />
'''MathML'''<br />
:<math forcemathmode="5">E=mc^2</math><br />
<br />
'''PNG''' (currently default in production)<br />
:<math forcemathmode="0">E=mc^2</math><br />
<br />
'''source'''<br />
:<math forcemathmode="3">E=mc^2</math> <br />
<br />
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].<br />
<br />
==Demos==<br />
<br />
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:<br />
<br />
<br />
* accessibility:<br />
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]<br />
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].<br />
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].<br />
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.<br />
<br />
==Test pages ==<br />
<br />
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:<br />
*[[Displaystyle]]<br />
*[[MathAxisAlignment]]<br />
*[[Styling]]<br />
*[[Linebreaking]]<br />
*[[Unique Ids]]<br />
*[[Help:Formula]]<br />
<br />
*[[Inputtypes|Inputtypes (private Wikis only)]]<br />
*[[Url2Image|Url2Image (private Wikis only)]]<br />
==Bug reporting==<br />
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .</div>Adminhttps://en.formulasearchengine.com/index.php?title=User:Admin&diff=270840User:Admin2015-02-08T15:44:40Z<p>Admin: </p>
<hr />
<div>===Inline===<br />
The sum <math>\textstyle \sum_{i=0}^\infty 2^{-i}</math> converges to<br />
<br />
something so that the line width is fine.<br />
This example should be written as<br />
<br />
:<code><nowiki><math display="inline">\sum_{i=0}^\infty 2^{-i}</math></nowiki></code><br />
<br />
The convention https://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style_(mathematics)#Using_HTML is really annoying.<br />
<br />
===display===<br />
called the <br />
:<math>\text{geometric series:}\quad \begin{align} \sum_{i=0}^\infty 2^{-i}=2 \end{align}</math><br />
should be able to be written as<br />
:<code><nowiki>:<math display="block">\text{geometric series:}\quad \sum_{i=0}^\infty 2^{-i}=2 </math></nowiki></code><br />
===tidy-test===<br />
<br />
<br />
{|<br />
|| foo<br />
<tr><td>bar</td></tr><br />
|}<br />
<br />
{| class="mw-collapsible mw-collapsed wikitable"<br />
! The header || remains visible<br />
|-<br />
| This content || is hidden<br />
|-<br />
| at first || load time<br />
|}<br />
<br />
<div class="NavFrame"><br />
<div class="NavHead">'''Title 1'''</div><br />
<div class="NavContent"><br />
Hello world.<br />
</div><br />
</div><br />
<br />
<br />
<div class="NavFrame"><br />
<div class="NavHead">'''Title 2'''</div><br />
<div class="NavContent" style="display: none;"><br />
Hello world.<br />
</div><br />
</div></div>Adminhttps://en.formulasearchengine.com/index.php?title=User:Admin&diff=270839User:Admin2015-02-08T15:44:03Z<p>Admin: </p>
<hr />
<div>===Inline===<br />
The sum <math>\textstyle \sum_{i=0}^\infty 2^{-i}</math> converges to<br />
<br />
something so that the line width is fine.<br />
This example should be written as<br />
<br />
:<code><nowiki><math display="inline">\sum_{i=0}^\infty 2^{-i}</math></nowiki></code><br />
<br />
The convention https://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style_(mathematics)#Using_HTML is really annoying.<br />
<br />
===display===<br />
called the <br />
:<math>\text{geometric series:}\quad \begin{align} \sum_{i=0}^\infty 2^{-i}=2 \end{align}</math><br />
should be able to be written as<br />
:<code><nowiki>:<math display="block">\text{geometric series:}\quad \sum_{i=0}^\infty 2^{-i}=2 </math></nowiki></code><br />
===tidy-test===<br />
<br />
<br />
{|<br />
|| foo<br />
<tr><td>bar</td></tr><br />
|}<br />
<br />
{| class="mw-collapsible mw-collapsed wikitable"<br />
! The header || remains visible<br />
|-<br />
| This content || is hidden<br />
|-<br />
| at first || load time<br />
|}</div>Adminhttps://en.formulasearchengine.com/index.php?title=User:Admin&diff=270838User:Admin2015-02-04T12:45:48Z<p>Admin: /* tidy-test */</p>
<hr />
<div>===Inline===<br />
The sum <math>\textstyle \sum_{i=0}^\infty 2^{-i}</math> converges to<br />
<br />
something so that the line width is fine.<br />
This example should be written as<br />
<br />
:<code><nowiki><math display="inline">\sum_{i=0}^\infty 2^{-i}</math></nowiki></code><br />
<br />
The convention https://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style_(mathematics)#Using_HTML is really annoying.<br />
<br />
===display===<br />
called the <br />
:<math>\text{geometric series:}\quad \begin{align} \sum_{i=0}^\infty 2^{-i}=2 \end{align}</math><br />
should be able to be written as<br />
:<code><nowiki>:<math display="block">\text{geometric series:}\quad \sum_{i=0}^\infty 2^{-i}=2 </math></nowiki></code><br />
===tidy-test===<br />
<br />
<br />
{|<br />
|| foo<br />
<tr><td>bar</td></tr><br />
|}</div>Adminhttps://en.formulasearchengine.com/index.php?title=User:Admin&diff=270837User:Admin2015-02-04T12:45:16Z<p>Admin: </p>
<hr />
<div>===Inline===<br />
The sum <math>\textstyle \sum_{i=0}^\infty 2^{-i}</math> converges to<br />
<br />
something so that the line width is fine.<br />
This example should be written as<br />
<br />
:<code><nowiki><math display="inline">\sum_{i=0}^\infty 2^{-i}</math></nowiki></code><br />
<br />
The convention https://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style_(mathematics)#Using_HTML is really annoying.<br />
<br />
===display===<br />
called the <br />
:<math>\text{geometric series:}\quad \begin{align} \sum_{i=0}^\infty 2^{-i}=2 \end{align}</math><br />
should be able to be written as<br />
:<code><nowiki>:<math display="block">\text{geometric series:}\quad \sum_{i=0}^\infty 2^{-i}=2 </math></nowiki></code><br />
===tidy-test===</div>Adminhttps://en.formulasearchengine.com/index.php?title=Main_Page&diff=173822Main Page2014-10-23T13:45:50Z<p>Admin: </p>
<hr />
<div>This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is currently active on [http://en.wikipedia.beta.wmflabs.org/wiki/Main_Page BETA Wikipedia] and will be availble in production soon.<br />
<br />
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]<br />
* Only registered users will be able to execute this rendering mode.<br />
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.<br />
<br />
Registered users will be able to choose between the following three rendering modes: <br />
<br />
'''MathML'''<br />
:<math forcemathmode="5">E=mc^2</math><br />
<br />
'''PNG''' (currently default in production)<br />
:<math forcemathmode="0">E=mc^2</math><br />
<br />
'''source'''<br />
:<math forcemathmode="3">E=mc^2</math> <br />
<br />
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].<br />
<br />
==Demos==<br />
<br />
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:<br />
<br />
* accessibility:<br />
** [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv Safari + VoiceOver (video only)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]<br />
** From our testing, ChromeVox and JAWS 16 Beta are not able to read the formulas generated by the MathML mode. There is ongoing work by NVDA and Orca developers to support MathML, but at that time no public release is available.<br />
* scaling: [https://commons.wikimedia.org/wiki/File:Zoomed-Windows8-Firefox32-PNG.png PNG], [https://commons.wikimedia.org/wiki/File:Zoomed-Windows8-Firefox32-SVG.png SVG], [https://commons.wikimedia.org/wiki/File:Zoomed-Windows8-Firefox32-MathML-LatinModern.png MathML]<br />
* math axis alignment: [https://commons.wikimedia.org/wiki/File:MathAxisAlignment-Windows8-Firefox32-PNG.png PNG], [https://commons.wikimedia.org/wiki/File:MathAxisAlignment-Windows8-Firefox32-SVG.png SVG], [https://commons.wikimedia.org/wiki/File:MathAxisAlignment-Windows8-Firefox32-MathML-LatinModern.png MathML]<br />
* [https://commons.wikimedia.org/wiki/File:MathML_on_a_high_contrast_screen.png MathML rendered on high contrast screen]<br />
* styling: [https://commons.wikimedia.org/wiki/File:Styling-Windows8-Firefox32-PNG.png PNG], [https://commons.wikimedia.org/wiki/File:Styling-Windows8-Firefox32-SVG.png SVG], [https://commons.wikimedia.org/wiki/File:Styling-Windows8-Firefox32-MathML-LatinModern.png MathML]<br />
* font family: [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-Cambria.png Cambria], [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-LatinModern.png Latin Modern], [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-TeXGyreBonum.png TeX Gyre Bonum], [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-TeXGyreSchola.png TeX Gyre Schola], [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-TeXGyrePagella.png TeX Gyre Pagella], [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-XITS.png XITS].<br />
* MathML add-ons: [https://commons.wikimedia.org/wiki/File:AddOnCopying-Windows8-Firefox32-MathML-LatinModern.png Copying TeX/MathML], [https://commons.wikimedia.org/wiki/File:AddOnZooming-Windows8-Firefox32-MathML-LatinModern.png Zooming].<br />
<br />
==Test pages ==<br />
<br />
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:<br />
*[[Displaystyle]]<br />
*[[MathAxisAlignment]]<br />
*[[Styling]]<br />
*[[Linebreaking]]<br />
*[[Unique Ids]]<br />
*[[Help:Formula]]<br />
<br />
*[[Inputtypes|Inputtypes (private Wikis only)]]<br />
*[[Url2Image|Url2Image (private Wikis only)]]<br />
==Bug reporting==<br />
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .</div>Adminhttps://en.formulasearchengine.com/index.php?title=Main_Page&diff=162271Main Page2014-10-22T17:17:53Z<p>Admin: </p>
<hr />
<div>This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is currently active on [http://en.wikipedia.beta.wmflabs.org/wiki/Main_Page BETA Wikipedia] and will be availble in production soon.<br />
<br />
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]<br />
* Only registered users will be able to execute this rendering mode.<br />
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.<br />
<br />
Registered users will be able to choose between the following three rendering modes: <br />
<br />
'''MathML'''<br />
:<math forcemathmode="5">E=mc^2</math><br />
<br />
'''PNG''' (currently default in production)<br />
:<math forcemathmode="0">E=mc^2</math><br />
<br />
'''source'''<br />
:<math forcemathmode="3">E=mc^2</math> <br />
<br />
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].<br />
<br />
==Demos==<br />
<br />
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:<br />
<br />
* accessibility:<br />
** [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv Safari + VoiceOver (video only)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]<br />
** From our testing, ChromeVox and JAWS 16 Beta are not able to read the formulas generated by the MathML mode. There is ongoing work by NVDA and Orca developers to support MathML, but at that time no public release is available.<br />
* scaling: [https://commons.wikimedia.org/wiki/File:Zoomed-Windows8-Firefox32-PNG.png PNG], [https://commons.wikimedia.org/wiki/File:Zoomed-Windows8-Firefox32-SVG.png SVG], [https://commons.wikimedia.org/wiki/File:Zoomed-Windows8-Firefox32-MathML-LatinModern.png MathML]<br />
* math axis alignment: [https://commons.wikimedia.org/wiki/File:MathAxisAlignment-Windows8-Firefox32-PNG.png PNG], [https://commons.wikimedia.org/wiki/File:MathAxisAlignment-Windows8-Firefox32-SVG.png SVG], [https://commons.wikimedia.org/wiki/File:MathAxisAlignment-Windows8-Firefox32-MathML-LatinModern.png MathML]<br />
* [https://commons.wikimedia.org/wiki/File:MathML_on_a_high_contrast_screen.png MathML rendered on high contrast screen]<br />
* styling: [https://commons.wikimedia.org/wiki/File:Styling-Windows8-Firefox32-PNG.png PNG], [https://commons.wikimedia.org/wiki/File:Styling-Windows8-Firefox32-SVG.png SVG], [https://commons.wikimedia.org/wiki/File:Styling-Windows8-Firefox32-MathML-LatinModern.png MathML]<br />
* font family: [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-Cambria.png Cambria], [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-LatinModern.png Latin Modern], [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-TeXGyreBonum.png TeX Gyre Bonum], [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-TeXGyreSchola.png TeX Gyre Schola], [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-TeXGyrePagella.png TeX Gyre Pagella], [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-XITS.png XITS].<br />
* MathML add-ons: [https://commons.wikimedia.org/wiki/File:AddOnCopying-Windows8-Firefox32-MathML-LatinModern.png Copying TeX/MathML], [https://commons.wikimedia.org/wiki/File:AddOnZooming-Windows8-Firefox32-MathML-LatinModern.png Zooming].<br />
<br />
==Test pages ==<br />
<br />
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:<br />
*[[Displaystyle]]<br />
*[[MathAxisAlignment]]<br />
*[[Styling]]<br />
*[[Linebreaking]]<br />
*[[Unique Ids]]<br />
*[[Help:Formula]]<br />
<br />
*[[Inputtypes|Inputtypes (private Wikis only)]]<br />
*[[Url2Image|Url2Image (private Wikis only)]]<br />
==Bug reporting==<br />
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .</div>Adminhttps://en.formulasearchengine.com/index.php?title=Main_Page&diff=145443Main Page2014-10-20T06:43:37Z<p>Admin: /* Demos */</p>
<hr />
<div>This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is currently active on [http://en.wikipedia.beta.wmflabs.org/wiki/Main_Page BETA Wikipedia] and will be availble in production soon.<br />
<br />
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]<br />
* Only registered users will be able to execute this rendering mode.<br />
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.<br />
<br />
Registered users will be able to choose between the following three rendering modes: <br />
<br />
'''MathML'''<br />
:<math forcemathmode="5">E=mc^2</math><br />
<br />
'''PNG''' (currently default in production)<br />
:<math forcemathmode="0">E=mc^2</math><br />
<br />
'''source'''<br />
:<math forcemathmode="3">E=mc^2</math> <br />
<br />
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].<br />
<br />
==Demos==<br />
<br />
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:<br />
<br />
* accessibility:<br />
** [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv Safari + VoiceOver (video only)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]<br />
** From our testing, ChromeVox and JAWS 16 Beta are not able to read the formulas generated by the MathML mode. There is ongoing work by NVDA and Orca developers to support MathML, but at that time no public release is available.<br />
* scaling: [https://commons.wikimedia.org/wiki/File:Zoomed-Windows8-Firefox32-PNG.png PNG], [https://commons.wikimedia.org/wiki/File:Zoomed-Windows8-Firefox32-SVG.png SVG], [https://commons.wikimedia.org/wiki/File:Zoomed-Windows8-Firefox32-MathML-LatinModern.png MathML]<br />
* math axis alignment: [https://commons.wikimedia.org/wiki/File:MathAxisAlignment-Windows8-Firefox32-PNG.png PNG], [https://commons.wikimedia.org/wiki/File:MathAxisAlignment-Windows8-Firefox32-SVG.png SVG], [https://commons.wikimedia.org/wiki/File:MathAxisAlignment-Windows8-Firefox32-MathML-LatinModern.png MathML]<br />
* [https://commons.wikimedia.org/wiki/File:MathML_on_a_high_contrast_screen.png MathML rendered on high contrast screen]<br />
* styling: [https://commons.wikimedia.org/wiki/File:Styling-Windows8-Firefox32-PNG.png PNG], [https://commons.wikimedia.org/wiki/File:Styling-Windows8-Firefox32-SVG.png SVG], [https://commons.wikimedia.org/wiki/File:Styling-Windows8-Firefox32-MathML-LatinModern.png MathML]<br />
* font family: [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-Cambria.png Cambria], [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-LatinModern.png Latin Modern], [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-TeXGyreBonum.png TeX Gyre Bonum], [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-TeXGyreSchola.png TeX Gyre Schola], [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-TeXGyrePagella.png TeX Gyre Pagella], [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-XITS.png XITS].<br />
* MathML add-ons: [https://commons.wikimedia.org/wiki/File:AddOnCopying-Windows8-Firefox32-MathML-LatinModern.png Copying TeX/MathML], [https://commons.wikimedia.org/wiki/File:AddOnZooming-Windows8-Firefox32-MathML-LatinModern.png Zooming].<br />
<br />
==Test pages ==<br />
<br />
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:<br />
*[[Displaystyle]]<br />
*[[MathAxisAlignment]]<br />
*[[Styling]]<br />
*[[Linebreaking]]<br />
*[[Unique Ids]]<br />
*[[Help:Formula]]<br />
<br />
*[[Inputtypes|Inputtypes (private Wikis only)]]<br />
*[[Url2Image|Url2Image (private Wikis only)]]<br />
==Bug reporting==<br />
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .</div>Adminhttps://en.formulasearchengine.com/index.php?title=Main_Page&diff=145442Main Page2014-10-19T14:27:04Z<p>Admin: </p>
<hr />
<div>This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is currently active on [http://en.wikipedia.beta.wmflabs.org/wiki/Main_Page BETA Wikipedia] and will be availble in production soon.<br />
<br />
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]<br />
* Only registered users will be able to execute this rendering mode.<br />
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.<br />
<br />
Registered users will be able to choose between the following three rendering modes: <br />
<br />
'''MathML'''<br />
:<math forcemathmode="5">E=mc^2</math><br />
<br />
'''PNG''' (currently default in production)<br />
:<math forcemathmode="0">E=mc^2</math><br />
<br />
'''source'''<br />
:<math forcemathmode="3">E=mc^2</math> <br />
<br />
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].<br />
<br />
==Demos==<br />
<br />
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:<br />
<br />
* accessibility:<br />
** [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv Safari + VoiceOver (video only)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]<br />
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]<br />
** From our testing, ChromeVox and JAWS 16 Beta are not able to read the formulas generated by the MathML mode. There is ongoing work by NVDA and Orca developers to support MathML, but at that time no public release is available.<br />
* scaling: [https://commons.wikimedia.org/wiki/File:Zoomed-Windows8-Firefox32-PNG.png PNG], [https://commons.wikimedia.org/wiki/File:Zoomed-Windows8-Firefox32-SVG.png SVG], [https://commons.wikimedia.org/wiki/File:Zoomed-Windows8-Firefox32-MathML-LatinModern.png MathML]<br />
* math axis alignment: [https://commons.wikimedia.org/wiki/File:MathAxisAlignment-Windows8-Firefox32-PNG.png PNG], [https://commons.wikimedia.org/wiki/File:MathAxisAlignment-Windows8-Firefox32-SVG.png SVG], [https://commons.wikimedia.org/wiki/File:MathAxisAlignment-Windows8-Firefox32-MathML-LatinModern.png MathML]<br />
* styling: [https://commons.wikimedia.org/wiki/File:Styling-Windows8-Firefox32-PNG.png PNG], [https://commons.wikimedia.org/wiki/File:Styling-Windows8-Firefox32-SVG.png SVG], [https://commons.wikimedia.org/wiki/File:Styling-Windows8-Firefox32-MathML-LatinModern.png MathML]<br />
* font family: [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-Cambria.png Cambria], [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-LatinModern.png Latin Modern], [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-TeXGyreBonum.png TeX Gyre Bonum], [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-TeXGyreSchola.png TeX Gyre Schola], [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-TeXGyrePagella.png TeX Gyre Pagella], [https://commons.wikimedia.org/wiki/File:Fourier-Windows8-Firefox32-MathML-XITS.png XITS].<br />
* MathML add-ons: [https://commons.wikimedia.org/wiki/File:AddOnCopying-Windows8-Firefox32-MathML-LatinModern.png Copying TeX/MathML], [https://commons.wikimedia.org/wiki/File:AddOnZooming-Windows8-Firefox32-MathML-LatinModern.png Zooming].<br />
<br />
==Test pages ==<br />
<br />
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:<br />
*[[Displaystyle]]<br />
*[[MathAxisAlignment]]<br />
*[[Styling]]<br />
*[[Linebreaking]]<br />
*[[Unique Ids]]<br />
*[[Help:Formula]]<br />
<br />
*[[Inputtypes|Inputtypes (private Wikis only)]]<br />
*[[Url2Image|Url2Image (private Wikis only)]]<br />
==Bug reporting==<br />
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .</div>Adminhttps://en.formulasearchengine.com/index.php?title=Main_Page&diff=134557Main Page2014-10-13T23:01:49Z<p>Admin: </p>
<hr />
<div>This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is currently active on [http://en.wikipedia.beta.wmflabs.org/wiki/Main_Page BETA Wikipedia] and will be availble in production soon.<br />
<br />
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]<br />
* Only registered users will be able to execute this rendering mode.<br />
* Note: you need not enter a email address (nor any pther private information). Please do not use a password that you use elsewhere.<br />
<br />
Registered users will be able to choose between the following three rendering modes: <br />
<br />
'''MathML'''<br />
:<math forcemathmode="5">E=mc^2</math><br />
<br />
'''PNG''' (currently default in production)<br />
:<math forcemathmode="0">E=mc^2</math><br />
<br />
'''source'''<br />
:<math forcemathmode="3">E=mc^2</math><br />
<br />
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span><br />
==Test pages ==<br />
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:<br />
*[[Displaystyle]]<br />
*[[MathAxisAlignment]]<br />
*[[Linebreaking]]<br />
*[[Unique Ids]]<br />
*[[Help:Formula]]<br />
<br />
*[[Inputtypes|Inputtypes (private Wikis only)]]<br />
*[[Url2Image|Url2Image (private Wikis only)]]<br />
==Bug reporting==<br />
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .</div>Adminhttps://en.formulasearchengine.com/index.php?title=Main_Page&diff=134548Main Page2014-10-13T22:59:13Z<p>Admin: </p>
<hr />
<div>This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is currently active on [http://en.wikipedia.beta.wmflabs.org/wiki/Main_Page BETA Wikipedia] and will be availble in production soon.<br />
<br />
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]<br />
* Only registered users will be able to execute this rendering mode.<br />
* Note: you need not enter a email address (nor any pther private information). Please do not use a password that you use elsewhere.<br />
<br />
Registered users will be able to choose between the following three rendering modes: <br />
<br />
'''MathML''' (can not yet be used on the Wikimedia production cluster)<br />
:<math forcemathmode="5">E=mc^2</math><br />
<br />
'''PNG''' (currently active in production)<br />
:<math forcemathmode="0">E=mc^2</math><br />
<br />
'''source''' (currently disabled in production via configuration)<br />
:<math forcemathmode="3">E=mc^2</math><br />
<br />
<span style="color: red">Follow this [[https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering|link]] to change your Math rendering settings.</span><br />
==Test pages ==<br />
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:<br />
*[[Displaystyle]]<br />
*[[MathAxisAlignment]]<br />
*[[Linebreaking]]<br />
*[[Unique Ids]]<br />
*[[Help:Formula]]<br />
<br />
*[[Inputtypes|Inputtypes (private Wikis only)]]<br />
*[[Url2Image|Url2Image (private Wikis only)]]<br />
==Bug reporting==<br />
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .</div>Adminhttps://en.formulasearchengine.com/index.php?title=Main_Page&diff=134547Main Page2014-10-13T22:58:07Z<p>Admin: </p>
<hr />
<div>This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is currently active on [http://en.wikipedia.beta.wmflabs.org/wiki/Main_Page BETA Wikipedia] and will be availble in production soon.<br />
<br />
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]<br />
* Only registered users will be able to execute this rendering mode.<br />
* Note: you need not enter a email address (nor any pther private information). Please do not use a password that you use elsewhere.<br />
<br />
Registered users will be able to choose between the following three rendering modes: <br />
<br />
'''MathML''' (can not yet be used on the Wikimedia production cluster)<br />
:<math forcemathmode="5">E=mc^2</math><br />
<br />
'''PNG''' (currently active in production)<br />
:<math forcemathmode="0">E=mc^2</math><br />
<br />
'''source''' (currently disabled in production via configuration)<br />
:<math forcemathmode="3">E=mc^2</math><br />
<br />
<span style="color: red">Follow this [[https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering|link]] to change your Math rendering settings.</span><br />
==Test pages ==<br />
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:<br />
*[[User:Physikerwelt/Displaystyle]]<br />
*[[User:Physikerwelt/Unique Ids]]<br />
*[[User:Physikerwelt/Help:Formula]]<br />
==Bug reporting==<br />
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .<br />
<br />
This page is a preview for the new Math rendering that will get live at Wikipedia soon.<br />
Registered user will be able to chose between<br />
;Source:<math forcemathmode="3">E=mc^2</math>(currently disabled via config)<br />
;PNG:<math forcemathmode="0">E=mc^2</math> (currently active)<br />
;MathML:<math forcemathmode="5">E=mc^2</math>(no Mathoid server can be accessed from the Wikimedia production cluster)<br />
* In a first step MathML and SVG will be available to registered users only.<br />
** If you want to test please register an account here [http://math-preview.wmflabs.org/w/index.php?title=Special:UserLogin&type=signup&returnto=Help:Formula Register]<br />
*** You don't have to enter a email address nor any private information do not use a password that you use elsewhere<br />
** Change your Math rendering settings to MathML [http://math-preview.wmflabs.org/wiki/Special:Preferences#mw-prefsection-rendering here]<br />
* Go to a [http://math-preview.wmflabs.org/wiki/Special:Random random page] or<br />
* one of the test pages listed below.<br />
==I found a bug==<br />
If you find any bugs please report a bug at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla] or write a mail to math_bugs (at) ckurs (dot) de<br />
==Test pages ==<br />
*[[Displaystyle]]<br />
*[[MathAxisAlignment]]<br />
*[[Linebreaking]]<br />
*[[Unique Ids]]<br />
*[[Help:Formula]]<br />
<br />
*[[Inputtypes|Inputtypes (private Wikis only)]]<br />
*[[Url2Image|Url2Image (private Wikis only)]]</div>Adminhttps://en.formulasearchengine.com/index.php?title=Main_Page&diff=130388Main Page2014-08-28T14:49:58Z<p>Admin: </p>
<hr />
<div>This page is a preview for the new Math rendering that will get live at Wikipedia soon.<br />
Registered user will be able to chose between<br />
;Source:<math forcemathmode="3">E=mc^2</math>(currently disabled via config)<br />
;PNG:<math forcemathmode="0">E=mc^2</math> (currently active)<br />
;MathML:<math forcemathmode="5">E=mc^2</math>(no Mathoid server can be accessed from the Wikimedia production cluster)<br />
* In a first step MathML and SVG will be available to registered users only.<br />
** If you want to test please register an account here [http://math-preview.wmflabs.org/w/index.php?title=Special:UserLogin&type=signup&returnto=Help:Formula Register]<br />
*** You don't have to enter a email address nor any private information do not use a password that you use elsewhere<br />
** Change your Math rendering settings to MathML [http://math-preview.wmflabs.org/wiki/Special:Preferences#mw-prefsection-rendering here]<br />
* Go to a [http://math-preview.wmflabs.org/wiki/Special:Random random page] or<br />
* one of the test pages listed below.<br />
==I found a bug==<br />
If you find any bugs please report a bug at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla] or write a mail to math_bugs (at) ckurs (dot) de<br />
==Test pages ==<br />
*[[Displaystyle]]<br />
*[[Unique Ids]]<br />
*[[Help:Formula]]<br />
<br />
*[[Inputtypes|Inputtypes (private Wikis only)]]<br />
*[[Url2Image|Url2Image (private Wikis only)]]</div>Adminhttps://en.formulasearchengine.com/index.php?title=Main_Page&diff=122798Main Page2014-08-27T00:22:34Z<p>Admin: </p>
<hr />
<div>This page is a preview for the new Math rendering that will get live at Wikipedia soon.<br />
Registered user will be able to chose between<br />
;Source:<math forcemathmode="3">E=mc^2</math>(currently disabled via config)<br />
;PNG:<math forcemathmode="0">E=mc^2</math> (currently active)<br />
;MathML:<math forcemathmode="5">E=mc^2</math>(no Mathoid server can be accessed from the Wikimedia production cluster)<br />
* In a first step MathML and SVG will be available to registered users only.<br />
** If you want to test please register an account here [http://math-preview.wmflabs.org/w/index.php?title=Special:UserLogin&type=signup&returnto=Help:Formula Register]<br />
*** You don't have to enter a email address nor any private information do not use a password that you use elsewhere<br />
** Change your Math rendering settings to MathML [http://math-preview.wmflabs.org/wiki/Special:Preferences#mw-prefsection-rendering here]<br />
* Go to a [http://math-preview.wmflabs.org/wiki/Special:Random random page] or<br />
* one of the test pages listed below.<br />
==I found a bug==<br />
If you find any bugs please report a bug at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla] or write a mail to math_bugs (at) ckurs (dot) de<br />
==Test pages ==<br />
*[[Displaystyle]]<br />
*[[Unique Ids]]<br />
*[[Help:Formula]]<br />
<br />
*[[Inputtypes|Inputtypes (private Wikis only)]]<br />
*[[Url2Image|Url2Image (private Wikis only)]]</div>Adminhttps://en.formulasearchengine.com/index.php?title=Albedo&diff=218454Albedo2014-07-31T13:58:34Z<p>Admin: Reverted edits by MediaWiki spam cleanup (talk) to last revision by Trappist the monk</p>
<hr />
<div>{{about|the reflectivity measurement|the inner fleshy part of a citrus fruit|Mesocarp}}<br />
{{Use dmy dates|date=June 2013}}<br />
[[File:Albedo-e hg.svg|thumb|Percentage of diffusely reflected sunlight in relation to various surface conditions]]<br />
<br />
'''Albedo''' ({{IPAc-en|æ|l|ˈ|b|iː|d|oʊ}}), or ''reflection coefficient'', derived from [[Latin]] ''albedo'' "whiteness" (or reflected sunlight) in turn from ''albus'' "white," is the [[diffuse reflection|diffuse reflectivity]] or reflecting power of a surface. It is the ratio of reflected radiation from the surface to incident radiation upon it. Its [[Dimensionless number|dimensionless]] nature lets it be expressed as a percentage and is measured on a scale from zero for no reflection of a perfectly black surface to 1 for perfect reflection of a white surface.<br />
<br />
Albedo depends on the [[frequency]] of the radiation. When quoted unqualified, it usually refers to some appropriate average across the spectrum of [[visible light]]. In general, the albedo depends on the directional distribution of incident radiation, except for [[Lambertian reflectance|Lambertian surfaces]], which scatter radiation in all directions according to a cosine function and therefore have an albedo that is independent of the incident distribution. In practice, a [[bidirectional reflectance distribution function]] (BRDF) may be required to accurately characterize the scattering properties of a surface, but albedo is very useful as a first approximation.<br />
<br />
The albedo is an important concept in [[climatology]], [[astronomy]], and calculating [[reflectivity]] of surfaces in [[Leadership in Energy and Environmental Design|LEED]] sustainable-rating systems for buildings. The average overall albedo of Earth, its ''planetary albedo'', is 30 to 35% because of cloud cover, but widely varies locally across the surface because of different geological and environmental features.<ref>Environmental Encyclopedia, 3rd ed., Thompson Gale, 2003, ISBN 0-7876-5486-8</ref><br />
<br />
The term was introduced into optics by [[Johann Heinrich Lambert]] in his 1760 work ''[[Photometria]]''.<br />
<br />
==Terrestrial albedo==<br />
{| class="wikitable" style="float:right; margin:10px"<br />
|+ Sample albedos<br />
|-<br />
! Surface<br />
! Typical<br>albedo<br />
|-<br />
| Fresh asphalt || 0.04<ref name="heat island">{{cite web<br />
| last=Pon | first=Brian | date=30 June 1999<br />
| url=http://eetd.lbl.gov/HeatIsland/Pavements/Albedo/<br />
| title=Pavement Albedo | publisher=Heat Island Group<br />
| accessdate=2007-08-27<br />
| archiveurl= http://web.archive.org/web/20070829153207/http://eetd.lbl.gov/HeatIsland/Pavements/Albedo/| archivedate= 29 August 2007<!--Added by DASHBot-->}}</ref><br />
|-<br />
| Worn asphalt || 0.12<ref name="heat island"/><br />
|-<br />
| Conifer forest<br>(Summer) || 0.08,<ref name="Betts 1">{{Cite journal<br />
| author=Alan K. Betts, John H. Ball<br />
| title=Albedo over the boreal forest<br />
| journal=Journal of Geophysical<br />
| year=1997<br />
| volume=102<br />
| issue=D24<br />
| pages=28,901–28,910<br />
| url=http://www.agu.org/pubs/crossref/1997/96JD03876.shtml<br />
| accessdate=2007-08-27<br />
| doi=10.1029/96JD03876<br />
|bibcode = 1997JGR...10228901B | archiveurl= http://web.archive.org/web/20070930184719/http://www.agu.org/pubs/crossref/1997/96JD03876.shtml| archivedate= 30 September 2007<!--Added by DASHBot-->}}</ref> 0.09 to 0.15<ref name="mmutrees"/><br />
|-<br />
| [[Deciduous trees]] || 0.15 to 0.18<ref name="mmutrees"/><br />
|-<br />
| Bare soil || 0.17<ref name="markvart">{{Cite book<br />
| author=Tom Markvart, Luis CastaŁżer | year=2003<br />
| title=Practical Handbook of Photovoltaics: Fundamentals and Applications<br />
| publisher=Elsevier | isbn=1-85617-390-9 }}</ref><br />
|-<br />
| Green grass || 0.25<ref name="markvart"/><br />
|-<br />
| Desert sand || 0.40<ref name="Tetzlaff">{{Cite book<br />
| first=G. | last=Tetzlaff | year=1983<br />
| title=Albedo of the Sahara<br />
| work=Cologne University Satellite Measurement of Radiation Budget Parameters<br />
| pages=60–63 }}</ref><br />
|-<br />
| New concrete || 0.55<ref name="markvart"/><br />
|-<br />
| Ocean ice|| 0.5–0.7<ref name="markvart"/><br />
|-<br />
| Fresh snow || 0.80–0.90<ref name="markvart"/><br />
|}<br />
Albedos of typical materials in visible light range from up to 0.9 for fresh snow to about 0.04 for charcoal, one of the darkest substances. Deeply shadowed cavities can achieve an effective albedo approaching the zero of a [[black body]]. When seen from a distance, the ocean surface has a low albedo, as do most forests, whereas desert areas have some of the highest albedos among landforms. Most land areas are in an albedo range of 0.1 to 0.4.<ref name="PhysicsWorld">{{cite web|url=http://scienceworld.wolfram.com/physics/Albedo.html |title=Albedo - from Eric Weisstein's World of Physics |publisher=Scienceworld.wolfram.com |date= |accessdate=2011-08-19}}</ref> The average albedo of the [[Earth]] is about 0.3.<ref name="Goode"/> This is far higher than for the ocean primarily because of the contribution of clouds.<br />
<br />
[[File:Ceres 2003 2004 clear sky total sky albedo.png|thumb|200px|left|2003–2004 mean annual clear-sky and total-sky albedo]]<br />
The Earth's surface albedo is regularly estimated via [[Earth observation]] satellite sensors such as [[NASA]]'s [[MODIS]] instruments on board the [[Terra (satellite)|Terra]] and [[Aqua (satellite)|Aqua]] satellites. As the total amount of reflected radiation cannot be directly measured by satellite, a [[mathematical model]] of the BRDF is used to translate a sample set of satellite reflectance measurements into estimates of [[directional-hemispherical reflectance]] and bi-hemispherical reflectance (e.g.<ref name="NASA"/>).<br />
<br />
The Earth's average surface temperature due to its albedo and the [[greenhouse effect]] is currently about 15&nbsp;°C. If the Earth was frozen entirely (and hence be more reflective) the average temperature of the planet would drop below −40&nbsp;°C.<ref name="washington" /> If only the continental land masses became covered by glaciers, the mean temperature of the planet would drop to about 0&nbsp;°C.<ref name="clim-past"/> In contrast, if the entire Earth is covered by water—a so-called aquaplanet—the average temperature on the planet would rise to just under 27&nbsp;°C.<ref name="Smith Robin"/><br />
<br />
===White-sky and black-sky albedo===<br />
It has been shown that for many applications involving terrestrial albedo, the albedo at a particular [[solar zenith angle]] ''θ''<sub>''i''</sub> can reasonably be approximated by the proportionate sum of two terms: the directional-hemispherical reflectance at that solar zenith angle, <math>{\bar \alpha(\theta_i)}</math>, and the bi-hemispherical reflectance, <math>\bar{ \bar \alpha}</math> the proportion concerned being defined as the proportion of diffuse illumination <math>{D}</math>.<br />
<br />
Albedo <math>{\alpha}</math> can then be given as:<br />
<br />
:<math>{\alpha}= (1-D) \bar \alpha(\theta_i) + D \bar{ \bar \alpha}.</math><br />
<br />
[[Directional-hemispherical reflectance]] is sometimes referred to as black-sky albedo and [[bi-hemispherical reflectance]] as white-sky albedo. These terms are important because they allow the albedo to be calculated for any given illumination conditions from a knowledge of the intrinsic properties of the surface.<ref name="BlueskyAlbedo"/><br />
<br />
==Astronomical albedo==<br />
The albedos of [[planet]]s, [[Natural satellite|satellites]] and [[asteroid]]s can be used to infer much about their properties. The study of albedos, their dependence on wavelength, lighting angle ("phase angle"), and variation in time comprises a major part of the astronomical field of [[photometry (astronomy)|photometry]]. For small and far objects that cannot be resolved by telescopes, much of what we know comes from the study of their albedos. For example, the absolute albedo can indicate the surface ice content of outer solar system objects, the variation of albedo with phase angle gives information about [[regolith]] properties, while unusually high radar albedo is indicative of high metallic content in [[asteroid]]s.<br />
<br />
[[Enceladus (moon)|Enceladus]], a moon of Saturn, has one of the highest known albedos of any body in the Solar system, with 99% of EM radiation reflected. Another notable high-albedo body is [[Eris (dwarf planet)|Eris]], with an albedo of 0.96.<ref name="sicardy"><br />
{{cite journal<br />
| title = Size, density, albedo and atmosphere limit of dwarf planet Eris from a stellar occultation<br />
| journal = European Planetary Science Congress Abstracts<br />
| volume = 6<br />
| year = 2011<br />
| url = http://meetingorganizer.copernicus.org/EPSC-DPS2011/EPSC-DPS2011-137-8.pdf<br />
| accessdate = 2011-09-14<br />
| bibcode = 2011epsc.conf..137S<br />
| author1 = Sicardy<br />
| first1 = B.<br />
| last2 = Ortiz<br />
| first2 = J. L.<br />
| last3 = Assafin<br />
| first3 = M.<br />
| last4 = Jehin<br />
| first4 = E.<br />
| last5 = Maury<br />
| first5 = A.<br />
| last6 = Lellouch<br />
| first6 = E.<br />
| last7 = Gil-Hutton<br />
| first7 = R.<br />
| last8 = Braga-Ribas<br />
| first8 = F.<br />
| last9 = Colas<br />
| first9 = F.<br />
| displayauthors=8<br />
| pages = 137<br />
}}<br />
</ref> Many small objects in the outer solar system<ref name="tnoalbedo">{{cite web<br />
|date=17 September 2008<br />
|title=TNO/Centaur diameters and albedos<br />
|publisher=Johnston's Archive<br />
|author=Wm. Robert Johnston<br />
|url=http://www.johnstonsarchive.net/astro/tnodiam.html<br />
|accessdate=2008-10-17| archiveurl= http://web.archive.org/web/20081022223827/http://www.johnstonsarchive.net/astro/tnodiam.html| archivedate= 22 October 2008<!--Added by DASHBot-->}}</ref> and [[asteroid belt]] have low albedos down to about 0.05.<ref name="astalbedo">{{cite web<br />
|date=28 June 2003<br />
|title=Asteroid albedos: graphs of data<br />
|publisher=Johnston's Archive<br />
|author=Wm. Robert Johnston<br />
|url=http://www.johnstonsarchive.net/astro/astalbedo.html<br />
|accessdate=2008-06-16| archiveurl= http://web.archive.org/web/20080517100307/http://www.johnstonsarchive.net/astro/astalbedo.html| archivedate= 17 May 2008<!--Added by DASHBot-->}}</ref> A typical [[comet nucleus]] has an albedo of 0.04.<ref name="dark">{{cite web<br />
|date=29 November 2001<br />
|title=Comet Borrelly Puzzle: Darkest Object in the Solar System<br />
|publisher=Space.com<br />
|author=Robert Roy Britt<br />
|url=http://www.space.com/scienceastronomy/solarsystem/borrelly_dark_011129.html<br />
|accessdate=2012-09-01| archiveurl= http://web.archive.org/web/20090122074028/http://www.space.com/scienceastronomy/solarsystem/borrelly_dark_011129.html| archivedate= 22 January 2009}}</ref> Such a dark surface is thought to be indicative of a primitive and heavily [[space weathering|space weathered]] surface containing some [[organic compound]]s.<br />
<br />
The overall albedo of the [[Moon]] is around 0.12, but it is strongly directional and non-Lambertian, displaying also a strong [[opposition effect]].<ref name="medkeff" /> While such reflectance properties are different from those of any terrestrial terrains, they are typical of the [[regolith]] surfaces of airless solar system bodies.<br />
<br />
Two common albedos that are used in astronomy are the (V-band) [[geometric albedo]] (measuring brightness when illumination comes from directly behind the observer) and the [[Bond albedo]] (measuring total proportion of electromagnetic energy reflected). Their values can differ significantly, which is a common source of confusion.<br />
<br />
In detailed studies, the directional reflectance properties of astronomical bodies are often expressed in terms of the five [[Hapke parameters]] which semi-empirically describe the variation of albedo with [[phase angle (astronomy)|phase angle]], including a characterization of the opposition effect of [[regolith]] surfaces.<br />
<br />
The correlation between astronomical (geometric) albedo, [[Absolute magnitude#Absolute magnitude for planets (H)|absolute magnitude]] and diameter is:<ref name="bruton">{{cite web<br />
|title=Conversion of Absolute Magnitude to Diameter for Minor Planets<br />
|publisher=Department of Physics & Astronomy (Stephen F. Austin State University)<br />
|author=Dan Bruton<br />
|url=http://www.physics.sfasu.edu/astro/asteroids/sizemagnitude.html<br />
|accessdate=2008-10-07| archiveurl= http://web.archive.org/web/20081210190134/http://www.physics.sfasu.edu/astro/asteroids/sizemagnitude.html| archivedate= 10 December 2008<!--Added by DASHBot-->}}</ref><br />
<math>A =\left ( \frac{1329\times10^{-H/5}}{D} \right ) ^2</math>,<br />
<br />
where <math>A</math> is the astronomical albedo, <math>D</math> is the diameter in kilometers, and <math>H</math> is the absolute magnitude.<br />
<br />
==Examples of terrestrial albedo effects==<br />
<br />
===Illumination===<br />
Although the albedo–temperature effect is best known in colder, whiter regions on Earth, the maximum albedo is actually found in the tropics where year-round illumination is greater. The maximum is additionally in the northern hemisphere, varying between three and twelve degrees north.<ref name=Winston>{{cite journal| first=Jay |last=Winston |title=The Annual Course of Zonal Mean Albedo as Derived From ESSA 3 and 5 Digitized Picture Data |journal=Monthly Weather Review |volume=99(11) |pages=818–827| bibcode=1971MWRv...99..818W| year=1971| doi=10.1175/1520-0493(1971)099<0818:TACOZM>2.3.CO;2| issue=11}}</ref> The minima are found in the subtropical regions of the northern and southern hemispheres, beyond which albedo increases without respect to illumination.<ref name=Winston/><br />
<br />
===Insolation effects ===<br />
The intensity of albedo temperature effects depend on the amount of albedo and the level of local [[insolation]]; high albedo areas in the [[arctic]] and [[antarctic]] regions are cold due to low insolation, where areas such as the [[Sahara Desert]], which also have a relatively high albedo, will be hotter due to high insolation. [[Tropical]] and [[sub-tropical]] [[rain forest]] areas have low albedo, and are much hotter than their [[temperate forest]] counterparts, which have lower insolation. Because insolation plays such a big role in the heating and cooling effects of albedo, high insolation areas like the tropics will tend to show a more pronounced fluctuation in local temperature when local albedo changes. {{citation needed|date=November 2013}}<br />
<br />
===Climate and weather===<br />
Albedo affects [[climate]] and drives [[weather]]. All weather is a result of the uneven heating of the Earth caused by different areas of the planet having different albedos. Essentially, for the driving of weather, there are two types of albedo regions on Earth: Land and ocean. Land and ocean regions produce the four basic different types of [[air masses]], depending on latitude and therefore [[insolation]]: Warm and dry, which form over tropical and sub-tropical land masses; warm and wet, which form over tropical and sub-tropical oceans; cold and dry which form over temperate, polar and sub-polar land masses; and cold and wet, which form over temperate, polar and sub-polar oceans. Different temperatures between the air masses result in different air pressures, and the masses develop into [[pressure systems]]. High pressure systems flow toward lower pressure, driving weather from north to south in the northern hemisphere, and south to north in the lower; however due to the spinning of the Earth, the [[Coriolis effect]] further complicates flow and creates several weather/climate bands and the [[jet streams]].<br />
<br />
===Albedo–temperature feedback===<br />
When an area's albedo changes due to snowfall, a snow–temperature [[feedback]] results. A layer of snowfall increases local albedo, reflecting away sunlight, leading to local cooling. In principle, if no outside temperature change affects this area (e.g. a warm [[air mass]]), the lowered albedo and lower temperature would maintain the current snow and invite further snowfall, deepening the snow–temperature feedback. However, because local [[weather]] is dynamic due to the change of [[seasons]], eventually warm air masses and a more direct angle of sunlight (higher [[insolation]]) cause melting. When the melted area reveals surfaces with lower albedo, such as grass or soil, the effect is reversed: the darkening surface lowers albedo, increasing local temperatures, which induces more melting and thus increasing the albedo further, resulting in still more heating.<br />
<br />
===Small-scale effects===<br />
Albedo works on a smaller scale, too. In sunlight, dark clothes absorb more heat and light-coloured clothes reflect it better, thus allowing some control over body temperature by exploiting the albedo effect of the colour of external clothing.<ref name="ranknfile-ue">{{cite web|url=http://www.ranknfile-ue.org/h&s0897.html |title=Health and Safety: Be Cool! (August 1997) |publisher=Ranknfile-ue.org |date= |accessdate=2011-08-19}}</ref><br />
<br />
===Solar photovoltaic effects===<br />
Albedo can effect the [[electrical energy]] output of solar [[photovoltaic system|photovoltaic device]]s (PV). For example, the effects of a spectrally responsive albedo are illustrated by the differences between the spectrally weighted albedo of solar PV technology based on hydrogenated amorphous silicon (a-Si:H) and crystalline silicon (c-Si)-based compared to traditional spectral-integrated albedo predictions. Research showed impacts of over 10%.<ref>Rob W. Andrews and Joshua M. Pearce, [http://dx.doi.org/10.1016/j.solener.2013.01.030 The effect of spectral albedo on amorphous silicon and crystalline silicon solar photovoltaic device performance], ''Solar Energy'', '''91''',233–241 (2013). DOI:10.1016/j.solener.2013.01.030 [http://www.academia.edu/3081684/The_effect_of_spectral_albedo_on_amorphous_silicon_and_crystalline_silicon_solar_photovoltaic_device_performance open access]</ref> More recently, the analysis was extended to the effects of spectral bias due to the specular reflectivity of 22 commonly occurring surface materials (both human-made and natural) and analyzes the albedo effects on the performance of seven PV materials covering three common PV system topologies: industrial (solar farms), commercial flat rooftops and residential pitched-roof applications.<ref>M.P. Brennan, A.L. Abramase, R.W. Andrews, [[J. M. Pearce]], [http://dx.doi.org/10.1016/j.solmat.2014.01.046 Effects of spectral albedo on solar photovoltaic devices], ''Solar Energy Materials and Solar Cells'', 124, pp. 111-116,(2014). DOI: http://dx.doi.org/10.1016/j.solmat.2014.01.046.</ref><br />
<br />
===Trees===<br />
Because forests are generally attributed a low albedo, (as the majority of the ultraviolet and visible spectrum is absorbed through [[photosynthesis]]), it has been erroneously assumed that removing forests would lead to cooling on the grounds of increased albedo. Through the [[evapotranspiration]] of water, trees discharge excess heat from the forest canopy. This water vapour rises resulting in [[cloud cover]] which also has a high albedo, thereby further increasing the net global cooling effect attributable to forests.{{Citation needed|date=July 2013}}<br />
<br />
In seasonally snow-covered zones, winter albedos of treeless areas are 10% to 50% higher than nearby forested areas because snow does not cover the trees as readily. [[Deciduous trees]] have an albedo value of about 0.15 to 0.18 whereas [[coniferous trees]] have a value of about 0.09 to 0.15.<ref name="mmutrees" /><br />
<br />
Studies by the [[Hadley Centre]] have investigated the relative (generally warming) effect of albedo change and (cooling) effect of [[carbon sequestration]] on planting forests. They found that new forests in tropical and midlatitude areas tended to cool; new forests in high latitudes (e.g. Siberia) were neutral or perhaps warming.<ref name="Betts" /><br />
<br />
===Snow===<br />
Snow albedos can be as high as 0.9; this, however, is for the ideal example: fresh deep snow over a featureless landscape. Over [[Antarctica]] they average a little more than 0.8. If a marginally snow-covered area warms, snow tends to melt, lowering the albedo, and hence leading to more snowmelt (the ice-albedo [[positive feedback]]). [[Cryoconite]], powdery windblown [[dust]] containing soot, sometimes reduces albedo on glaciers and ice sheets.<ref name = "Nat. Geo">[http://ngm.nationalgeographic.com/2010/06/melt-zone/jenkins-text/3 "Changing Greenland - Melt Zone"] page 3, of 4, article by Mark Jenkins in ''[[National Geographic (magazine)|National Geographic]]'' June 2010, accessed 8 July 2010</ref><br />
<br />
===Water===<br />
Water reflects light very differently from typical terrestrial materials. The reflectivity of a water surface is calculated using the [[Fresnel equations]] (see graph).<br />
[[File:water reflectivity.jpg|thumb|right|250px|Reflectivity of smooth water at 20&nbsp;°C (refractive index=1.333)]]<br />
At the scale of the wavelength of light even wavy water is always smooth so the light is reflected in a locally [[specular reflection|specular manner]] (not [[Diffuse reflection|diffusely]]). The glint of light off water is a commonplace effect of this. At small [[angle of incidence|angles of incident]] light, [[waviness]] results in reduced reflectivity because of the steepness of the reflectivity-vs.-incident-angle curve and a locally increased average incident angle.<ref name="Fresnel" /><br />
<br />
Although the reflectivity of water is very low at low and medium angles of incident light, it increases tremendously at high angles of incident light such as occur on the illuminated side of the Earth near the [[terminator (solar)|terminator]] (early morning, late afternoon and near the poles). However, as mentioned above, waviness causes an appreciable reduction. Since the light specularly reflected from water does not usually reach the viewer, water is usually considered to have a very low albedo in spite of its high reflectivity at high angles of incident light.<br />
<br />
Note that white caps on waves look white (and have high albedo) because the water is foamed up, so there are many superimposed bubble surfaces which reflect, adding up their reflectivities. Fresh ‘black’ ice exhibits Fresnel reflection.<br />
<br />
===Clouds===<br />
[[Cloud albedo]] has substantial influence over atmospheric temperatures. Different types of clouds exhibit different reflectivity, theoretically ranging in albedo from a minimum of near 0 to a maximum approaching 0.8. "On any given day, about half of Earth is covered by clouds, which reflect more sunlight than land and water. Clouds keep Earth cool by reflecting sunlight, but they can also serve as blankets to trap warmth."<ref name="livescience">{{cite web|url=http://www.livescience.com/environment/060124_earth_albedo.html |title=Baffled Scientists Say Less Sunlight Reaching Earth |publisher=LiveScience |date=24 January 2006 |accessdate=2011-08-19}}</ref><br />
<br />
Albedo and climate in some areas are affected by artificial clouds, such as those created by the [[contrail]]s of heavy commercial airliner traffic.<ref name="uww" /> A study following the burning of the Kuwaiti oil fields during Iraqi occupation showed that temperatures under the burning oil fires were as much as 10&nbsp;°C colder than temperatures several miles away under clear skies.<ref name="harvard">{{cite journal |title=The Kuwait oil fires as seen by Landsat |publisher=Adsabs.harvard.edu |date=30 May 1991|bibcode=1992JGR....9714565C |author1=Cahalan |first1=Robert F. |volume=97 |pages=14565 |journal=Journal of Geophysical Research |doi=10.1029/92JD00799}}</ref><br />
<br />
===Aerosol effects===<br />
[[Aerosols]] (very fine particles/droplets in the atmosphere) have both direct and indirect effects on the Earth’s radiative balance. The direct (albedo) effect is generally to cool the planet; the indirect effect (the particles act as [[cloud condensation nuclei]] and thereby change cloud properties) is less certain.<ref name="girda">{{cite web|url=http://www.grida.no/climate/ipcc_tar/wg1/231.htm#671 |title=Climate Change 2001: The Scientific Basis |publisher=Grida.no |date= |accessdate=2011-08-19| archiveurl= http://web.archive.org/web/20110629175429/http://www.grida.no/climate/ipcc_tar/wg1/231.htm| archivedate= 29 June 2011<!--Added by DASHBot-->}}</ref> As per <ref name="DOMINICK" /> the effects are:<br />
<blockquote><br />
<!-- Aerosol radiative forcing. --><br />
* ''Aerosol direct effect.'' Aerosols directly scatter and absorb radiation. The scattering of radiation causes atmospheric cooling, whereas absorption can cause atmospheric warming.<br />
* ''Aerosol indirect effect.'' Aerosols modify the properties of clouds through a subset of the aerosol population called [[cloud condensation nuclei]]. Increased nuclei concentrations lead to increased cloud droplet number concentrations, which in turn leads to increased cloud albedo, increased light scattering and radiative cooling (''first indirect effect''), but also leads to reduced precipitation efficiency and increased lifetime of the cloud (''second indirect effect'').<br />
</blockquote><br />
<br />
===Black carbon===<br />
Another albedo-related effect on the climate is from [[black carbon]] particles. The size of this effect is difficult to quantify: the [[Intergovernmental Panel on Climate Change]] estimates that the global mean radiative forcing for black carbon aerosols from fossil fuels is +0.2 W m<sup>−2</sup>, with a range +0.1 to +0.4 W m<sup>−2</sup>.<ref name="girda 1">{{cite web|url=http://www.grida.no/climate/ipcc_tar/wg1/233.htm |title=Climate Change 2001: The Scientific Basis |publisher=Grida.no |date= |accessdate=2011-08-19| archiveurl= http://web.archive.org/web/20110629180154/http://www.grida.no/climate/ipcc_tar/wg1/233.htm| archivedate= 29 June 2011<!--Added by DASHBot-->}}</ref> Black carbon is a bigger cause of the melting of the polar ice cap in the Arctic than carbon dioxide due to its effect on the albedo.<ref>James Hansen & Larissa Nazarenko, ''Soot Climate Forcing Via Snow and Ice Albedos'', 101 Proc. of the Nat'l. Acad. of Sci. 423 (13 January 2004) (“The efficacy of this forcing is »2 (i.e. for a given forcing it is twice as effective as CO<sub>2</sub> in altering global surface air temperature)”); ''compare'' Zender Testimony, ''supra'' note 7, at 4 (figure 3); See J. Hansen & L. Nazarenko, ''supra'' note 18, at 426. (“The efficacy for changes of Arctic sea ice albedo is >3. In additional runs not shown here, we found that the efficacy of albedo changes in Antarctica is also >3.”); ''See also'' Flanner, M.G., C.S. Zender, J.T. Randerson, and P.J. Rasch, ''Present-day climate forcing and response from black carbon in snow'', 112 J. GEOPHYS. RES. D11202 (2007) (“The forcing is maximum coincidentally with snowmelt onset, triggering strong snow-albedo feedback in local springtime. Consequently, the “efficacy” of black carbon/snow forcing is more than three times greater than forcing by CO<sub>2</sub>.”).</ref><br />
<br />
===Human activities===<br />
Human activities (e.g. deforestation, farming, and urbanization) change the albedo of various areas around the globe. However, quantification of this effect on the global scale is difficult.{{citation needed|date=November 2013}}<br />
<br />
==Other types of albedo==<br />
[[Single-scattering albedo]] is used to define scattering of electromagnetic waves on small particles. It depends on properties of the material ([[refractive index]]); the size of the particle or particles; and the wavelength of the incoming radiation.<br />
<br />
==See also==<br />
{{div col|colwidth=30em}}<br />
* [[Cool roof]]<br />
* [[Solar radiation management]]<br />
* [[Global dimming]]<br />
* [[Irradiance]]<br />
* [[Polar see-saw]]<br />
* [[Daisyworld]]<br />
{{div col end}}<br />
<br />
==References==<br />
{{Reflist|30em|refs=<br />
<ref name="Goode">{{Cite journal |last=Goode |first=P. R. |authorlink= |author2=''et al.'' |year=2001 |month= |title=Earthshine Observations of the Earth's Reflectance |journal=[[Geophysical Research Letters]] |volume=28 |issue=9 |pages=1671–1674 |id= |url=http://www.agu.org/journals/ABS/2001/2000GL012580.shtml |accessdate= |quote=|doi=10.1029/2000GL012580 |bibcode = 2001GeoRL..28.1671G }}</ref><br />
<br />
<ref name="NASA">{{cite web|url=http://modis.gsfc.nasa.gov/data/atbd/atbd_mod09.pdf|title=MODIS BRDF/Albedo Product: Algorithm Theoretical Basis Document, Version 5.0|accessdate=2009-06-02| archiveurl= http://web.archive.org/web/20090601063932/http://modis.gsfc.nasa.gov/data/atbd/atbd_mod09.pdf| archivedate= 1 June 2009<!--Added by DASHBot-->}}</ref><br />
<br />
<ref name="washington">{{cite web|url=http://www.atmos.washington.edu/~sgw/PAPERS/2002_Snowball.pdf|title=Snowball Earth: Ice thickness on the tropical ocean|accessdate=2009-09-20}}</ref><br />
<br />
<ref name="clim-past">{{cite web|url=http://www.clim-past.net/2/31/2006/cp-2-31-2006.pdf|title=Effect of land albedo, CO2, orography, and oceanic heat transport on extreme climates|accessdate=2009-09-20}}</ref><br />
<br />
<ref name="Smith Robin">{{cite web|url=http://www.mpimet.mpg.de/fileadmin/staff/smithrobin/IC_JClim-final.pdf|title=Global climate and ocean circulation on an aquaplanet ocean-atmosphere general circulation model|accessdate=2009-09-20| archiveurl= http://web.archive.org/web/20090920212836/http://www.mpimet.mpg.de/fileadmin/staff/smithrobin/IC_JClim-final.pdf| archivedate= 20 September 2009<!--Added by DASHBot-->}}</ref><br />
<ref name="medkeff">{{cite web<br />
| url = http://jeff.medkeff.com/astro/lunar/obs_tech/albedo.htm<br />
| title = Lunar Albedo<br />
| first = Jeff<br />
| last = Medkeff<br />
| authorlink = Jeffrey S. Medkeff<br />
| year = 2002<br />
| archiveurl = http://web.archive.org/web/20080523151225/http://jeff.medkeff.com/astro/lunar/obs_tech/albedo.htm<br />
| archivedate = 23 May 2008<br />
| accessdate = 5 July 2010<br />
| postscript =<br />
}}<br />
</ref><br />
<br />
<!-- <ref name="Dickinson">Dickinson, R. E., and P. J. Kennedy, 1992: ''Impacts on regional climate of Amazon deforestation''. Geophys. Res. Lett., '''19''', 1947–1950.</ref> --><br />
<br />
<!-- <ref name="mit">[http://web.mit.edu/12.000/www/m2006/final/characterization/abiotic_water.html http://web.mit.edu/12.000/www/m2006/final/characterization/abiotic_water.html] Project Amazonia: Characterization - Abiotic - Water</ref> --><br />
<br />
<ref name="mmutrees">{{cite web | url=http://www.ace.mmu.ac.uk/Resources/gcc/1-3-3.html | title=The Climate System | publisher=Manchester Metropolitan University | accessdate=2007-11-11| archiveurl= http://web.archive.org/web/20071121192518/http://www.ace.mmu.ac.uk/resources/gcc/1-3-3.html| archivedate= 21 November 2007<!--Added by DASHBot-->}}</ref><br />
<br />
<ref name="Betts">{{cite journal | doi = 10.1038/35041545 | year = 2000 | last1 = Betts | first1 = Richard A. | journal = Nature | volume = 408 | issue = 6809 | pages = 187–190 | pmid = 11089969 | title = Offset of the potential carbon sink from boreal forestation by decreases in surface albedo }}</ref><br />
<br />
<ref name="Fresnel">http://vih.freeshell.org/pp/01-ONW-St.Petersburg/Fresnel.pdf</ref><br />
<br />
<ref name="uww">http://facstaff.uww.edu/travisd/pdf/jetcontrailsrecentresearch.pdf</ref><br />
<br />
<ref name="DOMINICK">{{cite journal | doi = 10.1098/rsta.2008.0201 | title = Boreal forests, aerosols and the impacts on clouds and climate | year = 2008 | last1 = Spracklen | first1 = D. V | last2 = Bonn | first2 = B. | last3 = Carslaw | first3 = K. S | journal = Philosophical Transactions of the Royal Society A | volume = 366 | issue = 1885 | pages = 4613–4626 |url=http://homepages.see.leeds.ac.uk/~eardvs/papers/spracklen08c.pdf | format = PDF|bibcode = 2008RSPTA.366.4613S | pmid=18826917}}</ref><br />
<br />
<ref name="BlueskyAlbedo">{{Cite journal |last=Roman |first=M. O. |authorlink= |coauthors=C.B. Schaaf, P. Lewis, F. Gao, G.P. Anderson, J.L. Privette, A.H. Strahler, C.E. Woodcock, and M. Barnsley |year=2010 |month= |title=Assessing the Coupling between Surface Albedo derived from MODIS and the Fraction of Diffuse Skylight over Spatially-Characterized Landscapes |journal=Remote Sensing of Environment |volume=114 |pages=738–760 |id= |doi=10.1016/j.rse.2009.11.014 |accessdate= |quote= |issue=4 }}</ref><br />
<br />
}}<br />
<br />
==External links==<br />
{{wiktionary}}<br />
* [http://www.albedo-project.org/ Official Website of Albedo Project]<br />
* [http://www-c4.ucsd.edu/gap/ Global Albedo Project (Center for Clouds, Chemistry, and Climate)]<br />
* [http://www.eoearth.org/article/Albedo Albedo - Encyclopedia of Earth]<br />
* [http://www-modis.bu.edu/brdf/product.html NASA MODIS BRDF/albedo product site]<br />
*[http://www.eumetsat.int/Home/Main/Access_to_Data/Meteosat_Meteorological_Products/Product_List/SP_1125489019643?l=en Surface albedo derived from Meteosat observations]<br />
* [http://jeff.medkeff.com/astro/lunar/obs_tech/albedo.htm A discussion of Lunar albedos]<br />
* [http://www.tvu.com/metalreflectivityLR.jpg reflectivity of metals (chart)]<br />
<br />
{{Global warming}}<br />
<br />
[[Category:Climate forcing]]<br />
[[Category:Climatology]]<br />
[[Category:Electromagnetic radiation]]<br />
[[Category:Radiometry]]<br />
[[Category:Scattering, absorption and radiative transfer (optics)]]<br />
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<div>{{refimprove|date=July 2013}}<br />
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{{Use dmy dates|date=June 2013}}<br />
In [[mathematics]] and [[statistics]], the '''arithmetic mean''' ({{IPAc-en|pron|ˌ|æ|r|ɪ|θ|ˈ|m|ɛ|t|ɪ|k|_|ˈ|m|iː|n}}), or simply the [[mean]] or '''average''' when the context is clear, is the sum of a collection of numbers divided by the number of numbers in the collection.<ref>{{cite book | last = Jacobs | first = Harold R. | title = Mathematics: A Human Endeavor | edition = Third | year = 1994 | publisher = [[W. H. Freeman]] | page = 547 | isbn = 0-7167-2426-X}}</ref> The collection is often a set of results of an [[experiment (probability theory)|experiment]], or a set of results from a [[Survey methodology|survey]]. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics because it helps distinguish it from other [[average|mean]]s, such as the [[geometric mean]] and the [[harmonic mean]].<br />
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In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, [[sociology]], and history, and it is used in almost every academic field to some extent. For example, per capita income is the arithmetic average income of a nation's population.<br />
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While the arithmetic mean is often used to report [[central tendency|central tendencies]], it is not a [[robust statistic]], meaning that it is greatly influenced by [[outlier]]s (values that are very much larger or smaller than most of the values). Notably, for [[skewed distribution]]s, such as the [[distribution of income]] for which a few people's incomes are substantially greater than most people's, the arithmetic mean may not accord with one's notion of "middle", and robust statistics, such as the [[median]], may be a better description of central tendency.<br />
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In a more obscure usage, any sequence of values that form an [[Arithmetic progression|arithmetic sequence]] between two numbers ''x'' and ''y'' can be called "arithmetic means between ''x'' and ''y''."<ref>{{cite book | last = Foerster | first = Paul A. | title = Algebra and Trigonometry: Functions and Applications, Teacher's Edition | edition = Classics | year = 2006 | publisher = [[Prentice Hall]] | location = Upper Saddle River, NJ | page = 573 | url = http://www.amazon.com/Algebra-Trigonometry-Functions-Applications-Prentice/dp/0131657100 | isbn = 0-13-165711-9}}</ref><br />
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==Definition==<br />
Suppose we have a data set containing the values <math>a_1,\ldots,a_n.</math> The arithmetic mean <math>A</math> is defined by the formula<br />
:<math>A=\frac{1}{n}\sum_{i=1}^{n} a_i</math>.<br />
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If the data set is a [[statistical population]] (i.e., consists of every possible observation and not just a subset of them), then the mean of that population is called the '''population mean'''. If the data set is a [[sampling (statistics)|statistical sample]] (a subset of the population), we call the statistic resulting from this calculation a '''sample mean'''.<br />
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The arithmetic mean of a variable is often denoted by a bar, for example as in <math>\bar{x}</math> (read "x bar"), which is the mean of the <math>n</math> values <math>x_1,x_2,\ldots,x_n</math>.<ref name="JM">{{cite book| last = Medhi| first = Jyotiprasad| title = Statistical Methods: An Introductory Text| url = http://books.google.com/?id=bRUwgf_q5RsC| year = 1992| publisher = New Age International| isbn = 9788122404197| pages = 53–58 }}</ref><br />
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==Motivating properties==<br />
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The arithmetic mean has several properties that make it useful, especially as a measure of central tendency. These include:<br />
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* If numbers <math>x_1,\dotsc,x_n</math> have mean <math>\bar{x}</math>, then <math>(x_1-\bar{x}) + \dotsb + (x_n-\bar{x}) = 0</math>. Since <math>x_i-\bar{x}</math> is the distance from a given number to the mean, one way to interpret this property is as saying that the numbers to the left of the mean are balanced by the numbers to the right of the mean. The mean is the only single number for which the [[errors and residuals in statistics|residuals]] (deviations from the estimate) sum to zero.<br />
* If it is required to use a single number as a "typical" value for a set of known numbers <math>x_1,\dotsc,x_n</math>, then the arithmetic mean of the numbers does this best, in the sense of minimizing the sum of squared deviations from the typical value: the sum of <math>(x_i-\bar{x})^2</math>. (It follows that the sample mean is also the best single predictor in the sense of having the lowest [[root mean squared error]].)<ref name="JM"/> If the arithmetic mean of a population of numbers is desired, then the estimate of it that is [[unbiased estimate|unbiased]] is the arithmetic mean of a sample drawn from the population.<br />
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==Contrast with median==<br />
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The arithmetic mean may be contrasted with the median. The median is defined such that half the values are larger than, and half are smaller than, the median. If elements in the sample data [[arithmetic progression|increase arithmetically]], when placed in some order, then the median and arithmetic average are equal. For example, consider the data sample <math>{1,2,3,4}</math>. The average is <math>2.5</math>, as is the median. However, when we consider a sample that cannot be arranged so as to increase arithmetically, such as <math>{1,2,4,8,16}</math>, the median and arithmetic average can differ significantly. In this case, the arithmetic average is 6.2 and the median is 4. In general, the average value can vary significantly from most values in the sample, and can be larger or smaller than most of them.<br />
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There are applications of this phenomenon in many fields. For example, since the 1980s, the median income in the United States has increased more slowly than the arithmetic average of income.<br />
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==Generalizations==<br />
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===Weighted average===<br />
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A [[weighted average]], or weighted mean, is an average in which some data points count more strongly than others, in that they are given more weight in the calculation. For example, the arithmetic mean of <math>3</math> and <math>5</math> is <math>\frac{(3+5)}{2} = 4</math>, or equivalently <math>\left( \frac{1}{2} \cdot 3\right) + \left( \frac{1}{2} \cdot 5\right) = 4</math>. In contrast, a ''weighted'' mean in which the first number receives twice as much weight as the second (perhaps because it is assumed to appear twice as often in the general population from which these numbers were sampled) would be calculated as <math>\left( \frac{2}{3} \cdot 3\right) + \left(\frac{1}{3} \cdot 5\right) = \frac{11}{3}</math>. Here the weights, which necessarily sum to the value one, are <math>(2/3)</math> and <math>(1/3)</math>, the former being twice the latter. Note that the arithmetic mean (sometimes called the "unweighted average" or "equally weighted average") can be interpreted as a special case of a weighted average in which all the weights are equal to each other (equal to <math>\frac{1}{2}</math> in the above example, and equal to <math>\frac{1}{n}</math> in a situation with <math>n</math> numbers being averaged).<br />
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===Continuous probability distributions===<br />
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[[File:Comparison mean median mode.svg|thumb|300px|Comparison of mean, [[median]] and [[mode (statistics)|mode]] of two [[log-normal distribution]]s with different [[skewness]].]]<br />
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When a population of numbers, and any sample of data from it, could take on any of a continuous range of numbers, instead of for example just integers, then the [[probability]] of a number falling into one range of possible values could differ from the probability of falling into a different range of possible values, even if the lengths of both ranges are the same. In such a case, the set of probabilities can be described using a [[continuous probability distribution]]. The analog of a weighted average in this context, in which there are an infinitude of possibilities for the precise value of the variable, is called the ''mean of the probability distribution''. The most widely encountered probability distribution is called the [[normal distribution]]; it has the property that all measures of its central tendency, including not just the mean but also the aforementioned median and the [[Mode (statistics)|mode]], are equal to each other. This property does not hold however, in the cases of a great many probability distributions, such as the [[lognormal distribution]] illustrated here.<br />
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==Angles==<br />
{{Main|Mean of circular quantities}}<br />
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Particular care must be taken when using cyclic data, such as phases or [[angle]]s. Naïvely taking the arithmetic mean of 1° and 359° yields a result of 180°.<br />
This is incorrect for two reasons:<br />
* Firstly, angle measurements are only defined up to an additive constant of [[degree (angle)|360°]] (or 2π, if measuring in [[radian]]s). Thus one could as easily call these 1° and −1°, or 361° and 719°, each of which gives a different average.<br />
* Secondly, in this situation, 0° (equivalently, 360°) is geometrically a better ''average'' value: there is lower [[statistical dispersion|dispersion]] about it (the points are both 1° from it, and 179° from 180°, the putative average).<br />
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In general application, such an oversight will lead to the average value artificially moving towards the middle of the numerical range. A solution to this problem is to use the optimization formulation ([[viz.]], define the mean as the central point: the point about which one has the lowest dispersion), and redefine the difference as a modular distance (i.e., the distance on the circle: so the modular distance between 1° and 359° is 2°, not 358°).<br />
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==See also==<br />
* [[Average]]<br />
* [[Fréchet mean]]<br />
* [[Generalized mean]]<br />
* [[Geometric mean]]<br />
* [[Mode (statistics)|Mode]]<br />
* [[Sample mean and covariance]]<br />
* [[Summary statistics]]<br />
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==References==<br />
{{reflist}}<br />
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==Further reading==<br />
* {{cite book| last = Huff| first = Darrell| title = How to Lie with Statistics| year = 1993| publisher = W. W. Norton| isbn = 978-0-393-31072-6 }}<br />
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==External links==<br />
* [http://www.sengpielaudio.com/calculator-geommean.htm Calculations and comparisons between arithmetic and geometric mean of two numbers]<br />
* {{MathWorld | urlname= ArithmeticMean | title= Arithmetic Mean}}<br />
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{{Statistics|descriptive}}<br />
{{Portal bar|Statistics}}<br />
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{{DEFAULTSORT:Arithmetic Mean}}<br />
[[Category:Means]]<br />
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[[it:Media (statistica)#Media aritmetica]]</div>Adminhttps://en.formulasearchengine.com/index.php?title=Axiom_of_choice&diff=218480Axiom of choice2014-07-31T13:57:47Z<p>Admin: Reverted edits by MediaWiki spam cleanup (talk) to last revision by BG19bot</p>
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<div>{{about|the mathematical concept|the band named after it|Axiom of Choice (band)}}<br />
{{Use dmy dates|date=June 2013}}<br />
[[File:Axiom of choice.svg|thumb|250px|(S<sub>''i''</sub>) is a [[indexed family|family]] of sets indexed over the [[real number]]s '''R'''; that is, there is a set S<sub>''i''</sub> for each real number ''i'', with a small sample shown above. Each set contains a nonzero, and possibly infinite, number of elements. The axiom of choice allows us to arbitrarily select a single element from each set, forming a corresponding family of elements (''x''<sub>''i''</sub>) also indexed over the real numbers, with ''x''<sub>''i''</sub> drawn from S<sub>''i''</sub>. In general the collections may be indexed over any set <span style="font-family:serif;">''I''</span>, not just '''R'''.]]<br />
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In [[mathematics]], the '''axiom of choice''', or '''AC''', is an [[axiom]] of [[set theory]] equivalent to the statement that ''the [[cartesian product#Infinite products|cartesian product]] of a collection of non-empty sets is non-empty''. It states that for every indexed [[Family of sets|family]] <math>(S_i)_{i \in I}</math> of [[nonempty]] sets there exists an indexed family <math>(x_i)_{i \in I}</math> of elements such that <math>x_i \in S_i</math> for every <math>i \in I</math>. The axiom of choice was formulated in 1904 by [[Ernst Zermelo]] in order to formalize his proof of the [[well-ordering theorem]].<ref name="Zermelo, 1904">{{cite journal| first=Ernst| last=Zermelo| year=1904| url=http://gdz.sub.uni-goettingen.de/no_cache/en/dms/load/img/?IDDOC=28526 |format=reprint|title=Beweis, dass jede Menge wohlgeordnet werden kann| journal=Mathematische Annalen| volume=59| issue=4| pages=514–16|doi=10.1007/BF01445300}}</ref><br />
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Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin. In many cases such a selection can be made without invoking the axiom of choice; this is in particular the case if the number of bins is finite, or if a selection rule is available: a distinguishing property that happens to hold for exactly one object in each bin. To give an informal example, for any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection, but for an infinite collection of pairs of socks (assumed to have no distinguishing features), such a selection can be obtained only by invoking the axiom of choice.<br />
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Although originally controversial, the axiom of choice is now used without reservation by most mathematicians,<ref>Jech, 1977, p. 348''ff''; Martin-Löf 2008, p. 210.</ref> and it is included in Zermelo–Fraenkel set theory with the axiom of choice ([[ZFC]]), the standard form of [[axiomatic set theory]]. One motivation for this use is that a number of generally accepted mathematical results, such as [[Tychonoff's theorem]], require the axiom of choice for their proofs. Contemporary set theorists also study axioms that are not compatible with the axiom of choice, such as the [[axiom of determinacy]]. The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced.<br />
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==Statement==<br />
A [[choice function]] is a function ''f'', defined on a collection ''X'' of nonempty sets, such that for every set ''s'' in ''X'', ''f''(''s'') is an element of ''s''. With this concept, the axiom can be stated:<br />
:For any set ''X'' of nonempty sets, there exists a choice function ''f'' defined on ''X''.<br />
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Formally, this may be expressed as follows:<br />
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:<math>\forall X \left[ \emptyset \notin X \implies \exists f \colon X \rarr \bigcup X \quad \forall A \in X \, ( f(A) \in A ) \right] \,.</math><br />
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Thus the negation of the axiom of choice states that there exists a set of nonempty sets which has no choice function.<br />
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Each choice function on a collection ''X'' of nonempty sets is an element of the [[Cartesian product#Infinite products|Cartesian product]] of the sets in ''X''. This is not the most general situation of a Cartesian product of a [[indexed family|family]] of sets, where a same set can occur more than once as a factor; however, one can focus on elements of such a product that select the same element every time a given set appears as factor, and such elements correspond to an element of the Cartesian product of all ''distinct'' sets in the family. The axiom of choice asserts the existence of such elements; it is therefore equivalent to:<br />
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:Given any family of nonempty sets, their Cartesian product is a nonempty set.<br />
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=== Nomenclature ZF, AC, and ZFC ===<br />
In this article and other discussions of the '''Axiom of Choice''' the following abbreviations are common:<br />
*AC &ndash; the Axiom of Choice.<br />
*ZF &ndash; [[Zermelo–Fraenkel set theory]] omitting the Axiom of Choice.<br />
*ZFC &ndash; [[Zermelo–Fraenkel set theory]], extended to include the Axiom of Choice.<br />
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===Variants===<br />
There are many other equivalent statements of the axiom of choice. These are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it.<br />
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One variation avoids the use of choice functions by, in effect, replacing each choice function with its range.<br />
:Given any set ''X'' of [[pairwise disjoint]] non-empty sets, there exists at least one set ''C'' that contains exactly one element in common with each of the sets in ''X''.<ref>Herrlich, p. 9.</ref><br />
This guarantees for any [[partition of a set|partition]] of a set ''X'' the existence of a subset ''C'' of ''X'' containing exactly one element from each part of the partition.<br />
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Another equivalent axiom only considers collections ''X'' that are essentially powersets of other sets:<br />
:For any set A, the [[power set]] of A (with the empty set removed) has a choice function.<br />
Authors who use this formulation often speak of the ''choice function on A'', but be advised that this is a slightly different notion of choice function. Its domain is the powerset of ''A'' (with the empty set removed), and so makes sense for any set ''A'', whereas with the definition used elsewhere in this article, the domain of a choice function on a ''collection of sets'' is that collection, and so only makes sense for sets of sets. With this alternate notion of choice function, the axiom of choice can be compactly stated as<br />
:Every set has a choice function.<ref>[[Patrick Suppes]], "Axiomatic Set Theory", Dover, 1972 (1960), ISBN 0-486-61630-4, p. 240</ref><br />
which is equivalent to<br />
:For any set A there is a function ''f'' such that for any non-empty subset B of ''A'', ''f''(''B'') lies in ''B''.<br />
The negation of the axiom can thus be expressed as:<br />
:There is a set ''A'' such that for all functions ''f'' (on the set of non-empty subsets of ''A''), there is a ''B'' such that ''f''(''B'') does not lie in ''B''.<br />
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=== Restriction to finite sets ===<br />
The statement of the axiom of choice does not specify whether the collection of nonempty sets is finite or infinite, and thus implies that every [[finite set|finite collection]] of nonempty sets has a choice function. However, that particular case is a theorem of [[Zermelo–Fraenkel set theory]] without the axiom of choice (ZF); it is easily proved by [[mathematical induction]].<ref>Tourlakis (2003), pp. 209–210, 215–216.</ref> In the even simpler case of a collection of ''one'' set, a choice function just corresponds to an element, so this instance of the axiom of choice says that every nonempty set has an element; this holds trivially. The axiom of choice can be seen as asserting the generalization of this property, already evident for finite collections, to arbitrary collections.<br />
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==Usage==<br />
Until the late 19th century, the axiom of choice was often used implicitly, although it had not yet been formally stated. For example, after having established that the set ''X'' contains only non-empty sets, a mathematician might have said "let ''F(s)'' be one of the members of ''s'' for all ''s'' in ''X''." In general, it is impossible to prove that ''F'' exists without the axiom of choice, but this seems to have gone unnoticed until [[Zermelo]].<br />
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Not every situation requires the axiom of choice. For finite sets ''X'', the axiom of choice follows from the other axioms of set theory. In that case it is equivalent to saying that if we have several (a finite number of) boxes, each containing at least one item, then we can choose exactly one item from each box. Clearly we can do this: We start at the first box, choose an item; go to the second box, choose an item; and so on. The number of boxes is finite, so eventually our choice procedure comes to an end. The result is an explicit choice function: a function that takes the first box to the first element we chose, the second box to the second element we chose, and so on. (A formal proof for all finite sets would use the principle of [[mathematical induction]] to prove "for every natural number ''k'', every family of ''k'' nonempty sets has a choice function.") This method cannot, however, be used to show that every countable family of nonempty sets has a choice function, as is asserted by the [[axiom of countable choice]]. If the method is applied to an infinite sequence (''X''<sub>''i''</sub> : ''i''∈ω) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no "limiting" choice function can be constructed, in general, in ZF without the axiom of choice.<br />
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==Examples==<br />
The nature of the individual nonempty sets in the collection may make it possible to avoid the axiom of choice even for certain infinite collections. For example, suppose that each member of the collection ''X'' is a nonempty subset of the natural numbers. Every such subset has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set. This gives us a definite choice of an element from each set, and makes it unnecessary to apply the axiom of choice.<br />
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The difficulty appears when there is no natural choice of elements from each set. If we cannot make explicit choices, how do we know that our set exists? For example, suppose that ''X'' is the set of all non-empty subsets of the [[real number]]s. First we might try to proceed as if ''X'' were finite. If we try to choose an element from each set, then, because ''X'' is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of ''X''. Next we might try specifying the least element from each set. But some subsets of the real numbers do not have least elements. For example, the open interval (0,1) does not have a least element: if ''x'' is in (0,1), then so is ''x''/2, and ''x''/2 is always strictly smaller than ''x''. So this attempt also fails.<br />
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Additionally, consider for instance the unit circle ''S'', and the action on ''S'' by a group ''G'' consisting of all rational rotations. Namely, these are rotations by angles which are rational multiples of&nbsp;''π''. Here ''G'' is countable while ''S'' is uncountable. Hence ''S'' breaks up into uncountably many orbits under&nbsp;''G''. Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset ''X'' of ''S'' with the property that all of its translates by G are disjoint from&nbsp;''X''. The set of those translates partitions the circle into a countable collection of disjoint sets, which are all pairwise congruent. Since ''X'' is not measurable for any rotation-invariant countably additive finite measure on ''S'', finding an algorithm to select a point in each orbit requires the axiom of choice. See [[non-measurable set#Example|non-measurable set]] for more details.<br />
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The reason that we are able to choose least elements from subsets of the natural numbers is the fact that the natural numbers are [[well-order]]ed: every nonempty subset of the natural numbers has a unique least element under the natural ordering. One might say, "Even though the usual ordering of the real numbers does not work, it may be possible to find a different ordering of the real numbers which is a well-ordering. Then our choice function can choose the least element of every set under our unusual ordering." The problem then becomes that of constructing a well-ordering, which turns out to require the axiom of choice for its existence; every set can be well-ordered if and only if the axiom of choice holds.<br />
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==Criticism and acceptance==<br />
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A proof requiring the axiom of choice may establish the existence of an object without explicitly [[definable set|defining]] the object in the language of set theory. For example, while the axiom of choice implies that there is a [[well-ordering]] of the real numbers, there are models of set theory with the axiom of choice in which no well-ordering of the reals is definable. Similarly, although a subset of the real numbers that is not [[Lebesgue measure|Lebesgue measurable]] can be proven to exist using the axiom of choice, it is [[consistent]] that no such set is definable.<br />
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The axiom of choice produces these intangibles (objects that are proven to exist, but which cannot be explicitly constructed), which may conflict with some philosophical principles. Because there is no [[Canonical form|canonical]] well-ordering of all sets, a construction that relies on a well-ordering may not produce a canonical result, even if a canonical result is desired (as is often the case in [[category theory]]). This has been used as an argument against the use of the axiom of choice.<br />
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Another argument against the axiom of choice is that it implies the existence of counterintuitive objects. One example is the [[Banach–Tarski paradox]] which says that it is possible to decompose ("carve up") the 3-dimensional solid unit ball into finitely many pieces and, using only rotations and translations, reassemble the pieces into two solid balls each with the same volume as the original. The pieces in this decomposition, constructed using the axiom of choice, are [[non-measurable set]]s.<br />
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Despite these facts, most mathematicians accept the axiom of choice as a valid principle for proving new results in mathematics. The debate is interesting enough, however, that it is considered of note when a theorem in ZFC (ZF plus AC) is [[logical equivalence|logically equivalent]] (with just the ZF axioms) to the axiom of choice, and mathematicians look for results that require the axiom of choice to be false, though this type of deduction is less common than the type which requires the axiom of choice to be true.<br />
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It is possible to prove many theorems using neither the axiom of choice nor its negation; such statements will be true in any [[model theory|model]] of [[Zermelo–Fraenkel set theory]] (ZF), regardless of the truth or falsity of the axiom of choice in that particular model. The restriction to ZF renders any claim that relies on either the axiom of choice or its negation unprovable. For example, the Banach–Tarski paradox is neither provable nor disprovable from ZF alone: it is impossible to construct the required decomposition of the unit ball in ZF, but also impossible to prove there is no such decomposition. Similarly, all the statements listed below which require choice or some weaker version thereof for their proof are unprovable in ZF, but since each is provable in ZF plus the axiom of choice, there are models of ZF in which each statement is true. Statements such as the Banach–Tarski paradox can be rephrased as conditional statements, for example, "If AC holds, then the decomposition in the Banach–Tarski paradox exists." Such conditional statements are provable in ZF when the original statements are provable from ZF and the axiom of choice.<br />
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== In constructive mathematics ==<br />
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As discussed above, in ZFC, the axiom of choice is able to provide "[[nonconstructive proof]]s" in which the existence of an object is proved although no explicit example is constructed. ZFC, however, is still formalized in classical logic. The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed. The status of the axiom of choice varies between different varieties of constructive mathematics.<br />
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In [[Martin-Löf type theory]] and higher-order [[Heyting arithmetic]], the appropriate statement of the axiom of choice is (depending on approach) included as an axiom or provable as a theorem.<ref>[[Per Martin-Löf]], ''[http://www.cs.cmu.edu/afs/cs/Web/People/crary/819-f09/Martin-Lof80.pdf Intuitionistic type theory]'', 1980. <br />
[[Anne Sjerp Troelstra]], ''Metamathematical investigation of intuitionistic arithmetic and analysis'', Springer, 1973.</ref> [[Errett Bishop]] argued that the axiom of choice was constructively acceptable, saying<br />
:"A choice function exists in constructive mathematics, because a choice is implied by the very meaning of existence."<ref>[[Errett Bishop]] and [[Douglas S. Bridges]], ''Constructive analysis'', Springer-Verlag, 1985.</ref><br />
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In [[constructive set theory]], however, [[Diaconescu's theorem]] shows that the axiom of choice implies the [[Law of excluded middle]] (unlike in Martin-Löf type theory, where it does not). Thus the axiom of choice is not generally available in constructive set theory. A cause for this difference is that the axiom of choice in type theory does not have the [[extensionality]] properties that the axiom of choice in constructive set theory does.<ref>[[Per Martin-Löf]], "100 Years of Zermelo’s Axiom of Choice: What was the Problem with It?", ''The Computer Journal'' (2006) 49 (3): 345-350. doi: 10.1093/comjnl/bxh162</ref><br />
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Some results in constructive set theory use the [[axiom of countable choice]] or the [[axiom of dependent choice]], which do not imply the law of the excluded middle in constructive set theory. Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.<ref>Fred Richman, “Constructive mathematics without choice”, in: Reuniting the Antipodes—Constructive and Nonstandard Views of the Continuum (P. Schuster et al., eds), Synthèse Library 306, 199–205, Kluwer Academic Publishers, Amsterdam, 2001.</ref><br />
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==Independence==<br />
<br />
Assuming ZF is consistent, [[Kurt Gödel]] showed that the ''negation'' of the axiom of choice is not a theorem of ZF by constructing an [[inner model]] (the [[constructible universe]]) which satisfies ZFC and thus showing that ZFC is consistent. Assuming ZF is consistent, [[Paul Cohen (mathematician)|Paul Cohen]] employed the technique of [[forcing (mathematics)|forcing]], developed for this purpose, to show that the axiom of choice itself is not a theorem of ZF by constructing a much more complex model which satisfies ZF¬C (ZF with the negation of AC added as axiom) and thus showing that ZF¬C is consistent. Together these results establish that the axiom of choice is [[Independence (mathematical logic)|logically independent]] of ZF. The assumption that ZF is consistent is harmless because adding another axiom to an already inconsistent system cannot make the situation worse. Because of independence, the decision whether to use the axiom of choice (or its negation) in a proof cannot be made by appeal to other axioms of set theory. The decision must be made on other grounds.<br />
<br />
One argument given in favor of using the axiom of choice is that it is convenient to use it because it allows one to prove some simplifying propositions that otherwise could not be proved. Many theorems which are provable using choice are of an elegant general character: every [[Ideal (ring theory)|ideal]] in a ring is contained in a [[maximal ideal]], every [[vector space]] has a [[Basis (linear algebra)|basis]], and every [[Product topology|product]] of [[compact space]]s is compact. Without the axiom of choice, these theorems may not hold for mathematical objects of large cardinality.<br />
<br />
The proof of the independence result also shows that a wide class of mathematical statements, including all statements that can be phrased in the language of [[Peano arithmetic]], are provable in ZF if and only if they are provable in ZFC.<ref>This is because arithmetical statements are [[absoluteness (mathematical logic)|absolute]] to the [[constructible universe]] ''L''. [[Shoenfield's absoluteness theorem]] gives a more general result.</ref> Statements in this class include the statement that [[P = NP]], the [[Riemann hypothesis]], and many other unsolved mathematical problems. When one attempts to solve problems in this class, it makes no difference whether ZF or ZFC is employed if the only question is the existence of a proof. It is possible, however, that there is a shorter proof of a theorem from ZFC than from ZF.<br />
<br />
The axiom of choice is not the only significant statement which is independent of ZF. For example, the [[Continuum hypothesis#The generalized continuum hypothesis|generalized continuum hypothesis]] (GCH) is not only independent of ZF, but also independent of ZFC. However, ZF plus GCH implies AC, making GCH a strictly stronger claim than AC, even though they are both independent of ZF.<br />
<br />
==Stronger axioms==<br />
The [[axiom of constructibility]] and the [[Continuum hypothesis#The generalized continuum hypothesis|generalized continuum hypothesis]] each imply the axiom of choice and so are strictly stronger than it. In class theories such as [[Von Neumann–Bernays–Gödel set theory]] and [[Morse–Kelley set theory]], there is a possible axiom called the [[axiom of global choice]] which is stronger than the axiom of choice for sets because it also applies to proper classes. And the axiom of global choice follows from the [[axiom of limitation of size]].<br />
<br />
==Equivalents==<br />
There are important statements that, assuming the axioms of [[Zermelo–Fraenkel set theory|ZF]] but neither AC nor ¬AC, are equivalent to the axiom of choice. The most important among them are [[Zorn's lemma]] and the [[well-ordering theorem]]. In fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well-ordering theorem.<br />
<br />
*[[Set theory]]<br />
**[[Well-ordering theorem]]: Every set can be well-ordered. Consequently, every [[cardinal number|cardinal]] has an [[initial ordinal]].<br />
**[[Tarski's theorem]]: For every infinite set ''A'', there is a [[bijective map]] between the sets ''A'' and ''A''×''A''.<br />
**[[Trichotomy (mathematics)|Trichotomy]]: If two sets are given, then either they have the same cardinality, or one has a smaller cardinality than the other.<br />
**The [[Cartesian product#Infinite products|Cartesian product]] of any family of nonempty sets is nonempty.<br />
**[[König's theorem (set theory)|König's theorem]]: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals. (The reason for the term "colloquially" is that the sum or product of a "sequence" of cardinals cannot be defined without some aspect of the axiom of choice.)<br />
**Every [[surjective function]] has a [[Inverse function#Left and right inverses|right inverse]].<br />
<br />
*[[Order theory]]<br />
**[[Zorn's lemma]]: Every non-empty partially ordered set in which every chain (i.e. totally ordered subset) has an upper bound contains at least one maximal element.<br />
**[[Hausdorff maximal principle]]: In any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset. The restricted principle "Every partially ordered set has a maximal totally ordered subset" is also equivalent to AC over ZF.<br />
**[[Tukey's lemma]]: Every non-empty collection of [[finite character]] has a maximal element with respect to inclusion.<br />
**[[Antichain]] principle: Every partially ordered set has a maximal [[antichain]].<br />
<br />
*[[Abstract algebra]]<br />
**Every [[vector space]] has a [[basis (linear algebra)|basis]].<ref>{{cite journal|last=Blass|first=Andreas|title=Existence of bases implies the axiom of choice|journal=Contemporary mathematics|year=1984|volume=31}}</ref><br />
**Every unital [[ring (mathematics)|ring]] other than the trivial ring contains a [[maximal ideal]].<br />
**For every non-empty set ''S'' there is a [[binary operation]] defined on ''S'' that gives it a [[group structure and the axiom of choice|group structure]].<ref>[[András Hajnal|A. Hajnal]], A. Kertész: Some new algebraic equivalents of the axiom of choice, ''Publ. Math. Debrecen'', '''19'''(1972), 339&ndash;340, see also H. Rubin, J. Rubin, ''Equivalents of the axiom of choice, II'', [[North-Holland Publishing Company|North-Holland]], 1985, p. 111.</ref> (A [[cancellation property|cancellative]] binary operation is enough.)<br />
<br />
*[[Functional analysis]]<br />
**The closed unit ball of the dual of a [[normed vector space]] over the reals has an [[extreme point]].<br />
<br />
*[[Point-set topology]]<br />
**[[Tychonoff's theorem]]: Every [[product topology|product]] of [[Compact space|compact]] [[topological space]]s is compact.<br />
**In the product topology, the [[closure (topology)|closure]] of a product of subsets is equal to the product of the closures.<br />
<br />
*[[Mathematical logic]]<br />
**If ''S'' is a set of sentences of [[first-order logic]] and ''B'' is a consistent subset of ''S'', then ''B'' is included in a set that is maximal among consistent subsets of ''S''. The special case where ''S'' is the set of '''all''' first-order sentences in a given [[signature (logic)|signature]] is weaker, equivalent to the [[Boolean prime ideal theorem]]; see the section "Weaker forms" below.<br />
<br />
*[[Graph theory]]<br />
**Every [[connected graph]] has a [[spanning tree]].<ref>{{citation|title=Trees|first=Jean-Pierre|last=Serre|authorlink=Jean-Pierre Serre|page=23|publisher=Springer|series=Springer Monographs in Mathematics|year=2003}}; {{citation<br />
| last = Soukup | first = Lajos<br />
| contribution = Infinite combinatorics: from finite to infinite<br />
| doi = 10.1007/978-3-540-77200-2_10<br />
| location = Berlin<br />
| mr = 2432534<br />
| pages = 189–213<br />
| publisher = Springer<br />
| series = Bolyai Soc. Math. Stud.<br />
| title = Horizons of combinatorics<br />
| volume = 17<br />
| year = 2008}}. See in particular Theorem 2.1, [http://books.google.com/books?id=kIKW18ENfUMC&pg=PA192 pp.&nbsp;192–193].</ref><br />
<br />
=== Category theory ===<br />
There are several results in [[category theory]] which invoke the axiom of choice for their proof. These results might be weaker than, equivalent to, or stronger than the axiom of choice, depending on the strength of the technical foundations. For example, if one defines categories in terms of sets, that is, as sets of objects and morphisms (usually called a [[small category]]), or even locally small categories, whose hom-objects are sets, then there is no [[category of sets|category of all sets]], and so it is difficult for a category-theoretic formulation to apply to all sets. On the other hand, other foundational descriptions of category theory are considerably stronger, and an identical category-theoretic statement of choice may be stronger than the standard formulation, à la class theory, mentioned above.<br />
<br />
Examples of category-theoretic statements which require choice include:<br />
*Every small [[category (mathematics)|category]] has a [[skeleton (category theory)|skeleton]].<br />
*If two small categories are weakly equivalent, then they are [[equivalence of categories|equivalent]].<br />
*Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a [[adjoint functors|left-adjoint]] (the Freyd adjoint functor theorem).<br />
<br />
==Weaker forms==<br />
There are several weaker statements that are not equivalent to the axiom of choice, but are closely related. One example is the [[axiom of dependent choice]] (DC). A still weaker example is the [[axiom of countable choice]] (AC<sub>ω</sub> or CC), which states that a choice function exists for any countable set of nonempty sets. These axioms are sufficient for many proofs in elementary [[mathematical analysis]], and are consistent with some principles, such as the Lebesgue measurability of all sets of reals, that are disprovable from the full axiom of choice.<br />
<br />
Other choice axioms weaker than axiom of choice include the [[Boolean prime ideal theorem]] and the [[Uniformization (set theory)|axiom of uniformization]]. The former is equivalent in ZF to the existence of an [[ultrafilter]] containing each given filter, proved by Tarski in 1930.<br />
<br />
===Results requiring AC (or weaker forms) but weaker than it===<!-- This section is linked from [[Basis (linear algebra)]] --><br />
One of the most interesting aspects of the axiom of choice is the large number of places in mathematics that it shows up. Here are some statements that require the axiom of choice in the sense that they are not provable from ZF but are provable from ZFC (ZF plus AC). Equivalently, these statements are true in all models of ZFC but false in some models of ZF.<br />
<br />
*[[Set theory]]<br />
**Any [[union (set theory)|union]] of countably many [[countable sets]] is itself countable.<br />
**If the set ''A'' is [[infinite set|infinite]], then there exists an [[injective function|injection]] from the [[natural number]]s '''N''' to ''A'' (see [[Dedekind infinite]]).<br />
**Every infinite [[determinacy#Basic notions|game]] <math>G_S</math> in which <math>S</math> is a [[Borel set|Borel]] subset of [[Baire space (set theory)|Baire space]] is [[determinacy#Basic notions|determined]].<br />
<br />
*[[Measure theory]]<br />
**The [[Vitali set|Vitali theorem]] on the existence of [[non-measurable set]]s which states that there is a subset of the [[real numbers]] that is not [[Lebesgue measurable]].<br />
**The [[Hausdorff paradox]].<br />
**The [[Banach–Tarski paradox]].<br />
**The [[Lebesgue measure]] of a countable [[disjoint union]] of measurable sets is equal to the sum of the measures of the individual sets.<br />
<br />
*[[Algebra]]<br />
**Every [[field (mathematics)|field]] has an [[algebraic closure]].<br />
**Every [[field extension]] has a [[transcendence basis]].<br />
**[[Stone's representation theorem for Boolean algebras]] needs the [[Boolean prime ideal theorem]].<br />
**The [[Nielsen–Schreier theorem]], that every subgroup of a free group is free.<br />
**The additive groups of '''[[real numbers|R]]''' and '''[[complex numbers|C]]''' are isomorphic.<ref>http://www.cs.nyu.edu/pipermail/fom/2006-February/009959.html</ref><ref>http://journals.cambridge.org/action/displayFulltext?type=1&fid=4931240&aid=4931232</ref><br />
<br />
*[[Functional analysis]]<br />
**The [[Hahn–Banach theorem]] in [[functional analysis]], allowing the extension of [[linear map|linear functionals]]<br />
**The theorem that every [[Hilbert space]] has an orthonormal basis.<br />
**The [[Banach–Alaoglu theorem]] about [[compactness]] of sets of functionals.<br />
**The [[Baire category theorem]] about [[complete space|complete]] [[metric space]]s, and its consequences, such as the [[open mapping theorem (functional analysis)|open mapping theorem]] and the [[closed graph theorem]].<br />
**On every infinite-dimensional topological vector space there is a [[discontinuous linear map]].<br />
<br />
*[[General topology]]<br />
**A uniform space is compact if and only if it is complete and totally bounded.<br />
**Every [[Tychonoff space]] has a [[Stone–Čech compactification]].<br />
<br />
*[[Mathematical logic]]<br />
**[[Gödel's completeness theorem]] for first-order logic: every consistent set of first-order sentences has a completion. That is, every consistent set of first-order sentences can be extended to a maximal consistent set.<br />
<br />
==Stronger forms of the negation of AC==<br />
Now, consider stronger forms of the negation of AC. For example, if we abbreviate by BP the claim that every set of real numbers has the [[property of Baire]], then BP is stronger than ¬AC, which asserts the nonexistence of any choice function on perhaps only a single set of nonempty sets. Note that strengthened negations may be compatible with weakened forms of AC. For example, ZF + DC<ref>[[Axiom of dependent choice]]</ref> + BP is consistent, if ZF is.<br />
<br />
It is also consistent with ZF + DC that every set of reals is [[Lebesgue measurable]]; however, this consistency result, due to [[Robert M. Solovay]], cannot be proved in ZFC itself, but requires a mild [[large cardinal]] assumption (the existence of an [[inaccessible cardinal]]). The much stronger [[axiom of determinacy]], or AD, implies that every set of reals is Lebesgue measurable, has the property of Baire, and has the [[perfect set property]] (all three of these results are refuted by AC itself). ZF + DC + AD is consistent provided that a sufficiently strong large cardinal axiom is consistent (the existence of infinitely many [[Woodin cardinal]]s).<br />
<br />
==Statements consistent with the negation of AC==<br />
There are models of Zermelo-Fraenkel set theory in which the axiom of choice is false. We will abbreviate "Zermelo-Fraenkel set theory plus the negation of the axiom of choice" by ZF¬C. For certain models of ZF¬C, it is possible to prove the negation of some standard facts.<br />
Note that any model of ZF¬C is also a model of ZF, so for each of the following statements, there exists a model of ZF in which that statement is true.<br />
<br />
*There exists a model of ZF¬C in which there is a function ''f'' from the real numbers to the real numbers such that ''f'' is not continuous at ''a'', but ''f'' is [[Sequential continuity|sequentially continuous]] at ''a'', i.e., for any sequence {''x<sub>n</sub>''} converging to ''a'', lim<sub>''n''</sub> f(''x<sub>n</sub>'')=f(a).<br />
*There exists a model of ZF¬C which has an infinite set of real numbers without a countably infinite subset.<br />
*There exists a model of ZF¬C in which real numbers are a countable union of countable sets.<ref>Jech, Thomas (1973) "The axiom of choice", ISBN 0-444-10484-4, CH. 10, p. 142.</ref><br />
*There exists a model of ZF¬C in which there is a field with no algebraic closure.<br />
*In all models of ZF¬C there is a vector space with no basis.<br />
*There exists a model of ZF¬C in which there is a vector space with two bases of different cardinalities.<br />
*There exists a model of ZF¬C in which there is a free [[complete boolean algebra]] on countably many generators.<ref name="Stavi, 1974">{{cite journal| first= Jonathan | last=Stavi| year=1974| url=http://www.springerlink.com/content/d5710380t753621u/ |format=reprint|title=A model of ZF with an infinite free complete Boolean algebra| journal=Israel Journal of Mathematics| volume=20| issue= 2| pages=149–163|doi=10.1007/BF02757883}}</ref><br />
<br />
For proofs, see [[Thomas Jech]], ''The Axiom of Choice'', [[American Elsevier]] Pub. Co., New York, 1973.<br />
<br />
*There exists a model of ZF¬C in which every set in R<sup>''n''</sup> is [[measurable]]. Thus it is possible to exclude counterintuitive results like the [[Banach–Tarski paradox]] which are provable in ZFC. Furthermore, this is possible whilst assuming the [[Axiom of dependent choice]], which is weaker than AC but sufficient to develop most of [[real analysis]].<br />
*In all models of ZF¬C, the [[generalized continuum hypothesis]] does not hold.<br />
<br />
==Quotes==<br />
"The Axiom of Choice is obviously true, the [[Well-ordering theorem|well-ordering principle]] obviously false, and who can tell about [[Zorn's lemma]]?" —&nbsp;[[Jerry Bona]]<br />
:This is a joke: although the three are all mathematically equivalent, many mathematicians find the axiom of choice to be intuitive, the well-ordering principle to be counterintuitive, and Zorn's lemma to be too complex for any intuition.<br />
<br />
"The Axiom of Choice is necessary to select a set from an infinite number of socks, but not an infinite number of shoes." —&nbsp;[[Bertrand Russell]]<br />
:The observation here is that one can define a function to select from an infinite number of pairs of shoes by stating for example, to choose the left shoe. Without the axiom of choice, one cannot assert that such a function exists for pairs of socks, because left and right socks are (presumably) indistinguishable from each other.<br />
<br />
"Tarski tried to publish his theorem [the equivalence between AC and 'every infinite set ''A'' has the same cardinality as ''A''x''A''', see above] in [[Comptes rendus de l'Académie des sciences|Comptes Rendus]], but [[Maurice René Fréchet|Fréchet]] and [[Henri Lebesgue|Lebesgue]] refused to present it. Fréchet wrote that an implication between two well known [true] propositions is not a new result, and Lebesgue wrote that an implication between two false propositions is of no interest".<br />
:Polish-American mathematician [[Jan Mycielski]] relates this anecdote in a 2006 article in the Notices of the AMS.<br />
<br />
"The axiom gets its name not because mathematicians prefer it to other axioms." —&nbsp;[[A. K. Dewdney]]<br />
:This quote comes from the famous [[April Fools' Day]] article in the ''computer recreations'' column of the ''[[Scientific American]]'', April 1989.<br />
<br />
== Notes ==<br />
<references /><br />
<br />
==References==<br />
* Horst Herrlich, ''Axiom of Choice'', Springer Lecture Notes in Mathematics 1876, [[Springer Verlag]] Berlin Heidelberg (2006). ISBN 3-540-30989-6.<br />
*Paul Howard and Jean Rubin, "Consequences of the Axiom of Choice". Mathematical Surveys and Monographs 59; [[American Mathematical Society]]; 1998.<br />
*[[Thomas Jech]], "About the Axiom of Choice." ''Handbook of Mathematical Logic'', John Barwise, ed., 1977.<br />
* Per Martin-Löf, "100 years of Zermelo's axiom of choice: What was the problem with it?", in ''Logicism, Intuitionism, and Formalism: What Has Become of Them?'', Sten Lindström, Erik Palmgren, Krister Segerberg, and Viggo Stoltenberg-Hansen, editors (2008). ISBN 1-4020-8925-2<br />
*Gregory H Moore, "Zermelo's axiom of choice, Its origins, development and influence", [[Springer Science+Business Media|Springer]]; 1982. ISBN 0-387-90670-3, available as a [[Dover Publications]] reprint, 2013, ISBN 0-486-48841-1.<br />
*Herman Rubin, Jean E. Rubin: Equivalents of the axiom of choice. North Holland, 1963. Reissued by [[Elsevier]], April 1970. ISBN 0-7204-2225-6.<br />
*Herman Rubin, Jean E. Rubin: Equivalents of the Axiom of Choice II. North Holland/Elsevier, July 1985, ISBN 0-444-87708-8.<br />
*George Tourlakis, ''Lectures in Logic and Set Theory. Vol. II: Set Theory'', [[Cambridge University Press]], 2003. ISBN 0-511-06659-7<br />
*[[Ernst Zermelo]], "Untersuchungen über die Grundlagen der Mengenlehre I," ''Mathematische Annalen 65'': (1908) pp.&nbsp;261–81. [http://www.digizeitschriften.de/no_cache/home/jkdigitools/loader/?tx_jkDigiTools_pi1%5BIDDOC%5D=361762 PDF download via digizeitschriften.de]<br />
::Translated in: [[Jean van Heijenoort]], 2002. ''From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931''. New edition. [[Harvard University Press]]. ISBN 0-674-32449-8<br />
::*1904. "Proof that every set can be well-ordered," 139-41.<br />
::*1908. "Investigations in the foundations of set theory I," 199-215.<br />
<br />
==External links==<br />
*{{springer|title=Axiom of choice|id=p/a014270}}<br />
*[http://www.apronus.com/provenmath/choice.htm Axiom of Choice and Its Equivalents at ProvenMath] includes formal statement of the Axiom of Choice, Hausdorff's Maximal Principle, Zorn's Lemma and formal proofs of their equivalence down to the finest detail.<br />
*[http://www.math.purdue.edu/~hrubin/JeanRubin/Papers/conseq.html Consequences of the Axiom of Choice], based on the book by [http://www.emunix.emich.edu/~phoward/ Paul Howard] and Jean Rubin.<br />
*{{sep entry|axiom-choice|The Axiom of Choice|[[John Lane Bell]]}}<br />
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{{Set theory}}<br />
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{{DEFAULTSORT:Axiom Of Choice}}<br />
[[Category:Axiom of choice| ]]</div>Adminhttps://en.formulasearchengine.com/index.php?title=Alkali_metal&diff=218470Alkali metal2014-07-31T13:56:49Z<p>Admin: Reverted edits by MediaWiki spam cleanup (talk) to last revision by DePiep</p>
<hr />
<div>{{good article}}<br />
{{Periodic table (alkali metals)}}<br />
The '''alkali metals''' are a [[group (periodic table)|group]] in the [[periodic table]] consisting of the [[chemical element]]s [[lithium]] (Li), [[sodium]] (Na),<ref group=note>The symbol '''Na''' for sodium is derived from its [[Latin name]], ''natrium''; this is still the name for the element in some languages, such as German and Russian. In early English texts, the symbol So for the English name ''sodium'' is sometimes seen; this is wholly obsolete.</ref> [[potassium]] (K),<ref group=note>The symbol '''K''' for potassium is derived from its [[Latin name]], ''kalium''; this is still the name for the element in some languages, such as German and Russian. In early English texts, the symbol Po for the English name ''potassium'' is sometimes seen; this is wholly obsolete, and presently would collide with the symbol for [[polonium]] (also Po).</ref> [[rubidium]] (Rb), [[caesium]] (Cs),{{#tag:ref|''Caesium'' is the spelling recommended by the [[International Union of Pure and Applied Chemistry]] (IUPAC).<ref>{{RedBook2005|pages=248–49}}.</ref> The [[American Chemical Society]] (ACS) has used the spelling ''cesium'' since 1921,<ref>{{Cite book | editor1-first = Anne M. | editor1-last = Coghill | editor2-first = Lorrin R. | editor2-last = Garson | year = 2006 | title = The ACS Style Guide: Effective Communication of Scientific Information | edition = 3rd | publisher = American Chemical Society | location = Washington, D.C. | isbn = 0-8412-3999-1 | page = 127}}.</ref><ref>{{Cite journal | journal=Pure Appl. Chem. | volume=70 | issue=1 | last1=Coplen | pages = 237–257 | year = 1998 | first1=T. B. | url = http://old.iupac.org/reports/1998/7001coplen/history.pdf | last2=Peiser | first2=H. S. | title = History of the recommended atomic-weight values from 1882 to 1997: a comparison of differences from current values to the estimated uncertainties of earlier values | doi = 10.1351/pac199870010237}}.</ref> following ''Webster’s Third New International Dictionary''.|group=note}} and [[francium]] (Fr).<ref name="redbook">{{RedBook2005|pages=51}}.</ref> This group lies in the [[s-block]] of the periodic table<ref name="meta-synthesis3">{{cite web |url=http://www.meta-synthesis.com/webbook/35_pt/pt_database.php?Button=1900-1949+Formulations |title=The Internet Database of Periodic Tables |author=Leach, Mark R. |date=1999–2012 |work=meta-synthesis.com |accessdate=20 May 2012}}</ref> as all alkali metals have their outermost electron in an [[atomic orbital|s-orbital]].<ref name="rsc"/><ref>{{cite web |url=http://www.nist.gov/pml/data/upload/periodic_table_composite_2010_nobleed.pdf |title=Periodic Table: Atomic Properties of the Elements |date=September 2010 |work=nist.gov |publisher=[[National Institute of Standards and Technology]] |accessdate=17 February 2012}}</ref><ref name="RubberBible84th"/> The alkali metals provide the best example of [[periodic trends|group trends]] in properties in the periodic table,<ref name="rsc" /> with elements exhibiting well-characterized [[homology (chemistry)|homologous]] behaviour.<ref name="rsc"/><br />
<br />
The alkali metals have very similar properties: they are all shiny, [[hardness|soft]], highly [[reactivity (chemistry)|reactive]] metals at [[standard temperature and pressure]]<ref name="rsc">{{cite web |url=http://www.rsc.org/chemsoc/visualelements/PAGES/data/intro_groupi_data.html |title=Visual Elements: Group 1 – The Alkali Metals |author=[[Royal Society of Chemistry]] |work=Visual Elements |publisher=Royal Society of Chemistry |accessdate=13 January 2012}}</ref> and readily lose their [[valence electron|outermost electron]] to form [[cation]]s with [[electric charge|charge]] +1.<ref name="Greenwood&Earnshaw"/>{{rp|28}} They can all be cut easily with a knife due to their softness, exposing a shiny surface that tarnishes rapidly in air due to [[redox|oxidation]].<ref name="rsc"/> Because of their high reactivity, they must be stored under oil to prevent reaction with air,<ref name="OU">{{cite web |url=http://www.open.edu/openlearn/science-maths-technology/science/chemistry/alkali-metals |title=Alkali metals |author=The OpenLearn team |year=2012 |work=OpenLearn |publisher=The Open University |accessdate=9 July 2012}}</ref> and are found naturally only in [[salt (chemistry)|salts]] and never as the free element.<ref name="OU"/> In the modern [[International Union of Pure and Applied Chemistry|IUPAC]] nomenclature, the alkali metals comprise the '''group 1 elements''',{{#tag:ref|In both the old IUPAC and the [[Chemical Abstracts Service|CAS]] systems for group numbering, this group is known as '''group IA''' (pronounced as "group one A", as the "I" is a [[Roman numeral]]).<ref>{{cite journal |last1=Fluck |first1=E. |year=1988 |title=New Notations in the Periodic Table |journal=[[Pure and Applied Chemistry|Pure Appl. Chem.]] |volume=60 |issue=3 |pages=431–436 |publisher=[[International Union of Pure and Applied Chemistry|IUPAC]] |doi=10.1351/pac198860030431 |url=http://www.iupac.org/publications/pac/1988/pdf/6003x0431.pdf |accessdate=24 March 2012 }}</ref>|name=group-numbering|group=note}} excluding [[hydrogen]] (H), which is nominally a group 1 element<ref name="redbook"/><ref name="iupac-periodic-table">{{cite web |url=http://old.iupac.org/reports/periodic_table/IUPAC_Periodic_Table-21Jan11.pdf |title=IUPAC Periodic Table of the Elements |date=21 January 2011 |work= iupac.org |publisher=[[International Union of Pure and Applied Chemistry]] |accessdate=22 February 2012}}</ref> but not normally considered to be an alkali metal<ref name="iupac"/><ref name="Folden"/> as it rarely exhibits behaviour comparable to that of the alkali metals.<ref name="hydrogen-halogen"/> All the alkali metals react with water, with the heavier alkali metals reacting more vigorously than the lighter ones.<ref name="rsc"/><ref name="alkalibangs">{{cite web|last=Gray|first=Theodore|title=Alkali Metal Bangs|url=http://www.theodoregray.com/periodictable/AlkaliBangs/index.html|publisher=[[Theodore Gray]]|accessdate=13 May 2012}}</ref><br />
<br />
All the discovered alkali metals occur in nature: in order of [[abundance of the chemical elements|abundance]], sodium is the most abundant, followed by potassium, lithium, rubidium, caesium, and finally francium, which is very rare due to its extremely high [[radioactivity]] and thus occurs only in [[trace radioisotope|traces]] due to its presence in natural [[decay chain]]s.<ref name="webelements-occurrence"/><ref name="chemeducator"/> Experiments have been conducted to attempt the synthesis of [[ununennium]] (Uue), which is likely to be the next member of the group, but they have all met with failure.<ref name="link">{{cite journal|title=Search for superheavy elements using <sup>48</sup>Ca + <sup>254</sup>Es<sup>g</sup> reaction|first1=R.W. |last1=Lougheed |first2=J.H.|last2=Landrum|first3=E.K.|last3=Hulet|first4=J.F.|last4=Wild|first5=R.J.|last5=Dougan|first6=A.D.|last6=Dougan|first7=H.|last7=Gäggeler|first8=M|last8=Schädel|first9=K.J.|last9=Moody|first10=K.E.|last10=Gregorich|last11=Seaborg|journal=Physical Review C|year=1985|pages=1760–1763|volume=32|issue=5|bibcode = 1985PhRvC..32.1760L|doi=10.1103/PhysRevC.32.1760|first11=G. }}</ref> However, ununennium may not be an alkali metal due to [[relativistic quantum chemistry|relativistic effects]], which are predicted to have a large influence on the chemical properties of [[superheavy element]]s;<ref name="tanm">{{cite web |url=http://lch.web.psi.ch/files/lectures/TexasA&M/TexasA&M.pdf |title=Gas Phase Chemistry of Superheavy Elements |author=Gäggeler, Heinz W. |date=5–7 November 2007 |work=Lecture Course Texas A&M |accessdate=26 February 2012}}</ref> even if it does turn out to be an alkali metal, it is predicted to have some differences in physical and chemical properties from its lighter homologues.<ref name="Uue"/>{{rp|1729–1733}}<br />
<br />
Most alkali metals have many different applications. Two of the most well-known applications of the pure elements are rubidium and caesium [[atomic clock]]s,<ref name="atomic-clocks"/> of which caesium atomic clocks are the most accurate and precise representation of time.<ref name="pubs.usgs"/><ref name="nist-second"/> A common application of the compounds of sodium is the [[sodium-vapour lamp]], which emits very efficient light.<ref name="lamp1"/><ref name="lamp2"/> [[Table salt]], or sodium chloride, has been used since antiquity. Sodium and potassium are also [[essential element]]s, having major biological roles as [[electrolyte]]s,<ref name="webelements-potassium"/><ref name="webelements-sodium"/> and although the other alkali metals are not essential, they also have various effects on the body, both beneficial and harmful.<ref name="webelements-lithium"/><ref name="webelements-rubidium"/><ref name="webelements-caesium"/><ref name="rsc-francium"/><br />
<br />
==Characteristics==<br />
<br />
===Chemical===<br />
[[Image:Alkalimetalle.jpg|thumb|Series of alkali metals, stored in mineral oil to prevent oxidation. ("Natrium" is the German name for sodium.)]]<br />
Like other groups, the known members of this family show patterns in [[electronic configuration]], especially the outermost shells, resulting in trends in chemical behavior:<br />
<br />
{| class="wikitable" align="center"<br />
|-<br />
!''[[Atomic number|Z]]'' !! [[Chemical element|Element]] !! [[Electron shell|No. of electrons/shell]] !! [[Electron configuration|Electron<br>configuration]]<ref group="note">[[Noble gas notation]] is used for conciseness; the nearest noble gas that precedes the element in question is written first, and then the electron configuration is continued from that point forward.</ref><br />
|-<br />
| 3 || [[lithium]] || 2, 1 || &#91;[[Helium|He]]&#93; 2s<sup>1</sup><br />
|-<br />
| 11 || [[sodium]] || 2, 8, 1 || &#91;[[Neon|Ne]]&#93; 3s<sup>1</sup><br />
|-<br />
| 19 || [[potassium]] || 2, 8, 8, 1 || &#91;[[Argon|Ar]]&#93; 4s<sup>1</sup><br />
|-<br />
| 37 || [[rubidium]] || 2, 8, 18, 8, 1 || &#91;[[Krypton|Kr]]&#93; 5s<sup>1</sup><br />
|-<br />
| 55 || [[caesium]] || 2, 8, 18, 18, 8, 1 || &#91;[[Xenon|Xe]]&#93; 6s<sup>1</sup><br />
|-<br />
| 87 || [[francium]] || 2, 8, 18, 32, 18, 8, 1 || &#91;[[Radon|Rn]]&#93; 7s<sup>1</sup><br />
|}<br />
<br />
Most of the chemistry has been observed only for the first five members of the group. The chemistry of francium is not well established due to its extreme [[radioactive decay|radioactivity]];<ref name="rsc" /> thus, the presentation of its properties here is limited.<br />
<br />
[[File:Cesium water.theora.ogv|thumb|Caesium reacts explosively with water even at low temperatures]]<br />
All the alkali metals are highly reactive and are never found in elemental forms in nature.<ref name="krebs" /> Because of this, they are usually stored in [[mineral oil]] or [[kerosene]] (paraffin oil).<ref name="OU"/> They react aggressively with the [[halogen]]s to form the [[alkali metal halide]]s, which are white [[ionic crystal|ionic crystalline compounds]] that are all [[solubility|soluble]] in water except [[lithium fluoride]] ([[lithium|Li]][[fluorine|F]]).<ref name="rsc"/> The alkali metals also react with water to form strongly [[alkali]]ne [[hydroxide]]s and thus should be handled with great care. The heavier alkali metals react more vigorously than the lighter ones; for example, when dropped into water, caesium produces a larger explosion than potassium.<ref name="rsc"/><ref name="alkalibangs"/><ref name="pubs.usgs" /> The alkali metals have the lowest first [[ionization energy|ionisation energies]] in their respective periods of the [[periodic table]]<ref name="RubberBible84th">{{cite book | editor = Lide, D. R. | title = CRC Handbook of Chemistry and Physics | edition = 84th | location = Boca Raton, FL | publisher = CRC Press | year = 2003 }}</ref> because of their low [[effective nuclear charge]]<ref name="rsc"/> and the ability to attain a [[noble gas]] configuration by losing just one [[electron]]. The second ionisation energy of all of the alkali metals is very high<ref name="rsc"/><ref name="RubberBible84th" /><!--the second ionisation energy for francium is not given in [[ionization energies of the elements (data page)]]--> as it is in a full shell that is also closer to the nucleus;<ref name="rsc"/> thus, they almost always lose a single electron, forming cations.<ref name="Greenwood&Earnshaw"/>{{rp|28}} The [[alkalide]]s are an exception: they are unstable compounds which contain alkali metals in a −1 oxidation state, which is very unusual as before the discovery of the alkalides, the alkali metals were not expected to be able to form [[anion]]s and were thought to be able to appear in [[salt (chemistry)|salts]] only as cations. The alkalide anions have filled [[s-orbital|s-subshells]], which gives them more stability and allows them to exist. All the stable alkali metals except lithium are known to be able to form alkalides,<ref>{{cite journal | journal = [[J. Am. Chem. Soc.]] | author = J. L. Dye, J. M. Ceraso, Mei Lok Tak, B. L. Barnett, F. J. Tehan | title = Crystalline salt of the sodium anion (Na<sup>−</sup>) | year = 1974 | volume = 96 | issue = 2 | pages = 608–609 | doi = 10.1021/ja00809a060 }}</ref><ref>{{cite journal | author = F. J. Tehan, B. L. Barnett, J. L. Dye | title = Alkali anions. Preparation and crystal structure of a compound which contains the cryptated sodium cation and the sodium anion | journal = [[J. Am. Chem. Soc.]] | year = 1974 | volume = 96 | issue = 23 | pages = 7203–7208 | doi = 10.1021/ja00830a005 }}</ref><ref>{{cite journal | journal = [[Angew. Chem. Int. Ed. Engl.]] | year = 1979 | author = J. L. Dye | title = Compounds of Alkali Metal Anions | volume = 18 | issue = 8 | pages = 587–598 | doi = 10.1002/anie.197905871 }}</ref> and the alkalides have much theoretical interest due to their unusual [[stoichiometry]] and low [[ionization potential|ionisation potentials]]. Alkalides are chemically similar to the [[electride]]s, which are salts with trapped [[electron]]s acting as anions.<ref name="Redko">{{cite journal | author = M. Y. Redko, R. H. Huang, J. E. Jackson, J. F. Harrison, J. L. Dye | year = 2003 | title = Barium azacryptand sodide, the first alkalide with an alkaline Earth cation, also contains a novel dimer, (Na<sub>2</sub>)<sup>2−</sup> | journal = [[Journal of the American Chemical Society|J. Am. Chem. Soc.]] | volume = 125 | issue = 8 | pages = 2259–2263 | doi = 10.1021/ja027241m | pmid = 12590555 }}</ref> A particularly striking example of an alkalide is "inverse [[sodium hydride]]", H<sup>+</sup>Na<sup>−</sup>, as opposed to the usual sodium hydride, Na<sup>+</sup>H<sup>−</sup>:<ref name="HNa">{{cite journal | author = M. Y. Redko, M. Vlassa, J. E. Jackson, A. W. Misiolek, R. H. Huang RH, J. L. Dye | year = 2002 | title = "Inverse sodium hydride": a crystalline salt that contains H<sup>+</sup> and Na<sup>−</sup> | journal = [[Journal of the American Chemical Society|J. Am. Chem. Soc.]] | volume = 124 | issue = 21 | pages = 5928–5929 | doi = 10.1021/ja025655 }}</ref> it is unstable in isolation, due to its high energy resulting from the displacement of two electrons from hydrogen to sodium, although several derivatives are predicted to be [[metastability|metastable]] or stable.<ref name="HNa"/><ref name="HNa-theory">{{cite journal|url=http://simons.hec.utah.edu/papers/266.pdf|title=Inverse Sodium Hydride: A Theoretical Study|author=Agnieszka Sawicka, Piotr Skurski, and Jack Simons|journal=J. Am. Chem. Soc.|year=2003|volume=125|pages=3954–3958|doi=10.1021/ja021136v|pmid=12656631|issue=13}}</ref><br />
<br />
The chemistry of lithium shows several differences from that of the rest of the group as the small Li<sup>+</sup> cation [[chemical polarity|polarises]] [[anion]]s and gives its compounds a more [[covalent bond|covalent]] character.<ref name="rsc" /> Lithium and [[magnesium]] have a [[diagonal relationship]]:<ref name="rsc" /> because of this, lithium has some similarities to magnesium. For example, lithium forms a stable [[nitride]], a property common among all the [[alkaline earth metal]]s (magnesium's group) but unique among the alkali metals.<ref name="alkalireact"/> In addition, among their respective groups, only lithium and magnesium form [[covalent bond|covalent]] [[organometallic compound]]s (e.g. Li[[methyl group|Me]] and MgMe<sub>2</sub>).<ref name="Shriver&Atkins">{{cite book |title=Inorganic Chemistry |first1=Duward |last1=Shriver |first2=Peter |last2=Atkins |publisher=W. H. Freeman |year=2006 |isbn=978-0716748786 |page=259 |accessdate=10 November 2012 |url=http://www.google.ru/books?id=NwOTQAAACAAJ}}</ref> Lithium fluoride is the only alkali metal halide that is not soluble in water,<ref name="rsc"/> and [[lithium hydroxide]] is the only alkali metal hydroxide that is not [[deliquescent]].<ref name="rsc"/> Francium is also predicted show some differences due to its high [[atomic weight]], causing its electrons to travel at considerable fractions of the speed of light and thus making [[relativistic quantum chemistry|relativistic effects]] more prominent. In contrast to the trend of decreasing [[electronegativity|electronegativities]] and [[ionisation energy|ionisation energies]] of the alkali metals, francium's electronegativity and ionisation energy are predicted to be higher than caesium's due to the relativistic stabilisation of the 7s electrons; also, its [[atomic radius]] is expected to be abnormally low.<!--Haire says this happens for Uue because of the analogous effect for 8s - seems likely for Fr too--><ref name="Uue">{{cite book| title = The Chemistry of the Actinide and Transactinide Elements| editor1-last = Morss|editor2-first = Norman M.| editor2-last = Edelstein| editor3-last = Fuger|editor3-first = Jean| last1 = Hoffman|first1 = Darleane C.| last2=Lee|first2=Diana M. |last3=Pershina|first3=Valeria | chapter = Transactinides and the future elements| publisher = [[Springer Science+Business Media]]| year = 2006| isbn = 1-4020-3555-1| location = Dordrecht, The Netherlands| edition = 3rd| ref = CITEREFHaire2006}}</ref>{{rp|1729}}<ref name="andreev"/><br />
<br />
====Compounds and reactions====<br />
<br />
=====Reaction with water (alkali metal hydroxides)=====<br />
{{external media<br />
| align = left<br />
| video1 = [http://www.youtube.com/watch?v=QSZ-3wScePM Reactions of the alkali metals with water], conducted by The [[Open University]]<br />
}}<!--mention things like MIT's Sodium Drop and perhaps Brainiac's faked explosions with Gray's tests--><br />
[[File:Large Sodium Explosion.jpg|thumb|right|alt=A large orange-yellow explosion|A reaction of 3 [[pound (mass)|pounds]] (≈ 1.4 kg) of sodium with water]]<br />
All the alkali metals react vigorously or explosively with cold water, producing an [[aqueous solution]] of the strongly [[base (chemistry)|basic]] alkali metal [[hydroxide]] and releasing hydrogen gas.<ref name="alkaliwater">{{cite web |url=http://www.chemguide.co.uk/inorganic/group1/reacth2o.html#top |title=Reaction of the Group 1 Elements with Water |author=Clark, Jim |year=2005 |work=chemguide |accessdate=18 June 2012}}</ref> This reaction becomes more vigorous going down the group: lithium reacts steadily with [[effervescence]], but sodium and potassium can ignite and rubidium and caesium sink in water and generate hydrogen gas so rapidly that shock waves form in the water that may shatter glass containers.<ref name="rsc"/> When an alkali metal is dropped into water, it produces an explosion, of which there are two separate stages. The metal reacts with the water first, breaking the hydrogen bonds in the water and producing [[hydrogen]] gas; this takes place faster for the more reactive heavier alkali metals. Second, the heat generated by the first part of the reaction often ignites the hydrogen gas, causing it to burn explosively into the surrounding air. This secondary hydrogen gas explosion produces the visible flame above the bowl of water, lake or other body of water, not the initial reaction of the metal with water (which tends to happen mostly under water).<ref name="alkalibangs"/><br />
<br />
=====Reaction with the group 14 elements=====<br />
{{double image|right|Potassium-graphite-xtal-3D-SF-A.png|150|Potassium-graphite-xtal-3D-SF-B.png|150|Side ''(left)'' and top ''(right)'' views of the [[graphite intercalation compound]] KC<sub>8</sub>}}<br />
Lithium and sodium react with [[carbon]] to form [[acetylide]]s, Li<sub>2</sub>C<sub>2</sub> and Na<sub>2</sub>C<sub>2</sub>, which can also be obtained by reaction of the metal with [[acetylene]]. Potassium, rubidium, and caesium react with [[graphite]]; their atoms are [[intercalation (chemistry)|intercalated]] between the hexagonal graphite layers, forming [[graphite intercalation compound]]s of formulae MC<sub>60</sub> (dark grey, almost black), MC<sub>48</sub> (dark grey, almost black), MC<sub>36</sub> (blue), MC<sub>24</sub> (steel blue), and MC<sub>8</sub> (bronze) (M = K, Rb, or Cs). These compounds are over 200 times more electrically conductive than pure graphite, suggesting that the valence electron of the alkali metal is transferred to the graphite layers (e.g. {{chem|M|+|C|8|-}}).<ref name=generalchemistry/> Upon heating of KC<sub>8</sub>, the elimination of potassium atoms results in the conversion in sequence to KC<sub>24</sub>, KC<sub>36</sub>, KC<sub>48</sub> and finally KC<sub>60</sub>. KC<sub>8</sub> is a very strong [[reducing agent]] and is pyrophoric and explodes on contact with water.<ref name="InorgChem">{{cite book| title = Inorganic Chemistry, 3rd Edition| chapter = Chapter 14: The group 14 elements| author1 = Catherine E. Housecroft| author2 = Alan G. Sharpe| publisher = Pearson| year = 2008| isbn = 978-0-13-175553-6| page = 386}}</ref><ref>[http://physics.nist.gov/TechAct.2001/Div846/div846h.html NIST Ionizing Radiation Division 2001 - Technical Highlights]</ref> While the large alkali metals (K, Rb, and Cs) initially form MC<sub>8</sub>, the smaller ones initially form MC<sub>6</sub>.<ref name=cac6>{{cite journal|author=N. Emery ''et al.''|title=Review: Synthesis and superconducting properties of CaC6|journal=Sci. Technol. Adv. Mater.|volume=9|year=2008|pages=044102|doi=10.1088/1468-6996/9/4/044102|bibcode=2008STAdM...9d4102E|issue=4|first2=Claire|first3=Jean-François|first4=Philippe}}</ref><br />
<br />
When the alkali metals react with the heavier elements in the [[carbon group]], ionic substances with cage-like structures are formed, such as the [[silicide]] M<sub>4</sub>[[silicon|Si]]<sub>4</sub> (M = K, Rb, or Cs), which contains M<sup>+</sup> and tetrahedral {{chem|Si|4|4-}} ions.<ref name=generalchemistry/> The chemistry of alkali metal [[germanide]]s, involving the germanide ion [[germanium|Ge]]<sup>4−</sup> and other cluster ([[Zintl ion|Zintl]]) ions such as {{chem|Ge|4|2-}}, {{chem|Ge|9|4-}}, {{chem|Ge|9|2-}}, and [(Ge<sub>9</sub>)<sub>2</sub>]<sup>6−</sup>, is largely analogous to that of the corresponding silicides.<ref name="Greenwood&Earnshaw"/> Alkali metal [[stannide]]s are mostly ionic, sometimes with the stannide ion ([[tin|Sn]]<sup>4−</sup>),<ref name = "Kauzlarich">S.M. Kauzlarich,(1994), Zintl Compounds, Encyclopedia of Inorganic Chemistry, John Wiley & sons, ISBN 0-471-93620-0</ref> and sometimes with more complex Zintl ions such as {{chem|Sn|9|4-}}, which appears in tetrapotassium nonastannide (K<sub>4</sub>Sn<sub>9</sub>).<ref name = "Hoch">{{cite journal|doi=10.1107/S0108270102002032|title=Tetrapotassium nonastannide, K4Sn9|year=2002|last1=Hoch|first1=Constantin|last2=Wendorff|first2=Marco|last3=Röhr|first3=Caroline|journal=Acta Crystallographica Section C Crystal Structure Communications|volume=58|issue=4|pages=i45}}</ref> The monatomic [[plumbide]] ion ([[lead|Pb]]<sup>4−</sup>) is unknown, and indeed its formation is predicted to be energetically unfavourable; alkali metal plumbides have complex Zintl ions, such as {{chem|Pb|9|4-}}.<ref name="Greenwood&Earnshaw"/><br />
<br />
=====Reaction with the pnictogens (alkali metal pnictides)=====<br />
[[File:Lithium-nitride-xtal-CM-3D-polyhedra.png|thumb|right|[[Unit cell]] [[ball-and-stick model]] of [[lithium nitride]]<ref>{{cite journal|title=Structure of Lithium Nitride and Transition-Metal-Doped Derivatives, Li<sub>3−''x''−''y''</sub>M<sub>''x''</sub>N (M = Ni, Cu): A Powder Neutron Diffraction Study|author=Duncan H. Gregory, Paul M. O'Meara, Alexandra G. Gordon, Jason P. Hodges, Simine Short, and James D. Jorgensen|journal=Chem. Mater.|year=2002|volume=14|issue=5|pages=2063–2070|doi=10.1021/cm010718t}}</ref>]]<br />
Lithium, the lightest of the alkali metals, is the only alkali metal which reacts with [[nitrogen]] at [[standard conditions]], and its [[nitride]] is the only stable alkali metal nitride. Nitrogen is an [[reactivity (chemistry)|unreactive]] gas because breaking the strong [[triple bond]] in the [[dinitrogen]] molecule (N<sub>2</sub>) requires a lot of energy. The formation of an alkali metal nitride would consume the ionisation energy of the alkali metal (forming M<sup>+</sup> ions), the energy required to break the triple bond in N<sub>2</sub> and the formation of N<sup>3−</sup> ions, and all the energy released from the formation of an alkali metal nitride is from the [[lattice energy]] of the alkali metal nitride. The lattice energy is maximised with small, highly charged ions; the alkali metals do not form highly charged ions, only forming ions with a charge of +1, so only lithium, the smallest alkali metal, can release enough lattice energy to make the reaction with nitrogen [[exothermic]], forming [[lithium nitride]]. The reactions of the other alkali metals with nitrogen would not release enough lattice energy and would thus be [[endothermic]], so they do not form nitrides at standard conditions.<ref name="alkalireact">{{cite web |url=http://www.chemguide.co.uk/inorganic/group1/reacto2.html#top |title=Reaction of the Group 1 Elements with Oxygen and Chlorine |author=Clark, Jim |year=2005 |work=chemguide |accessdate=27 June 2012}}</ref> ([[Sodium nitride]] (Na<sub>3</sub>N) and [[potassium nitride]] (K<sub>3</sub>N), while existing, are extremely unstable, being prone to decomposing back into their constituent elements, and cannot be produced by reacting the elements with each other at standard conditions.)<ref name=Jansen1>{{cite journal|title=Synthesis and structure of Na<sub>3</sub>N|author=Fischer, D., Jansen, M.|journal= Angew Chem|volume=41|issue=10|page=1755|year=2002|doi=10.1002/1521-3773(20020517)41:10<1755::AID-ANIE1755>3.0.CO;2-C}}</ref><ref name="Jansen2">{{cite journal|title=Synthesis and structure of K<sub>3</sub>N|author=Fischer, D.; Cancarevic, Z.; Schön, J. C.; Jansen, M. Z. |journal=Z. anorg allgem Chemie|volume= 630|issue=1|page=156|doi=10.1002/zaac.200300280|year=2004}}. [http://pubs.acs.org/cen/topstory/8020/8020notw9.html 'Elusive Binary Compound Prepared'] ''Chemical & Engineering News'' '''80''' No. 20 (20 May 2002)</ref><br />
<br />
All the alkali metals react readily with [[phosphorus]] and [[arsenic]] to form phosphides and arsenides with the formula M<sub>3</sub>Pn (where M represents an alkali metal and Pn represents a [[pnictogen]]). This is due to the greater size of the P<sup>3−</sup> and As<sup>3−</sup> ions, so that less lattice energy needs to be released for the salts to form.<ref name=generalchemistry/> These are not the only phosphides and arsenides of the alkali metals: for example, potassium has nine different known phosphides, with formulae K<sub>3</sub>P, K<sub>4</sub>P<sub>3</sub>, K<sub>5</sub>P<sub>4</sub>, KP, K<sub>4</sub>P<sub>6</sub>, K<sub>3</sub>P<sub>7</sub>, K<sub>3</sub>P<sub>11</sub>, KP<sub>10.3</sub>, and KP<sub>15</sub>.<ref name = "Schnering">H.G. Von Schnering, W. Hönle ''Phosphides - Solid-state Chemistry'' Encyclopedia of Inorganic Chemistry Ed. R. Bruce King (1994) John Wiley & Sons ISBN 0-471-93620-0</ref> While most metals form arsenides, only the alkali and alkaline earth metals form mostly ionic arsenides. The structure of Na<sub>3</sub>As is complex with unusually short Na–Na distances of 328–330 pm which are shorter than in sodium metal, and this indicates that even with these electropositive metals the bonding cannot be straightforwardly ionic.<ref name="Greenwood&Earnshaw"/> Other alkali metal arsenides not conforming to the formula M<sub>3</sub>As are known, such as LiAs, which has a metallic lustre and electrical conductivity indicating the presence of some [[metallic bond]]ing.<ref name="Greenwood&Earnshaw"/> The [[antimonide]]s are unstable and reactive as the [[antimony|Sb]]<sup>3−</sup> ion is a strong reducing agent; reaction of them with acids form the toxic and unstable gas [[stibine]] (SbH<sub>3</sub>).<ref>{{cite book|title=Outlines of Chemistry&nbsp;– A Textbook for College Students|author=Kahlenberg, Louis|publisher=READ BOOKS|year=2008|isbn=1-4097-6995-X|pages=324–325}}</ref> [[Bismuth]]ides are not even wholly ionic; they are [[intermetallic compound]]s containing partially metallic and partially ionic bonds.<ref>{{cite web |url=http://xray.chem.ualberta.ca/mar/ |title=Welcome to Arthur Mar's Research Group |date=1999–2013 |work=University of Alberta |publisher=University of Alberta |accessdate=24 June 2013}}</ref><br />
<br />
=====Reaction with the chalcogens (alkali metal chalcogenides)=====<br />
{{see also|Alkali metal oxide}}<br />
{{double image|right|Rb9O2 cluster.png|150|Cs11O3 cluster.png|150|{{chem|Rb|9|O|2}} cluster, composed of two regular [[octahedron|octahedra]] connected to each other by one face|{{chem|Cs|11|O|3}} cluster, composed of three regular octahedra where each octahedron is connected to both of the others by one face each. All three octahedra have one edge in common.|The ball-and-stick diagram shows two regular octahedra which are connected to each other by one face. All nine vertices of the structure are purple spheres representing rubidium, and at the centre of each octahedron is a small red sphere representing oxygen.|The ball-and-stick diagram shows three regular octahedra where each octahedron is connected to both of the others by one face each. All three octahedra have one edge in common. All eleven vertices of the structure are violet spheres representing caesium, and at the centre of each octahedron is a small red sphere representing oxygen.}}<br />
All the alkali metals react vigorously with [[oxygen]] at standard conditions. They form various types of oxides, such as simple [[oxide]]s (containing the O<sup>2−</sup> ion), [[peroxide]]s (containing the {{chem|O|2|2-}} ion, where there is a [[single bond]] between the two oxygen atoms), [[superoxide]]s (containing the {{chem|O|2|-}} ion), and many others. Lithium burns in air to form [[lithium oxide]], but sodium reacts with oxygen to form a mixture of [[sodium oxide]] and [[sodium peroxide]]. Potassium forms a mixture of [[potassium peroxide]] and [[potassium superoxide]], while rubidium and caesium form the superoxide exclusively. Their reactivity increases going down the group: while lithium, sodium and potassium merely burn in air, rubidium and caesium are [[pyrophoric]] (spontaneously catch fire in air).<ref name="alkalireact"/><br />
<br />
The smaller alkali metals tend to polarise the more complex anions (the peroxide and superoxide) due to their small size. This attracts the electrons in the more complex anions towards one of its constituent oxygen atoms, forming an oxide ion and an oxygen atom. This causes lithium to form the oxide exclusively on reaction with oxygen at room temperature. This effect becomes drastically weaker for the larger sodium and potassium, allowing them to form the less stable peroxides. Rubidium and caesium, at the bottom of the group, are so large that even the least stable superoxides can form. Because the superoxide releases the most energy when formed, the superoxide is preferentially formed for the larger alkali metals where the more complex anions are not polarised. (The oxides and peroxides for these alkali metals do exist, but do not form upon direct reaction of the metal with oxygen at standard conditions.)<ref name="alkalireact"/> In addition, the small size of the Li<sup>+</sup> and O<sup>2−</sup> ions contributes to their forming a stable ionic lattice structure. Under controlled conditions, however, all the alkali metals, with the exception of francium, are known to form their oxides, peroxides, and superoxides. The alkali metal peroxides and superoxides are powerful [[oxidizing agent]]s. [[Sodium peroxide]] and [[potassium superoxide]] react with [[carbon dioxide]] to form the alkali metal carbonate and oxygen gas, which allows them to be used in [[submarine]] air purifiers; the presence of [[water vapour]], naturally present in breath, makes the removal of carbon dioxide by potassium superoxide even more efficient.<ref name=generalchemistry/><ref>{{cite journal |last1=Lindsay |first1=D. M. |last2=Garland |first2=D. A. |year=1987 |title=ESR spectra of matrix-isolated lithium superoxide |journal=The Journal of Physical Chemistry |volume=91 |issue=24 |pages=6158–61 |doi=10.1021/j100308a020}}</ref><br />
<br />
Rubidium and caesium can form even more complicated oxides than the superoxides. Rubidium can form Rb<sub>6</sub>O and Rb<sub>9</sub>O<sub>2</sub> upon oxidation in air, while caesium forms an immense variety of oxides, such as the [[ozonide]] CsO<sub>3</sub><ref>{{cite journal|doi =10.1007/BF00845494|title =Synthesis of cesium ozonide through cesium superoxide|year =1963|last1 =Vol'nov|first1 =I. I.|last2 =Matveev|first2 =V. V.|journal =Bulletin of the Academy of Sciences, USSR Division of Chemical Science|volume =12|pages =1040–1043|issue =6}}</ref><ref>{{cite journal|doi =10.1070/RC1971v040n02ABEH001903|title =Alkali and Alkaline Earth Metal Ozonides|year =1971|last1 =Tokareva|first1 =S. A.|journal =Russian Chemical Reviews|volume =40|pages =165–174|bibcode = 1971RuCRv..40..165T|issue =2}}</ref> and several brightly coloured [[suboxide]]s,<ref name=Simon>{{Cite journal|last = Simon|first = A.|title = Group 1 and 2 Suboxides and Subnitrides — Metals with Atomic Size Holes and Tunnels|journal = Coordination Chemistry Reviews |year = 1997|volume = 163|pages = 253–270|doi = 10.1016/S0010-8545(97)00013-1}}</ref> such as {{chem|Cs|7|O}}, {{chem|Cs|4|O}}, {{chem|Cs|11|O|3}}, {{chem|Cs|3|O}} (dark-green<ref>{{cite journal|doi =10.1021/j150537a023|year =1956|last1 =Tsai|first1 =Khi-Ruey|last2 =Harris|first2 =P. M.|last3 =Lassettre|first3 =E. N.|journal =Journal of Physical Chemistry|volume =60|pages =345–347|title=The Crystal Structure of Tricesium Monoxide|issue =3}}</ref>), CsO, {{chem|Cs|3|O|2}},<ref>{{cite journal|doi =10.1007/s11669-009-9636-5|title =Cs-O (Cesium-Oxygen)|year =2009|last1 =Okamoto|first1 =H.|journal =Journal of Phase Equilibria and Diffusion|volume =31|page =86}}</ref> as well as {{chem|Cs|7|O|2}}.<ref>{{cite journal|doi = 10.1021/jp036432o|title = Characterization of Oxides of Cesium|year = 2004|last1 = Band|first1 = A.|last2 = Albu-Yaron|first2 = A.|last3 = Livneh|first3 = T.|last4 = Cohen|first4 = H.|last5 = Feldman|first5 = Y.|last6 = Shimon|first6 = L.|last7 = Popovitz-Biro|first7 = R.|last8 = Lyahovitskaya|first8 = V.|last9 = Tenne|first9 = R.|displayauthors=9|journal = The Journal of Physical Chemistry B|volume = 108|pages = 12360–12367|issue = 33}}</ref><ref>{{cite journal|doi =10.1002/zaac.19472550110|title =Untersuchungen ber das System Cäsium-Sauerstoff|year =1947|last1 =Brauer|first1 =G.|journal =Zeitschrift für anorganische Chemie|volume =255|page =101}}</ref> The latter may be heated under vacuum to generate {{chem|Cs|2|O}}.<ref>{{cite web|url = http://pubs.usgs.gov/of/2004/1432/2004-1432.pdf|format = PDF|publisher = United States Geological Survey|accessdate =27 December 2009|title = Mineral Commodity Profile: Cesium|first1 = William C.|last1 = Butterman|first2 = William E.|last2 = Brooks|first3 = Robert G.|last3 = Reese, Jr.|year=2004}}</ref><br />
<br />
The alkali metals can also react analogously with the heavier chalcogens ([[sulfur]], [[selenium]], [[tellurium]], and [[polonium]]), and all the alkali metal chalcogenides are known (with the exception of francium's). Reaction with an excess of the chalcogen can similarly result in lower chalcogenides, with chalcogen ions containing chains of the chalcogen atoms in question. For example, sodium can react with sulfur to form the [[sulfide]] ([[sodium sulfide|Na<sub>2</sub>S]]) and various [[polysulfide]]s with the formula Na<sub>2</sub>S<sub>''x''</sub> (''x'' from 2 to 6), containing the {{chem|S|''x''|2-}} ions.<ref name=generalchemistry/> Due to the basicity of the Se<sup>2−</sup> and Te<sup>2−</sup> ions, the alkali metal [[selenide]]s and [[telluride (chemistry)|tellurides]] are alkaline in solution; when reacted directly with selenium and tellurium, alkali metal polyselenides and polytellurides are formed along with the selenides and tellurides with the {{chem|Se|''x''|2-}} and {{chem|Te|''x''|2-}} ions.<ref name="house2008">{{cite book|title = Inorganic chemistry| first = James E.|last = House| publisher = Academic Press| year = 2008| isbn = 0-12-356786-6| page = 524}}</ref> The alkali metal [[polonide]]s are all ionic compounds containing the Po<sup>2−</sup> ion; they are very chemically stable and can be produced by direct reaction of the elements at around 300–400&nbsp;°C.<ref name="Greenwood&Earnshaw"/><ref name="AEC-chem">{{Cite book | last = Moyer | first = Harvey V. | contribution = Chemical Properties of Polonium | pages = 33–96 | title = Polonium | url = http://www.osti.gov/bridge/servlets/purl/4367751-nEJIbm/ | editor-last = Moyer | editor-first = Harvey V. | id = TID-5221 | doi = 10.2172/4367751 | year = 1956 | location = Oak Ridge, Tenn. | publisher = United States Atomic Energy Commission | postscript = }}.</ref><ref name="Bagnall">{{Cite journal | first = K. W. | last = Bagnall | title = The Chemistry of Polonium | journal = Adv. Inorg. Chem. Radiochem. | year = 1962 | volume = 4 | pages = 197–229 | url = http://books.google.com/?id=8qePsa3V8GQC&pg=PA197#v=onepage&q&f=false | isbn = 978-0-12-023604-6 | doi = 10.1016/S0065-2792(08)60268-X | series = Advances in Inorganic Chemistry and Radiochemistry | postscript =}}.</ref><br />
<br />
=====Reaction with hydrogen and the halogens (alkali metal hydrides and halides)=====<br />
{{main|Alkali metal halide}}<br />
The alkali metals are among the most [[electropositive]] elements on the periodic table and thus tend to [[ionic bond|bond ionically]] to the most [[electronegative]] elements on the periodic table, the [[halogen]]s, forming [[salt (chemistry)|salts]] known as the alkali metal halides. This includes [[sodium chloride]], otherwise known as common salt. The reactivity becomes higher from lithium to caesium and drops from [[fluorine]] to [[iodine]]. All of the alkali metal halides have the formula MX where M is an alkali metal and X is a halogen. They are all white ionic crystalline solids.<ref name="rsc"/><ref name="alkalireact"/> All the alkali metal halides are [[solubility|soluble]] in water except for [[lithium fluoride]] (LiF), which is insoluble in water due to its very high [[lattice enthalpy]]. The high lattice enthalpy of lithium fluoride is due to the small sizes of the Li<sup>+</sup> and F<sup>−</sup> ions, causing the [[electrostatic interaction]]s between them to be strong.<ref name="rsc"/> The alkali metals also react similarly with hydrogen to form ionic alkali metal hydrides.<ref name="generalchemistry">{{cite book |last1=Averill |first1=Bruce A. |last2=Eldredge |first2=Patricia |title=Chemistry: Principles, Patterns, and Applications with Student Access Kit for Mastering General Chemistry |url=http://2012books.lardbucket.org/books/general-chemistry-principles-patterns-and-applications-v1.0/section_25_03.html |accessdate=24 June 2013 |year=2007 |publisher=Prentice Hall |edition=1st |isbn=9780805337990 |chapter=21.3: The Alkali Metals}}</ref><br />
<br />
=====Coordination complexes=====<br />
{{double image|right|18-crown-6-potassium-3D-balls-A.png|150|Cryptate of pottasium cation.jpg|150|[[18-crown-6]] coordinating a potassium ion|Structure of [[2.2.2-Cryptand]] encapsulating a potassium cation (purple). At crystalline state, obtained with an X-ray diffraction.<ref>{{cite journal | author = Alberto, R.; Ortner, K.; Wheatley, N.; Schibli, R.; Schubiger, A. P. | title = Synthesis and properties of boranocarbonate: a convenient in situ CO source for the aqueous preparation of [<sup>99m</sup>Tc(OH<sub>2</sub>)<sub>3</sub>(CO)<sub>3</sub>]<sup>+</sup> | journal = [[J. Am. Chem. Soc.]] | year = 2001 | volume = 121 | pages = 3135–3136 | doi = 10.1021/ja003932b | issue = 13}}</ref>}}<br />
Alkali metal cations do not usually form [[coordination complex]]es with simple [[Lewis base]]s due to their low charge of just +1 and their relatively large size; thus the Li<sup>+</sup> ion forms most complexes and the heavier alkali metal ions form less and less. In [[aqueous solution]], the alkali metal ions exist as octahedral hexahydrate complexes ([M(H<sub>2</sub>O)<sub>6</sub>)]<sup>+</sup>), with the exception of the lithium ion, which due to its small size forms tetrahedral tetrahydrate complexes ([Li(H<sub>2</sub>O)<sub>4</sub>)]<sup>+</sup>); the alkali metals form these complexes because their ions are attracted by electrostatic forces of attraction to the polar water molecules. Because of this, [[anhydrous]] salts containing alkali metal cations are often used as [[desiccant]]s.<ref name=generalchemistry/> Alkali metals also readily form complexes with [[crown ether]]s (e.g. [[12-crown-4]] for Li<sup>+</sup>, [[15-crown-5]] for Na<sup>+</sup>, and [[18-crown-6]] for K<sup>+</sup>) and [[cryptand]]s due to electrostatic attraction.<ref name=generalchemistry/><br />
<br />
=====Ammonia solutions=====<br />
Unlike most metals, the alkali metals dissolve slowly in liquid [[ammonia]], forming hydrogen gas and the [[metal amide#Alkali metal amides|alkali metal amide]] (MNH<sub>2</sub>, where M represents an alkali metal). The process may be speeded up by a [[catalyst]]. The amide salt is quite insoluble and readily precipitates out of solution, leaving intensely coloured ammonia solutions of the alkali metals. The colour is due to the presence of [[solvated electron]]s, which contribute to the high electrical conductivity of these solutions. At low concentrations (below 3 M), the solution is dark blue and has ten times the conductivity of aqueous [[sodium chloride]]; at higher concentrations (above 3 M), the solution is copper-coloured and has approximately the conductivity of liquid metals like [[mercury (element)|mercury]].<ref name="Greenwood&Earnshaw"/><ref name=generalchemistry/><ref name="c&w">{{cite book |last=Cotton |first=F.A. |first2=G.|last2=Wilkinson |title=Advanced Inorganic Chemistry |year=1972 |publisher=John Wiley and Sons Inc |location= |isbn=0-471-17560-9 }}</ref> In addition to the alkali metal amide salt and solvated electrons, such ammonia solutions also contain the alkali metal cation (M<sup>+</sup>), the neutral alkali metal atom (M), [[diatomic molecule|diatomic]] alkali metal molecules (M<sub>2</sub>) and alkali metal anions (M<sup>−</sup>). These are unstable and eventually become the more thermodynamically stable alkali metal amide and hydrogen gas. Solvated electrons are powerful [[reducing agent]]s and are often used in chemical synthesis.<ref name=generalchemistry/><br />
<br />
=====Organometallic chemistry=====<br />
[[File:Methyllithium-tetramer-2-3D-balls.png|thumb|right|200px|Structure of the methyllithium tetramer, (CH<sub>3</sub>Li)<sub>4</sub>]]<br />
Being the smallest alkali metal, lithium forms the widest variety of and most stable [[organometallic compound]]s, which are bonded covalently. [[Organolithium reagent|Organolithium]] compounds are electrically non-conducting volatile solids or liquids that melt at low temperatures, and tend to form [[oligomer]]s with the structure (RLi)<sub>''x''</sub> where R is the organic group. As the electropositive nature of lithium puts most of the [[charge density]] of the bond on the carbon atom, effectively creating a [[carbanion]], organolithium compounds are extremely powerful [[base (chemistry)|base]]s and [[carbon nucleophile|nucleophile]]s. For use as bases, [[butyllithium]]s are often used and are commercially available. An example of an organolithium compound is [[methyllithium]] ((CH<sub>3</sub>Li)<sub>''x''</sub>), which exists in tetrameric (''x'' = 4) and hexameric (''x'' = 6) forms.<ref name=generalchemistry/><ref name=Brown1957 >{{ cite journal | author = Brown, T. L.; Rogers, M. T. | title = The Preparation and Properties of Crystalline Lithium Alkyls | journal = Journal of the American Chemical Society | year = 1957 | volume = 79 | issue = 8 | pages = 1859–1861 | doi = 10.1021/ja01565a024 }}</ref><br />
<br />
The application of [[organosodium chemistry|organosodium]] compounds in chemistry is limited in part due to competition from [[organolithium compound]]s, which are commercially available and exhibit more convenient reactivity. The principal organosodium compound of commercial importance is [[sodium cyclopentadienide]]. [[Sodium tetraphenylborate]] can also be classified as an organosodium compound since in the solid state sodium is bound to the aryl groups. Organometallic compounds of the higher alkali metals are even more reactive than organosodium compounds and of limited utility. A notable reagent is [[Schlosser's base]], a mixture of [[n-Butyllithium|''n''-butyllithium]] and [[potassium tert-butoxide|potassium ''tert''-butoxide]]. This reagent reacts with [[propene]] to form the compound [[allylpotassium]] (KCH<sub>2</sub>CHCH<sub>2</sub>). [[cis-2-butene|''cis''-2-Butene]] and [[trans-2-butene|''trans''-2-butene]] equilibrate when in contact with alkali metals. Whereas [[isomerization]] is fast with lithium and sodium, it is slow with the higher alkali metals. The higher alkali metals also favor the [[steric hindrance|sterically]] congested conformation.<ref>{{cite journal | title = Superbases for organic synthesis | author = Manfred Schlosser | journal = Pure and Appl. Chem. | volume = 60 | issue = 11 | pages = 1627–1634 | year = 1988 | doi = 10.1351/pac198860111627}}</ref> Several crystal structures of organopotassium compounds have been reported, establishing that they, like the sodium compounds, are polymeric.<ref name=Klett>{{cite journal|doi=10.1002/ejic.201000983|title=Synthesis and Structures of \(Trimethylsilyl)methyl]sodium and -potassium with Bi- and Tridentate N-Donor Ligands|year=2011|last1=Clegg|first1=William|last2=Conway|first2=Ben|last3=Kennedy|first3=Alan R.|last4=Klett|first4=Jan|last5=Mulvey|first5=Robert E.|last6=Russo|first6=Luca|journal=European Journal of Inorganic Chemistry|volume=2011|issue=5|pages=721}}</ref> Organosodium, organopotassium, organorubidium and organocaesium compounds are all mostly ionic and are insoluble (or nearly so) in nonpolar solvents.<ref name=generalchemistry/><br />
<br />
===Physical===<br />
The alkali metals are all silver-coloured except for caesium, which has a golden tint.<ref name="theodoregray-caesium">{{cite web |url=http://www.theodoregray.com/periodictable/Elements/055/index.s7.html |title=Facts, pictures, stories about the element Cesium in the Periodic Table |author=[[Theodore Gray|Gray, Theodore]] |work=The Wooden Periodic Table Table |accessdate=13 January 2012}}</ref> All are soft and have low [[density|densities]],<ref name="rsc"/> [[melting point]]s,<ref name="rsc"/> and [[boiling point]]s.<ref name="rsc"/><br />
<br />
The table below is a summary of the key physical and atomic properties of the alkali metals. Data marked with question marks are either uncertain or are estimations partially based on [[periodic trends]] rather than observations.<!--cite the data pages--><br />
<br />
{| class="wikitable" style="text-align:center"<br />
!Alkali metal !!Standard [[atomic weight]]<br>([[unified atomic mass unit|u]]){{#tag:ref|The number given in [[bracket|parentheses]] refers to the [[Standard uncertainty|measurement uncertainty]]. This uncertainty applies to the [[significant figure|least significant figure]](s) of the number prior to the parenthesized value (ie. counting from rightmost digit to left). For instance, {{val|1.00794|(7)}} stands for {{val|1.00794|0.00007}}, while {{val|1.00794|(72)}} stands for {{val|1.00794|0.00072}}.<ref>{{cite web|url=http://physics.nist.gov/cgi-bin/cuu/Info/Constants/definitions.html|title=Standard Uncertainty and Relative Standard Uncertainty|work=[[CODATA]] reference|publisher=[[National Institute of Standards and Technology]]|accessdate=26 September 2011}}</ref>|group=note}}<ref name="atomicweights2007">{{cite journal |last1=Wieser |first1=Michael E. |last2=Berglund |first2=Michael |year=2009 |title=Atomic weights of the elements 2007 (IUPAC Technical Report) |journal=[[Pure and Applied Chemistry|Pure Appl. Chem.]] |volume=81 |issue=11 |pages= 2131–2156 |publisher=[[International Union of Pure and Applied Chemistry|IUPAC]] |doi=10.1351/PAC-REP-09-08-03 |url=http://iupac.org/publications/pac/pdf/2009/pdf/8111x2131.pdf |accessdate=7 February 2012 }}</ref><ref name="atomicweights2009">{{cite journal |last1=Wieser |first1=Michael E. |last2=Coplen |first2=Tyler B. |year=2011 |title=Atomic weights of the elements 2009 (IUPAC Technical Report) |journal=[[Pure and Applied Chemistry|Pure Appl. Chem.]] |volume=83 |issue=2 |pages=359–396 |publisher=[[International Union of Pure and Applied Chemistry|IUPAC]] |doi=10.1351/PAC-REP-10-09-14 |url=http://iupac.org/publications/pac/pdf/2011/pdf/8302x0359.pdf |accessdate=11 February 2012 }}</ref> !![[Melting point]] !![[Boiling point]]<ref name="RubberBible84th"/> !![[Density]]<br>(g/cm<sup>3</sup>) !![[Electronegativity]]<br>([[Pauling scale|Pauling]]) !! First [[ionization energy|ionisation energy]]<br>([[Kilojoule per mole|kJ·mol<sup>−1</sup>]]) !! [[Atomic radius]]<br>([[picometre|pm]]) !! colspan="2" | [[Flame test]] colour<br />
|-<br />
|[[Lithium]]||6.94(1){{#tag:ref|The value listed is the conventional value suitable for trade and commerce; the actual value may range from 6.938 to 6.997 depending on the isotopic composition of the sample.<ref name="atomicweights2009"/>|group=note}}||453.69&nbsp;[[Kelvin|K]],<br>180.54&nbsp;[[Celsius|°C]],<br>356.97&nbsp;[[Fahrenheit|°F]]||1615&nbsp;K,<br>1342&nbsp;°C,<br>2448&nbsp;°F||0.534||0.98||520.2||152||Red<ref name="rsc" /><ref name="flametests">{{cite web |url=http://www.chemguide.co.uk/inorganic/group1/flametests.html |title=Flame Tests |author=Clark, Jim |year=2005 |work=chemguide |accessdate=29 January 2012}}</ref>||[[File:FlammenfärbungLi.png|40px]]<br />
|-<br />
|[[Sodium]]||22.98976928(2)||370.87&nbsp;K,<br>97.72&nbsp;°C,<br>207.9&nbsp;°F||1156&nbsp;K,<br>883&nbsp;°C,<br>1621&nbsp;°F||0.968||0.93||495.8||186||Strong persistent orange or yellow<ref name="rsc" /><ref name="flametests" />||[[File:Flametest--Na.swn.jpg|40px]]<br />
|-<br />
|[[Potassium]]||39.0983(1)||336.53&nbsp;K,<br>63.38&nbsp;°C,<br>146.08&nbsp;°F||1032&nbsp;K,<br>759&nbsp;°C,<br>1398&nbsp;°F||0.89||0.82||418.8||227||[[Lilac (color)|Lilac]] or pink<ref name="rsc" /><ref name="flametests" />||[[File:FlammenfärbungK.png|40px]]<br />
|-<br />
|[[Rubidium]]||85.4678(3)||312.467&nbsp;K,<br>39.31&nbsp;°C,<br>102.76&nbsp;°F||961&nbsp;K,<br>688&nbsp;°C,<br>1270&nbsp;°F||1.532||0.82||403.0||248||Red or [[red-violet|reddish-violet]]<ref name="rsc" /><ref name="flametests" />||&nbsp;<br />
|-<br />
|[[Caesium]]||132.9054519(2)||301.59&nbsp;K,<br>28.44&nbsp;°C,<br>83.19&nbsp;°F||944&nbsp;K,<br>671&nbsp;°C,<br>1240&nbsp;°F||1.93||0.79||375.7||265||Blue or violet<ref name="rsc" /><ref name="flametests" />||&nbsp;<br />
|-<br />
|[[Francium]]||[223]{{#tag:ref|The element does not have any stable [[nuclide]]s, and a value in brackets indicates the [[mass number]] of the longest-lived [[isotope]] of the element.<ref name="atomicweights2007"/><ref name="atomicweights2009"/>|group=note}}||?&nbsp;300&nbsp;K,<br>?&nbsp;27&nbsp;°C,<br>?&nbsp;80&nbsp;°F||?&nbsp;950&nbsp;K,<br>?&nbsp;677&nbsp;°C,<br>?&nbsp;1250&nbsp;°F<ref name="apsidium-Fr">{{cite web |url=http://www.apsidium.com/elements/087.htm |archiveurl=http://web.archive.org/web/20080509155648/http://www.apsidium.com/elements/087.htm |archivedate=9 May 2008 |title=Francium |author=Klehr, Wolfram |date=21 May 2007 |work=apsidium.com |accessdate=25 April 2012}}</ref>||?&nbsp;1.87||?&nbsp;0.7||380||?||?||&nbsp;<br />
|}<br />
<br />
====Periodic trends====<br />
{{see also|Periodic trends}}<!--please do not change the layout without a preview, because the current layout (on 1152 x 864, at least) minimises whitespace and is better for understanding (because the tables and graphs can be seen at the opposite sides)--><br />
The alkali metals are more similar to each other than the elements in any other [[group (periodic table)|group]] are to each other.<ref name="rsc" /> For instance, when moving down the table, all known alkali metals show increasing [[atomic radius]],<ref name="chemguide"/> decreasing [[electronegativity]],<ref name="chemguide">{{cite web |url=http://www.chemguide.co.uk/inorganic/group1/properties.html |title=Atomic and Physical Properties of the Group 1 Elements |author=Clark, Jim |year=2005 |work=chemguide |accessdate=30 January 2012}}</ref> increasing [[Reactivity (chemistry)|reactivity]],<ref name="rsc" /> and decreasing melting and boiling points.<ref name="chemguide" /> In general, their [[density|densities]] increase when moving down the table, with the exception that potassium is less dense than sodium.<ref name="chemguide" /><br />
<br />
=====Atomic and ionic radii=====<br />
{{main|Atomic radius}}<br />
[[File:Effective Nuclear Charge.svg|thumb|250px|[[Effective nuclear charge]] on an atomic electron]]<br />
<div style="float: left; margin: 1px; font-size:85%;"><br />
:{| class="wikitable sortable"<br />
|+ [[Atomic radius|Atomic]] and [[ionic radius|ionic radii]] of the alkali metals<ref name="rsc"/><ref group="note">The values are in [[picometre]]s (pm). The shade of the box ranges from red to yellow as the radius increases. The atomic and ionic radii are displayed on the same scale of colour.</ref><br />
! Alkali metal<br />
! Atomic radius<br>([[picometre|pm]])<br />
! Ionic radius<br>([[picometre|pm]])<br />
|-<br />
| [[Lithium]]<br />
| style="background:#ff9200;"| <center>152</center><br />
| style="background:#ff4100;"| <center>68</center><br />
|-<br />
| [[Sodium]]<br />
| style="background:#ffb200;"| <center>186</center><br />
| style="background:#ff5e00;"| <center>98</center><br />
|-<br />
| [[Potassium]]<br />
| style="background:#ffda00;"| <center>227</center><br />
| style="background:#ff7f00;"| <center>133</center><br />
|-<br />
| [[Rubidium]]<br />
| style="background:#fe0;"| <center>248</center><br />
| style="background:#ff8e00;"| <center>148</center><br />
|-<br />
| [[Caesium]]<br />
| style="background:#ff0;"| <center>265</center><br />
| style="background:#ffa000;"| <center>167</center><br />
|}<!--Francium is predicted to be 270 pm (apsidium), if needed--><br />
</div><br />
The atomic radii of the alkali metals increase going down the group.<ref name="chemguide"/> Because of the [[shielding effect]], when an atom has more than one [[electron shell]], each electron feels electric repulsion from the other electrons as well as electric attraction from the nucleus.<ref name=shielding>{{cite book|first=Theodore|last=L. Brown|first2=H. Eugene |last2=LeMay, Jr.|first3=Bruce E.|last3=Bursten|first4=Julia R. |last4=Burdge|year=2003|title=Chemistry: The Central Science|edition=8th|publisher=Pearson Education|location=US|isbn=0-13-061142-5|url=http://www.pearsoneducation.net/brown}}{{dead link|date=May 2012}}</ref> In the alkali metals, the [[valence electron|outermost electron]] only feels a net charge of +1, as some of the [[nuclear charge]] (which is equal to the [[atomic number]]) is cancelled by the inner electrons; the number of inner electrons of an alkali metal is always one less than the nuclear charge. Therefore, the only factor which affects the atomic radius of the alkali metals is the number of electron shells. Since this number increases down the group, the atomic radius must also increase down the group.<ref name="chemguide"/><br />
<br />
The [[ionic radius|ionic radii]] of the alkali metals are much smaller than their atomic radii. This is because the outermost electron of the alkali metals is in a different [[electron shell]] than the inner electrons, and thus when it is removed the resulting atom has one fewer electron shell and is smaller. Additionally, the [[effective nuclear charge]] has increased, and thus the electrons are attracted more strongly towards the nucleus and the ionic radius decreases.<ref name="rsc"/><br />
{{clear}}<br />
<br />
=====First ionisation energy=====<br />
{{main|Ionisation energy}}<br />
[[File:First Ionization Energy.svg|thumb|300px|right|Periodic trend for ionisation energy: each period begins at a minimum for the alkali metals, and ends at a maximum for the [[noble gas]]es.]]<br />
<div style="float: left; margin: 1px; font-size:85%;"><br />
:{| class="wikitable sortable"<br />
|+ First ionisation energies of the alkali metals<ref name="huheey">J.E. Huheey, E.A. Keiter, and R.L. Keiter in ''Inorganic Chemistry : Principles of Structure and Reactivity'', 4th edition, HarperCollins, New York, USA, 1993.</ref><ref name="macmillan">A.M. James and M.P. Lord in ''Macmillan's Chemical and Physical Data'', Macmillan, London, UK, 1992.</ref><!--change these to use the cite book template--><ref group="note">The shade of the box ranges from red to yellow as the ionisation energy decreases.</ref> <br />
! Alkali metal<br />
! First<br>ionisation energy<br>([[Kilojoule per mole|kJ/mol]])<br />
|-<br />
| [[Lithium]]<br />
| style="background:#ffb800;"| <center>520.2</center><br />
|-<br />
| [[Sodium]]<br />
| style="background:#ffc100;"| <center>495.8</center><br />
|-<br />
| [[Potassium]]<br />
| style="background:#ffe500;"| <center>418.8</center><br />
|-<br />
| [[Rubidium]]<br />
| style="background:#fe0;"| <center>403.0</center><br />
|-<br />
| [[Caesium]]<br />
| style="background:#ff0;"| <center>375.7</center><br />
|-<br />
| [[Francium]]<br />
| style="background:#fffc00;"| <center>380{{#tag:ref|A different source gives 4.0712 ± 0.00004&nbsp;[[electron volt|eV]] (392.811(4)&nbsp;kJ/mol).<ref name="andreev">{{cite journal | author = Andreev, S.V.; Letokhov, V.S.; Mishin, V.I., | title = Laser resonance photoionization spectroscopy of Rydberg levels in Fr | url = http://link.aps.org/abstract/PRL/v59/p1274 | journal = [[Physical Review Letters|Phys. Rev. Lett.]] | year = 1987 | volume = 59 | pages = 1274–76 | doi = 10.1103/PhysRevLett.59.1274 | pmid=10035190 | bibcode=1987PhRvL..59.1274A | issue = 12}}</ref> |name=Fr-ionisation|group=note}}</center><br />
|}<br />
</div><br />
<br />
The first ionisation energy of an [[chemical element|element]] or [[molecule]] is the energy required to move the most loosely held electron from one [[mole (unit)|mole]] of gaseous atoms of the element or molecules to form one mole of gaseous ions with [[electric charge]] +1. The factors affecting the first ionisation energy are the [[nuclear charge]], the amount of [[shielding effect|shielding]] by the inner electrons and the distance from the most loosely held electron from the nucleus, which is always an outer electron in [[main group element]]s. The first two factors change the effective nuclear charge the most loosely held electron feels. Since the outermost electron of alkali metals always feel the same effective nuclear charge (+1), the only factor which affects the first ionisation energy is the distance from the outermost electron to the nucleus. Since this distance increases down the group, the outermost electron feels less attraction from the nucleus and thus the first ionisation energy decreases.<ref name="chemguide"/> (This trend is broken in francium due to the [[relativistic quantum chemistry|relativistic]] stabilization and contraction of the 7s orbital, bringing francium's valence electron closer to the nucleus than would be expected from non-relativistic calculations. This makes francium's outermost electron feel more attraction from the nucleus, increasing its first ionisation energy slightly beyond that of caesium.)<ref name="Uue"/>{{Rp|1729}}<!--Also explain why the alkali metals have the lowest ionization energies in their period.--><br />
<br />
The second ionisation energy of the alkali metals is much higher than the first as the second-most loosely held electron is part of a fully filled [[electron shell]] and is thus difficult to remove.<ref name="rsc"/><br />
{{clear}}<br />
<br />
=====Reactivity=====<br />
{{main|Reactivity (chemistry)}}<br />
The reactivities of the alkali metals increase going down the group. This is the result of a combination of two factors: the first ionisation energies and [[atomisation energy|atomisation energies]] of the alkali metals. Because the first ionisation energy of the alkali metals decreases down the group, it is easier for the outermost electron to be removed from the atom and participate in [[chemical reaction]]s, thus increasing reactivity down the group. The atomisation energy measures the strength of the [[metallic bond]] of an element, which falls down the group as the atoms increase in [[atomic radius|radius]] and thus the metallic bond must increase in length, making the delocalised electrons further away from the attraction of the nuclei of the heavier alkali metals. Adding the atomisation and first ionisation energies gives a quantity closely related to (but not equal to) the [[activation energy]] of the reaction of an alkali metal with another substance. This quantity decreases going down the group, and so does the activation energy; thus, chemical reactions can occur faster and the reactivity increases down the group.<ref name="alkaliwater"/><br />
{{clear}}<br />
<br />
=====Electronegativity=====<br />
{{main|Electronegativity}}<br />
[[File:Periodic variation of Pauling electronegativities.png|thumb|300px|right|The variation of Pauling electronegativity (y-axis) as one descends the [[main group element|main groups]] of the periodic table from the [[period 2 element|second]] to the [[period 6 element|sixth period]]]]<br />
<div style="float: left; margin: 1px; font-size:85%;"><br />
:{| class="wikitable sortable"<br />
|+ [[Electronegativity|Electronegativities]] of the alkali metals<ref name="RubberBible84th"/><ref group="note">The shade of the box ranges from red to yellow as the electronegativity decreases.</ref><br />
! Alkali metal<br />
! Electronegativity<br />
|-<br />
| [[Lithium]]<br />
| style="background:#ffe900;"| <center>0.98</center><br />
|-<br />
| [[Sodium]]<br />
| style="background:#ffed00;"| <center>0.93</center><br />
|-<br />
| [[Potassium]]<br />
| style="background:#fff600;"| <center>0.82</center><br />
|-<br />
| [[Rubidium]]<br />
| style="background:#fff600;"| <center>0.82</center><br />
|-<br />
| [[Caesium]]<br />
| style="background:#fff800;"| <center>0.79</center><br />
|-<br />
| [[Francium]]<br />
| style="background:#ff0;"| <center>? 0.7{{#tag:ref|[[Linus Pauling]] estimated the electronegativity of francium at 0.7 on the [[Pauling scale]], the same as caesium;<ref>{{cite book| last = Pauling| first = Linus| title = The Nature of the Chemical Bond|edition = Third| authorlink = Linus Pauling| publisher = Cornell University Press| year = 1960| isbn = 978-0-8014-0333-0| page = 93}}</ref> the value for caesium has since been refined to 0.79, although there are no experimental data to allow a refinement of the value for francium.<ref>{{cite journal |author= Allred, A. L. |year= 1961 |journal= J. Inorg. Nucl. Chem.|volume= 17 |issue= 3–4 |pages= 215–221 |title= Electronegativity values from thermochemical data |doi= 10.1016/0022-1902(61)80142-5}}</ref> Francium has a slightly higher ionization energy than caesium,<ref name="andreev"/> 392.811(4)&nbsp;kJ/mol as opposed to 375.7041(2)&nbsp;kJ/mol for caesium, as would be expected from [[Relativistic quantum chemistry|relativistic effects]], and this would imply that caesium is the less electronegative of the two.|name=Fr-electronegativity|group=note}}</center><br />
|}<br />
</div><br />
<br />
Electronegativity is a [[chemical property]] that describes the tendency of an [[atom]] or a [[functional group]] to attract [[electron]]s (or [[electron density]]) towards itself.<ref name="definition">{{GoldBookRef|file=E01990|title=Electronegativity}}</ref> If the bond between [[sodium]] and [[chlorine]] in [[sodium chloride]] were [[covalent bond|covalent]], the pair of shared electrons would be attracted to the chlorine because the effective nuclear charge on the outer electrons is +7 in chlorine but is only +1 in sodium. The electron pair is attracted so close to the chlorine atom that they are practically transferred to the chlorine atom (an [[ionic bond]]). However, if the sodium atom was replaced by a lithium atom, the electrons will not be attracted as close to the chlorine atom as before because the lithium atom is smaller, making the electron pair more strongly attracted to the closer effective nuclear charge from lithium. Hence, the larger alkali metal atoms (further down the group) will be less electronegative as the bonding pair is less strongly attracted towards them.<ref name="chemguide"/><br />
<br />
Because of the higher electronegativity of lithium, some of its compounds have a more covalent character. For example, [[lithium iodide]] ([[lithium|Li]][[iodine|I]]) will dissolve in [[organic solvent]]s, a property of most covalent compounds.<ref name="chemguide"/> [[Lithium fluoride]] (Li[[fluorine|F]]) is the only [[alkali halide]] that is not soluble in water,<ref name="rsc"/> and [[lithium hydroxide]] (Li[[hydroxide|OH]]) is the only alkali metal hydroxide that is not [[deliquescent]].<ref name="rsc"/><br />
{{clear}}<br />
<br />
=====Melting and boiling points=====<br />
{{main|Melting point|Boiling point}}<br />
<div style="float: left; margin: 1px; font-size:85%;"><br />
:{| class="wikitable sortable"<br />
|+ Melting and boiling points of the alkali metals<ref name="RubberBible84th"/><ref group="note">The shade of the box ranges from red to yellow as the melting and boiling points decrease. The melting and boiling points are displayed on different scales of colour.</ref><br />
! Alkali metal<br />
! Melting point<br />
! Boiling point<ref name="RubberBible84th"/><br />
|-<br />
| [[Lithium]]<br />
| style="background:#ffa800;"| <center>{{sort|454|453.69 K (180.54 °C)}}</center><br />
| style="background:#ff9500;"| <center>{{sort|1615|1615 K (1342 °C)}}</center><br />
|-<br />
| [[Sodium]]<br />
| style="background:#ffce00;"| <center>{{sort|371|370.87 K (97.72 °C)}}</center><br />
| style="background:#ffd000;"| <center>{{sort|1156|1156 K (883 °C)}}</center><br />
|-<br />
| [[Potassium]]<br />
| style="background:#ffe300;"| <center>{{sort|337|336.53 K (63.38 °C)}}</center><br />
| style="background:#ffe900;"| <center>{{sort|1032|1032 K (759 °C)}}</center><br />
|-<br />
| [[Rubidium]]<br />
| style="background:#fff500;"| <center>{{sort|312|312.46 K (39.31 °C)}}</center><br />
| style="background:#fffa00;"| <center>{{sort|0961|961 K (688 °C)}}</center><br />
|-<br />
| [[Caesium]]<br />
| style="background:#fffd00;"| <center>{{sort|302|301.59 K (28.44 °C)}}</center><br />
| style="background:#ff0;"| <center>{{sort|0944|944 K (671 °C)}}</center><br />
|-<br />
| [[Francium]]<br />
| style="background:#ff0;"| <center>{{sort|300|? 300 K (? 27 °C)}}{{#tag:ref|Francium's melting point was claimed to have been calculated to be around 27&nbsp;°C (80&nbsp;°F, 300&nbsp;K).<ref name="losalamos">{{cite web| title = Francium| publisher = Los Alamos National Laboratory|date = 15 December 2003| url = http://periodic.lanl.gov/87.shtml|accessdate =19 February 2012}}</ref> However, the melting point is uncertain because of the element's extreme rarity and radioactivity. Thus, the estimated boiling point value of 677&nbsp;°C (1250&nbsp;°F, 950&nbsp;K) is also uncertain. Because radioactive elements give off heat, francium would almost certainly be a liquid at [[standard temperature and pressure|standard conditions]]{{Vague|which set of conditions?|date=March 2012}} if enough were to be produced.|name=Fr-melt|group=note}}</center><br />
| style="background:#fffd00;"| <center>{{sort|0950|? 950 K (? 677 °C)}}<ref name="apsidium-Fr"/><ref name="Fr-melt" group="note"/></center><br />
|}<br />
</div><br />
The melting point of a substance is the point where it changes [[states of matter|state]] from [[solid]] to [[liquid]] while the boiling point of a substance (in liquid state) is the point where the [[vapor pressure]] of the liquid equals the environmental pressure surrounding the liquid<ref>{{cite book|author=David. E. Goldberg|title=3,000 Solved Problems in Chemistry|edition=1st|publisher=McGraw-Hill|year=1988|isbn=0-07-023684-4}} Section 17.43, page 321</ref><ref>{{cite book|author=Louis Theodore, R. Ryan Dupont and Kumar Ganesan (Editors)|title=Pollution Prevention: The Waste Management Approach to the 21st Century|publisher=CRC Press|year=1999|isbn=1-56670-495-2}} Section 27, p. 15</ref> and all the liquid changes state to [[gas]]. As a metal is heated to its melting point, the [[metallic bond]]s keeping the atoms in place weaken so that the atoms can move around, and the metallic bonds eventually break completely at the metal's boiling point.<ref name="chemguide"/><ref name="metallic-bonding">{{cite web |url=http://www.chemguide.co.uk/atoms/bonding/metallic.html |title=Metallic Bonding |author=Clark, Jim |year=2000 |work=chemguide |accessdate=23 March 2012}}</ref> Therefore, the falling melting and boiling points of the alkali metals indicate that the strength of the metallic bonds of the alkali metals decreases down the group.<ref name="chemguide"/> This is because metal atoms are held together by the electromagnetic attraction from the positive ions to the delocalised electrons.<ref name="chemguide"/><ref name="metallic-bonding"/> As the atoms increase in size going down the group (because their atomic radius increases), the nuclei of the ions move further away from the delocalised electrons and hence the metallic bond becomes weaker so that the metal can more easily melt and boil, thus lowering the melting and boiling points.<ref name="chemguide"/> (The increased nuclear charge is not a relevant factor due to the shielding effect.)<ref name="chemguide"/><br />
{{clear}}<br />
<br />
=====Density=====<br />
{{main|Density}}<br />
<div style="float: left; margin: 1px; font-size:85%;"><br />
:{| class="wikitable sortable"<br />
|+ [[Density|Densities]] of the alkali metals<ref name="RubberBible84th"/><ref group="note">The shade of the box ranges from red to yellow as the density increases.</ref><br />
! Alkali metal<br />
! Density (g/cm<sup>3</sup>)<br />
|-<br />
| [[Lithium]]<br />
| style="background:#ff4500;"| <center>0.534</center><br />
|-<br />
| [[Sodium]]<br />
| style="background:#ff7f00;"| <center>0.968</center><br />
|-<br />
| [[Potassium]]<br />
| style="background:#ff7500;"| <center>0.89</center><br />
|-<br />
| [[Rubidium]]<br />
| style="background:#ffca00;"| <center>1.532</center><br />
|-<br />
| [[Caesium]]<br />
| style="background:#ffff00;"| <center>1.93</center><br />
|-<br />
| [[Francium]]<br />
| style="background:#fff700;"| <center>{{sort|1.87|? 1.87}}</center><br />
|}<br />
</div><br />
The alkali metals all have the same [[crystal structure]] ([[body-centered cubic|body-centred cubic]])<ref name="Greenwood&Earnshaw"/> and thus the only relevant factors are the number of atoms that can fit into a certain volume and the mass of one of the atoms, since density is defined as mass per unit volume. The first factor depends on the volume of the atom and thus the atomic radius, which increases going down the group; thus, the volume of an alkali metal atom increases going down the group. The mass of an alkali metal atom also increases going down the group. Thus, the trend for the densities of the alkali metals depends on their atomic weights and atomic radii; if figures for these two factors are known, the ratios between the densities of the alkali metals can then be calculated. The resultant trend is that the densities of the alkali metals increase down the table, with an exception at potassium. Due to having the lowest atomic weight of all the elements in their period and having the largest atomic radius for their periods, the alkali metals are the least dense metals in the periodic table.<ref name="chemguide"/> Lithium, sodium, and potassium are the only three metals in the periodic table that are less dense than water.<ref name="rsc"/><br />
{{clear}}<br />
<br />
===Nuclear===<br />
<div style="float: right; margin: 5px;"><br />
{|class="sortable wikitable" align="top" style="text-align:center"<br />
|+Primordial isotopes of the alkali metals<br />
|-<br />
! Z<br><br />
! Alkali metal<br><br />
! <small>[[stable isotope|Stable]]</small><br><br />
! <small>''[[primordial element|Decays]]''</small><br><br />
! class="unsortable" colspan="3"|<small>''unstable: italics''<div style="background:pink">odd-odd isotopes coloured pink</div></small><br />
|-<br />
| 3 ||[[lithium]] || [[isotopes of lithium|2]] || — || {{SimpleNuclide|lithium|7}}||bgcolor="pink"|{{SimpleNuclide|lithium|6}}||&nbsp;<br />
|-<br />
| 11 ||[[sodium]] || [[isotopes of sodium|1]] || — ||{{SimpleNuclide|sodium|23}}||&nbsp;||&nbsp;<br />
|-<br />
| 19 ||[[potassium]] || [[isotopes of potassium|2]] || 1 ||{{SimpleNuclide|potassium|39}}||{{SimpleNuclide|potassium|41}}||bgcolor="pink"|''{{SimpleNuclide|potassium|40}}''<br />
|-<br />
| 37 ||[[rubidium]] || [[isotopes of rubidium|1]] || 1 ||{{SimpleNuclide|rubidium|85}}||''{{SimpleNuclide|rubidium|87}}''||&nbsp;<br />
|-<br />
| 55 ||[[caesium]] || [[isotopes of caesium|1]] || — ||{{SimpleNuclide|caesium|133}}||&nbsp;||&nbsp;<br />
|-<br />
| 87 ||[[francium]] || [[isotopes of francium|—]] || — ||colspan="3"|''No primordial isotopes''<br />
|}</div><br />
All the alkali metals have odd atomic numbers; hence, their isotopes must be either [[odd-odd nuclei|odd-odd]] (both proton and [[neutron number]] are odd) or [[odd-even nuclei|odd-even]] ([[proton number]] is odd, but neutron number is even). Odd-odd nuclei have even [[mass number]]s, while odd-even nuclei have odd mass numbers. Odd-odd [[primordial nuclide]]s are rare because most odd-odd nuclei are highly unstable with respect to [[beta decay]], because the decay products are even-even, and are therefore more strongly bound, due to [[Semi-empirical mass formula#Pairing term|nuclear pairing effects]].<ref name="Lide02">{{cite book | author=Various authors|editor-last=Lide |editor-first=David R. | year=2002 | title=Handbook of Chemistry & Physics | edition=88th | publisher=CRC | url=http://www.hbcpnetbase.com/ | accessdate=2008-05-23 | isbn=0-8493-0486-5 | oclc=179976746 }}</ref><br />
<br />
Due to the great rarity of odd-odd nuclei, almost all the primordial isotopes of the alkali metals are odd-even (the exceptions being the light stable isotope lithium-6 and the long-lived [[radioisotope]] potassium-40). For a given odd mass number, there can be only a single [[beta-decay stable isobars|beta-stable nuclide]], since there is not a difference in binding energy between even-odd and odd-even comparable to that between even-even and odd-odd, leaving other nuclides of the same mass number ([[isobar (nuclide)|isobar]]s) free to [[beta decay]] toward the lowest-mass nuclide. An effect of the instability of an odd number of either type of nucleons is that odd-numbered elements, such as the alkali metals, tend to have fewer stable isotopes than even-numbered elements. Of the 26 [[monoisotopic element]]s that have only a single stable isotope, all but one have an odd atomic number and all but one also have an even number of neutrons. [[Beryllium]] is the single exception to both rules, due to its low atomic number.<ref name="Lide02"/><br />
<br />
All of the alkali metals except lithium and caesium have at least one naturally occurring [[radioisotope]]: [[sodium-22]] and [[sodium-24]] are [[trace radioisotope]]s produced [[cosmogenic]]ally,<ref>{{cite web |url=http://www.nucleonica.net/unc.aspx |title=Universal Nuclide Chart |date=2007&ndash;2012 |work=Nucleonica |publisher=Institute for Transuranium Elements |accessdate=2011-04-17}}</ref> potassium-40 and [[rubidium-87]] have very long [[half-life|half-lives]] and thus occur naturally,<ref name="nuclideschart"/> and all [[isotopes of francium]] are [[radioactive decay|radioactive]].<ref name="nuclideschart"/> Caesium was also thought to be radioactive in the early 20th century,<ref name="Patt1926">{{cite journal | doi = 10.1021/cr60009a003 | title = The Radioactivity of the Alkali Metals | year = 1926 | last1 = Patton | first1 = I. Jocelyn | last2 = Waldbauer | first2 = L. J. | journal = Chemical Reviews | volume = 3 | page = 81}}</ref><ref name="Kenn1908">{{cite journal | doi =10.1080/14786440908636519 | title = On the radioactivity of potassium and other alkali metals | year = 1908 | last1 = McLennan | first1 = J. C. | last2 = Kennedy | first2 = W. T. | journal = Philosophical Magazine | series = 6 | volume = 16 | issue = 93 | pages = 377–395}}</ref> although it has no naturally occurring radioisotopes.<ref name="nuclideschart">{{cite web|url=http://www.nndc.bnl.gov/chart/ |title=Interactive Chart of Nuclides|publisher=Brookhaven National Laboratory|author=Sonzogni, Alejandro|location=National Nuclear Data Center|accessdate=4 October 2012}}</ref> (Francium had not been discovered yet at that time.) The natural radioisotope of potassium, potassium-40, makes up about 0.012% of natural potassium,<ref>{{cite web |url=http://www.ead.anl.gov/pub/doc/potassium.pdf |title=Potassium-40 |date=August 2005 |work=Human Health Fact Sheet |publisher=[[Argonne National Laboratory]], Environmental Science Division |accessdate=7 February 2012}}</ref> and thus natural potassium is weakly radioactive. This natural radioactivity became a basis for a mistaken claim of the discovery for element 87 (the next alkali metal after caesium) in 1925.<ref name="fontani" /><ref name="vanderkrogt-Fr">{{cite web| last = Van der Krogt| first = Peter| title = Francium| work = Elementymology & Elements Multidict| date = 10 January 2006| url = http://elements.vanderkrogt.net/element.php?sym=Fr| accessdate =8 April 2007}}</ref><br />
<br />
[[Caesium-137]], with a half-life of 30.17&nbsp;years, is one of the two principal [[medium-lived fission product]]s, along with [[strontium-90]], which are responsible for most of the [[radioactivity]] of [[spent nuclear fuel]] after several years of cooling, up to several hundred years after use. It constitutes most of the radioactivity still left from the [[Chernobyl accident]]. <sup>137</sup>Cs undergoes high-energy beta decay and eventually becomes stable [[barium-137]]. It is a strong emitter of gamma radiation. <sup>137</sup>Cs has a very low rate of neutron capture and cannot be feasibly disposed of in this way, but must be allowed to decay.<ref name="Cs-137">{{cite web|title=Radionuclide Half-Life Measurements|url=http://www.nist.gov/pml/data/halflife-html.cfm|author=National Institute of Standards and Technology|accessdate=2011-11-07}}</ref> <sup>137</sup>Cs has been used as a [[Flow tracer|tracer]] in hydrologic studies, analogous to the use of [[tritium]].<ref>http://www.bt.cdc.gov/radiation/isotopes/cesium.asp</ref> Small amounts of [[caesium-134]] and caesium-137 were released into the environment during nearly all [[nuclear weapon test]]s and some [[nuclear accident]]s, most notably the [[Goiânia accident]] and the [[Chernobyl disaster]]. As of 2005, caesium-137 is the principal source of radiation in the [[zone of alienation]] around the [[Chernobyl nuclear power plant]].<ref name="IAEA">{{cite book |title=The Radiological Accident in Goiânia |publisher=[[IAEA]] |year=1988 |url=http://www-pub.iaea.org/MTCD/publications/PubDetAR.asp?pubId=3684}}</ref><br />
<br />
==Extensions==<br />
{{see also|Ununennium}}<br />
[[File:Atomic radius of alkali metals and alkaline earth metals.svg|thumb|right|250px|[[Empirical]] (Na–Cs, Mg–Ra) and predicted (Fr–Uhp, Ubn–Uhh) atomic radius of the alkali and alkaline earth metals from the [[period 3 element|third]] to the [[period 9 element|ninth period]], measured in [[angstrom]]s<ref name="Uue"/>{{rp|1730}}<ref name="pyykko"/>]]<br />
Although francium is the heaviest alkali metal that has been discovered, there has been some theoretical work predicting the physical and chemical characteristics of the hypothetical heavier alkali metals. Being the first [[period 8 element]], the undiscovered element [[ununennium]] (element 119) is predicted to be the next alkali metal after francium and behave much like their lighter [[Congener (chemistry)|congener]]s; however, it is also predicted to differ from the lighter alkali metals in some properties.<ref name="Uue"/>{{rp|1729–1730}} Its chemistry is predicted to be closer to that of potassium<ref name=EB/> or rubidium<ref name="Uue"/>{{rp|1729–1730}} instead of caesium or francium. This is unusual as [[periodic trends]], ignoring relativistic effects would predict ununennium to be even more reactive than caesium and francium. This lowered [[reactivity (chemistry)|reactivity]] is due to the relativistic stabilisation of ununennium's valence electron, increasing ununennium's first ionisation energy and decreasing the [[metallic radius|metallic]] and [[ionic radius|ionic radii]];<ref name="EB"/> this effect is already seen for francium.<ref name="Uue"/>{{rp|1729–1730}} This assumes that ununennium will behave chemically as an alkali metal, which, although likely, may not be true due to relativistic effects.<ref name="tanm"/> The relativistic stabilisation of the 8s orbital also increases ununennium's [[electron affinity]] far beyond that of caesium and francium; indeed, ununennium is expected to have an electron affinity higher than all the alkali metals lighter than it. Relativistic effects also cause a very large drop in the [[polarisability]] of ununennium.<ref name="Uue"/>{{rp|1729–1730}} On the other hand, ununennium is predicted to continue the trend of melting points decreasing going down the group, being expected to have a melting point between 0&nbsp;°C and 30&nbsp;°C.<ref name="Uue"/>{{rp|1724}}<br />
<br />
[[File:Electron affinity of alkali metals.svg|thumb|left|200px|Empirical (Na–Fr) and predicted (Uue) electron affinity of the alkali metals from the third to the [[period 8 element|eighth period]], measured in [[electron volt]]s<ref name="Uue"/>{{rp|1730}}<ref name="pyykko"/>]]<br />
The stabilisation of ununennium's valence electron and thus the contraction of the 8s orbital cause its atomic radius to be lowered to 240&nbsp;[[picometer|pm]],<ref name="Uue"/>{{rp|1729–1730}} very close to that of rubidium (247&nbsp;pm),<ref name="rsc"/> so that the chemistry of ununennium in the +1 oxidation state should be more similar to the chemistry of rubidium than to that of francium. On the other hand, the ionic radius of the Uue<sup>+</sup> ion is predicted to be larger than that of Rb<sup>+</sup>, because the 7p orbitals are destabilised and are thus larger than the p-orbitals of the lower shells. Ununennium may also show the +3 [[oxidation state]],<ref name="Uue"/>{{rp|1729–1730}} which is not seen in any other alkali metal,<ref name="Greenwood&Earnshaw"/>{{rp|28}} in addition to the +1 oxidation state that is characteristic of the other alkali metals and is also the main oxidation state of all the known alkali metals: this is because of the destabilisation and expansion of the 7p<sub>3/2</sub> spinor, causing its outermost electrons to have a lower ionisation energy than what would otherwise be expected.<ref name="Greenwood&Earnshaw"/>{{rp|28}}<ref name="Uue"/>{{rp|1729–1730}} Indeed, many ununennium compounds are expected to have a large [[covalent]] character, due to the involvement of the 7p<sub>3/2</sub> electrons in the bonding.<ref name="Thayer">{{cite book|title=Chemistry of heavier main group elements|last=Thayer|first=John S.|doi=10.1007/9781402099755_2|year=2010|pages=81, 84}}</ref><br />
<br />
[[File:Ionization energy of alkali metals and alkaline earth metals.svg|thumb|right|250px|Empirical (Na–Fr, Mg–Ra) and predicted (Uue–Uhp, Ubn–Uhh) ionisation energy of the alkali and alkaline earth metals from the third to the ninth period, measured in electron volts<ref name="Uue"/>{{rp|1730}}<ref name="pyykko"/>]]<br />
Not as much work has been done predicting the properties of the alkali metals beyond ununennium. Although a simple extrapolation of the periodic table would put element 169, unhexennium, under ununennium, Dirac-Fock calculations predict that the next alkali metal after ununennium may actually be element 165, unhexpentium, which is predicted to have the electron configuration [Uuo] 5g<sup>18</sup> 6f<sup>14</sup> 7d<sup>10</sup> 8s<sup>2</sup> 8p<sub>1/2</sub><sup>2</sup> 9s<sup>1</sup>.<ref name="Uue"/>{{rp|1729–1730}}<ref name="pyykko">{{Cite journal|last1=Pyykkö|first1=Pekka|title=A suggested periodic table up to Z ≤ 172, based on Dirac–Fock calculations on atoms and ions|journal=Physical Chemistry Chemical Physics|volume=13|issue=1|pages=161–8|year=2011|pmid=20967377|doi=10.1039/c0cp01575j|bibcode = 2011PCCP...13..161P }}</ref> Further calculations show that unhexpentium would follow the trend of increasing ionisation energy beyond caesium, having an ionisation energy comparable to that of sodium, and that it should also continue the trend of decreasing atomic radii beyond caesium, having an atomic radius comparable to that of potassium.<ref name="Uue"/>{{rp|1729–1730}} However, the 7d electrons of unhexpentium may also be able to participate in chemical reactions along with the 9s electron, possibly allowing oxidation states beyond +1 and perhaps even making unhexpentium behave more like a [[boron group]] element than an alkali metal.<ref name="Uue"/>{{rp|1732–1733}}<br />
<br />
The probable properties of the alkali metals beyond unhexpentium have not been explored yet as of 2012. In periods 8 and above of the periodic table, relativistic and shell-structure effects become so strong that extrapolations from lighter congeners become completely inaccurate. In addition, the relativistic and shell-structure effects (which stabilise the s-orbitals and destabilise and expand the d-, f-, and g-orbitals of higher shells) have opposite effects, causing even larger difference between relativistic and non-relativistic calculations of the properties of elements with such high atomic numbers.<ref name="Uue"/>{{rp|1732–1733}} Due to the alkali and [[alkaline earth metal]]s both being [[s-block]] elements, these predictions for the trends and properties of ununennium and unhexpentium also mostly apply to the corresponding alkaline earth metals [[unbinilium]] (Ubn) and unhexhexium (Uhh).<ref name="Uue"/>{{rp|1729–1733}}<br />
<br />
==Other similar substances==<br />
<br />
===Hydrogen===<br />
{{Main|Hydrogen}}<!--why does it not fit into the periodic trends of the alkali metals?--><br />
[[File:Hydrogen discharge tube.jpg|thumb|Hydrogen gas glowing in a [[discharge tube]]]]<br />
<br />
The element [[hydrogen]], with one electron per neutral atom, is usually placed at the top of Group 1 of the periodic table for convenience, but hydrogen is not normally considered to be an alkali metal;<ref name="iupac"/> when it is considered to be an alkali metal, it is because of its atomic properties and not its chemical properties.<ref name="Folden">{{cite web |url=http://cyclotron.tamu.edu/smp/The%20Heaviest%20Elements%20in%20the%20Universe.pdf |title=The Heaviest Elements in the Universe |author=Folden, Cody |date=31 January 2009 |work=Saturday Morning Physics at Texas A&M |accessdate=9 March 2012}}</ref> Under typical conditions, pure hydrogen exists as a [[diatomic]] gas consisting of two atoms per [[molecule]] (H<sub>2</sub>);<ref>{{Cite book| author = Emsley, J. | title = The Elements | publisher = Oxford: Clarendon Press | year = 1989 | pages = 22–23| id= }}</ref><!--It is uncertain if this reference (from [[diatomic molecule]]) refers to astatine usually not being considered with the other halogens or the list of elements that form diatomic molecules.--> however, the alkali metals only form diatomic molecules (such as [[dilithium]], Li<sub>2</sub>) at high temperatures, when they are in the [[gas]]eous state.<ref>''Chemical Bonding'', Mark J. Winter, Oxford University Press, '''1994''', ISBN 0-19-855694-2</ref><br />
<br />
Hydrogen, like the alkali metals, has one valence electron<ref name="hydrogen-halogen" /> and reacts easily with the [[halogen]]s,<ref name="hydrogen-halogen" /> but the similarities end there.<ref name="hydrogen-halogen" /> Its placement above lithium is primarily due to its [[electron configuration]] and not its chemical properties.<ref name="iupac">{{cite web|url=http://old.iupac.org/reports/periodic_table/ |title=International Union of Pure and Applied Chemistry > Periodic Table of the Elements |publisher=IUPAC |accessdate=1 May 2011}}</ref><ref name="hydrogen-halogen" /> It is sometimes placed above [[carbon]] due to their similar electronegativities<ref name="hydrogen"/> or [[fluorine]] due to their similar chemical properties.<ref name="hydrogen-halogen" /><ref name="hydrogen">{{cite journal |last=Cronyn |first=Marshall W. |title=The Proper Place for Hydrogen in the Periodic Table |journal=Journal of Chemical Education |volume=80 |issue=8 |date=August 2003 |pages=947–951 |doi=10.1021/ed080p947 |url=http://www.reed.edu/reed_magazine/summer2009/columns/noaa/downloads/CronynHydrogen.pdf |bibcode=2003JChEd..80..947C}}</ref><br />
<br />
The first ionisation energy of hydrogen (1312.0 [[kilojoule per mole|kJ/mol]]) is much higher than that of the alkali metals.<ref name="huheey"/><ref name="macmillan"/> As only one additional electron is required to fill in the outermost shell of the hydrogen atom, hydrogen often behaves like a halogen, forming the negative [[hydride]] ion, and is sometimes considered to be a halogen.<ref name="hydrogen-halogen">{{cite web |url=http://hydrogentwo.com/hydrogen-halogen.html |title=Hydrogen is a Halogen |author=Vinson, Greg |year=2008 |work=HydrogenTwo.com |accessdate=14 January 2012}}</ref> (The alkali metals can also form negative ions, known as [[alkalide]]s, but these are little more than laboratory curiosities, being unstable.)<ref name="HNa"/><ref name="HNa-theory"/> Under extremely high [[pressure]]s, such as those found at the cores of [[Jupiter]] and [[Saturn]], hydrogen does become metallic and behaves like an alkali metal; in this phase, it is known as [[metallic hydrogen]].<ref>{{cite journal|last1=Wigner|first1=E.|last2=Huntington|first2=H.B.|year=1935|title=On the possibility of a metallic modification of hydrogen|journal=[[Journal of Chemical Physics]]|volume=3 |page=764|doi=10.1063/1.1749590|bibcode = 1935JChPh...3..764W|issue=12 }}</ref><br />
<br />
===Ammonium===<br />
{{Main|Ammonium}}<br />
<br />
The [[ammonium]] ion ({{chem|NH|4|+}}) has very similar properties to the heavier alkali metals, acting as an alkali metal intermediate between potassium and rubidium,<ref>{{cite web |url=http://www.meta-synthesis.com/webbook/35_pt/pt_database.php?PT_id=429|title=2002 Inorganic Chemist's Periodic Table|author=Mark R. Leach |accessdate=16 October 2012}}</ref> and is often considered a close relative.<ref name = "Holleman&Wiberg">{{Holleman&Wiberg}}</ref><ref name="Stevenson" /><ref name="Bernal&Massey" /> For example, most alkali metal [[salt (chemistry)|salts]] are [[solubility|soluble]] in water, a property which ammonium salts share.<ref>{{cite web|title=Solubility Rules!|url=http://www.chem.sc.edu/faculty/morgan/resources/solubility/|accessdate=4 January 2014}}</ref> Ammonium is expected to behave stably as a metal ({{chem|NH|4|+}} ions in a sea of electrons) at very high pressures (though less than the typical pressure where transitions from insulating to metallic behaviour occur around, 100&nbsp;[[pascal (unit)|GPa]]), and could possibly occur inside the [[Gas giant#Uranus and Neptune|ice giants]] [[Uranus]] and [[Neptune]], which may have significant impacts on their interior magnetic fields.<ref name="Stevenson">{{cite journal |last1=Stevenson |first1=D. J. |date=20 November 1975 |title=Does metallic ammonium exist? |journal=[[Nature (journal)|Nature]] |volume=258 |issue= 5532 |pages=222–223 |publisher=[[Nature Publishing Group]] |doi=10.1038/258222a0 |url=http://www.nature.com/nature/journal/v258/n5532/abs/258222a0.html |accessdate=13 January 2012 |bibcode = 1975Natur.258..222S }}</ref><ref name="Bernal&Massey">{{cite journal |last1=Bernal |first1=M. J. M. |last2=Massey |first2=H. S. W. |date=3 February 1954 |title=Metallic Ammonium |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=114 |pages=172–179 |publisher=[[Wiley-Blackwell]] for the [[Royal Astronomical Society]] |bibcode=1954MNRAS.114..172B |url=http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1954MNRAS.114..172B&amp;data_type=PDF_HIGH&amp;whole_paper=YES&amp;type=PRINTER&amp;filetype=.pdf }}</ref> It has been estimated that the transition from a mixture of [[ammonia]] and dihydrogen molecules to metallic ammonium may occur at pressures just below 25&nbsp;GPa.<ref name="Stevenson"/><br />
<br />
===Thallium===<br />
{{Main|Thallium}}<br />
[[File:Thallium pieces in ampoule.jpg|thumb|right|Very pure thallium pieces in a glass [[ampoule]], stored under [[argon]] gas]]<br />
<br />
Thallium displays the +1 [[oxidation state]]<ref name="Greenwood&Earnshaw"/>{{rp|28}} that all the known alkali metals display,<ref name="Greenwood&Earnshaw"/>{{rp|28}} and thallium compounds with thallium in its +1 [[oxidation state]] closely resemble the corresponding potassium or [[silver]] compounds due to the similar ionic radii of the Tl<sup>+</sup> (164&nbsp;[[picometer|pm]]), K<sup>+</sup> (152&nbsp;pm) and Ag<sup>+</sup> (129&nbsp;pm) ions.<ref name=Shannon>{{cite journal|doi=10.1107/S0567739476001551|title=Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides|author=R. D. Shannon|journal=Acta Cryst|volume=A32|year=1976|pages=751–767|bibcode = 1976AcCrA..32..751S|issue=5 }}</ref><ref name=Crookes/> It was sometimes considered an alkali metal in [[continental Europe]] (but not in England) in the years immediately following its discovery,<ref name=Crookes>{{cite journal |last1=Crookes |first1=William |authorlink=William Crookes |year=1864 |title=On Thallium |journal=The Journal of the Chemical Society, London |volume=XVII |pages=112–152 |publisher=Harrison & Sons |url=http://books.google.ru/books?id=H58wAAAAYAAJ |accessdate=13 January 2012 |doi=10.1039/js8641700112 }}</ref>{{rp|126}} and was placed just after caesium as the sixth alkali metal in [[Dmitri Mendeleev]]'s 1869 [[periodic table]] and [[Julius Lothar Meyer]]'s 1868 periodic table.<ref name="meta-synthesis2">{{cite web |url=http://www.meta-synthesis.com/webbook/35_pt/pt_database.php?Button=pre-1900+Formulations |title=The Internet Database of Periodic Tables |author=Leach, Mark R. |date=1999–2012 |work=meta-synthesis.com |accessdate=6 April 2012}}</ref> (Mendeleev's 1871 periodic table and Meyer's 1870 periodic table put thallium in its current position in the [[boron group]] and leave the space below caesium blank.)<ref name="meta-synthesis2"/> However, thallium also displays the oxidation state +3,<ref name="Greenwood&Earnshaw"/>{{rp|28}} which no known alkali metal displays<ref name="Greenwood&Earnshaw"/>{{rp|28}} (although ununennium, the undiscovered seventh alkali metal, is predicted to possibly display the +3 oxidation state).<ref name="Uue"/>{{rp|1729–1730}} The sixth alkali metal is now considered to be francium.<ref name="redbook"/><br />
<br />
==History==<br />
{{empty section|date=January 2013}}<!--talk about Döbereiner and all that--><br />
<br />
===Etymology===<br />
{{expand section|date=January 2013}}<br />
The alkali metals are so called because their hydroxides are all strong [[alkali]]s when dissolved in water.<ref name="rsc"/><br />
<br />
===Discovery===<br />
<br />
====Lithium====<br />
[[File:Petalite.jpg|thumb|alt=A sample of petalite|Petalite, the lithium mineral from which lithium was first isolated]]<br />
<br />
[[Petalite]] ([[Lithium|Li]][[Aluminium|Al]][[Silicon|Si]]<sub>4</sub>[[Oxygen|O]]<sub>10</sub>) was discovered in 1800 by the [[Brazil]]ian chemist [[José Bonifácio de Andrada]] in a mine on the island of [[Utö, Sweden]].<ref name=mindat>{{cite web|url=http://www.mindat.org/min-3171.html |title=Petalite: Petalite mineral information and data |last1=Ralph |first1=Jolyon |last2=Chau |first2=Ida |date=24 August 2011 |accessdate=27 November 2011}}</ref><ref name=webelementshistory>{{cite web|url=http://www.webelements.com/lithium/history.html|title=WebElements Periodic Table of the Elements {{pipe}} Lithium {{pipe}} historical information |last=Winter |first=Mark |accessdate=27 November 2011}}</ref><ref name=discovery>{{Cite book|title=Discovery of the Elements |last=Weeks |first=Mary|year=2003 |page=124 |publisher=Kessinger Publishing |location=Whitefish, Montana, United States |isbn=0-7661-3872-0 |url=http://books.google.com/?id=SJIk9BPdNWcC|accessdate=10 August 2009}}</ref> However, it was not until 1817 that [[Johan August Arfwedson]], then working in the laboratory of the chemist [[Jöns Jacob Berzelius]], [[discovery of the chemical elements|detected]] the presence of a new element while analyzing petalite ore.<ref name=uwis>{{cite web|url=http://genchem.chem.wisc.edu/lab/PTL/PTL/BIOS/arfwdson.htm |archiveurl=http://web.archive.org/web/20080605152857/http://genchem.chem.wisc.edu/lab/PTL/PTL/BIOS/arfwdson.htm |archivedate=5 June 2008 |title=Johan Arfwedson |accessdate=10 August 2009}}</ref><ref name=vanderkrogt>{{cite web|publisher = Elementymology & Elements Multidict|title = Lithium| first = Peter|last =van der Krogt|url =http://elements.vanderkrogt.net/element.php?sym=Li|accessdate =5 October 2010}}</ref> This new element formed compounds similar to those of sodium and potassium, though its [[lithium carbonate|carbonate]] and [[lithium hydroxide|hydroxide]] were less [[solubility|soluble in water]] and more [[Base (chemistry)|alkaline]] than the other alkali metals.<ref name=compounds>{{cite web|url=http://www.chemguide.co.uk/inorganic/group1/compounds.html|title=Compounds of the Group 1 Elements |accessdate=10 August 2009 |last=Clark |first=Jim |year=2005 |work=chemguide}}</ref> Berzelius gave the unknown material the name "''lithion''/''lithina''", from the [[Ancient Greek|Greek]] word ''λιθoς'' (transliterated as ''lithos'', meaning "stone"), to reflect its discovery in a solid mineral, as opposed to potassium, which had been discovered in plant ashes, and sodium, which was known partly for its high abundance in animal blood. He named the metal inside the material "''lithium''".<ref name=krebs>{{Cite book|last = Krebs|first = Robert E.|year = 2006|title = The History and Use of Our Earth's Chemical Elements: A Reference Guide|publisher = Greenwood Press|location = Westport, Conn.|isbn = 0-313-33438-2}}</ref><ref name=webelementshistory/><ref name=vanderkrogt/><br />
<br />
====Sodium====<br />
[[File:SodiumHydroxide.jpg|thumb|upright|alt=A sample of caustic soda (sodium hydroxide)|Caustic soda (sodium hydroxide), the sodium compound from which sodium was first isolated]]<br />
<br />
Sodium compounds have been known since ancient times; salt ([[sodium chloride]]) has been an important commodity in human activities, as testified by the English word ''salary'', referring to ''salarium'', the wafers of salt sometimes given to Roman soldiers along with their other wages.{{Citation needed|date=February 2012}} In [[medieval]] Europe a compound of sodium{{Clarify|date=January 2012}} with the [[Latin]] name of ''sodanum'' was used as a headache remedy.{{Citation needed|date=February 2012}} Pure sodium was not isolated until 1807 by [[Humphry Davy]] through the [[electrolysis]] of [[caustic soda]] (now called sodium hydroxide),<ref name=Davy1807>{{cite journal |first=Humphry |last=Davy |authorlink=Humphry Davy |title=On some new phenomena of chemical changes produced by electricity, in particular the decomposition of the fixed alkalies, and the exhibition of the new substances that constitute their bases; and on the general nature of alkaline bodies |year=1808 |volume=98 |journal=Philosophical Transactions of the Royal Society of London |pages=1–44 |url= http://books.google.com/?id=gpwEAAAAYAAJ&pg=PA57|doi=10.1098/rstl.1808.0001}}</ref> a very similar method to the one used to isolate potassium earlier that year.<br />
<br />
====Potassium====<br />
[[File:Potassium hydroxide.jpg|thumb|upright|alt=A sample of caustic potash|Caustic potash (potassium hydroxide), the potassium compound from which potassium was first isolated]]<br />
<br />
While potash has been used since ancient times, it was not understood for most of its history to be a fundamentally different substance from sodium mineral salts. [[Georg Ernst Stahl]] obtained experimental evidence which led him to suggest the fundamental difference of sodium and potassium salts in 1702,<ref name="1702Suspect">{{cite book|url = http://books.google.com/books?id=b-ATAAAAQAAJ&pg=PA167|page = 167|title = Chymische Schriften|last1 = Marggraf|first = Andreas Siegmund|year = 1761}}</ref> and [[Henri Louis Duhamel du Monceau]] was able to prove this difference in 1736.<ref>{{cite journal|url = http://gallica.bnf.fr/ark:/12148/bpt6k3533j/f73.image.r=Memoires%20de%20l%27Academie%20royale%20des%20Sciences.langEN|journal = Memoires de l'Academie royale des Sciences| title = Sur la Base de Sel Marine| last = du Monceau|first = H. L. D.| pages = 65–68| language = French}}</ref> The exact chemical composition of potassium and sodium compounds, and the status as chemical element of potassium and sodium, was not known then, and thus [[Antoine Lavoisier]] did include the alkali in his list of chemical elements in 1789.<ref name="weeks">{{cite journal|doi = 10.1021/ed009p1035|title = The discovery of the elements. IX. Three alkali metals: Potassium, sodium, and lithium|year = 1932|last1 = Weeks|first1 = Mary Elvira|authorlink1=Mary Elvira Weeks|journal = Journal of Chemical Education|volume = 9|issue = 6|page = 1035|bibcode = 1932JChEd...9.1035W}}</ref><ref name="disco">{{cite journal|jstor = 228541|pages = 247–258|last1 = Siegfried|first1 = R.|title = The Discovery of Potassium and Sodium, and the Problem of the Chemical Elements|volume = 54|issue = 2|journal = Isis|year = 1963|doi = 10.1086/349704}}</ref> Pure potassium was first isolated in 1807 in England by Sir [[Humphry Davy]], who derived it from [[Potassium hydroxide|caustic potash]] (KOH, potassium hydroxide) by the use of electrolysis of the molten salt with the newly invented [[voltaic pile]]. Potassium was the first metal that was isolated by electrolysis.<ref name=Enghag2004>{{cite book|author=Enghag, P.|year=2004|title=Encyclopedia of the elements|publisher=Wiley-VCH Weinheim|isbn=3-527-30666-8|chapter=11. Sodium and Potassium}}</ref> Later that same year, Davy reported extraction of sodium from the similar substance [[caustic soda]] (NaOH, lye) by a similar technique, demonstrating the elements, and thus the salts, to be different.<ref name=Davy1807>{{cite journal|first=Humphry|last=Davy|title=On some new phenomena of chemical changes produced by electricity, in particular the decomposition of the fixed alkalies, and the exhibition of the new substances that constitute their bases; and on the general nature of alkaline bodies|pages=1–44|year=1808|volume=98|journal=Philosophical Transactions of the Royal Society of London|url=http://books.google.com/?id=gpwEAAAAYAAJ&pg=PA57&q|doi=10.1098/rstl.1808.0001}}</ref><ref name="weeks" /><ref name="disco"/><ref name="200disco">{{cite journal|doi = 10.1134/S1061934807110160|title = History of the discovery of potassium and sodium (on the 200th anniversary of the discovery of potassium and sodium)|year = 2007|last1 = Shaposhnik|first1 = V. A.|journal = Journal of Analytical Chemistry|volume = 62|issue = 11|pages = 1100–1102}}</ref><br />
<br />
====Rubidium====<br />
[[File:Lepidolite-76774.jpg|thumb|upright|alt=A sample of lepidolite|Lepidolite, the rubidium mineral from which rubidium was first isolated]]<br />
<br />
Rubidium was discovered in 1861 in Heidelberg, Germany by [[Robert Bunsen]] and [[Gustav Kirchhoff]], the first people to suggest finding new elements by [[Spectroscopy|spectrum analysis]], in the mineral [[lepidolite]] through the use of a [[spectroscope]]. Because of the bright red lines in its [[emission spectrum]], they chose a name derived from the [[Latin]] word ''rubidus'', meaning dark red or bright red.<ref name="Weeks">{{Cite journal|title = The discovery of the elements. XIII. Some spectroscopic discoveries |pages = 1413–1434|last = Weeks|first = Mary Elvira|authorlink=Mary Elvira Weeks|doi=10.1021/ed009p1413|journal = [[Journal of Chemical Education]] |volume =9 |issue =8 |year = 1932 |bibcode=1932JChEd...9.1413W}}</ref><ref name="BuKi1861">{{Cite journal|title = Chemische Analyse durch Spectralbeobachtungen |pages = 337–381 |first1 = G.|last1 = Kirchhoff |first2 = R.|last2 = Bunsen|authorlink1 = Gustav Kirchhoff|authorlink2 = Robert Bunsen|doi = 10.1002/andp.18611890702 |journal = [[Annalen der Physik|Annalen der Physik und Chemie]] |volume = 189 |issue = 7|year = 1861 |bibcode=1861AnP...189..337K}}</ref> Rubidium's discovery succeeded that of caesium, also discovered by Bunsen and Kirchhoff through spectroscopy.<ref name="caesium" /><br />
<br />
====Caesium====<br />
In 1860, [[Robert Bunsen]] and [[Gustav Kirchhoff]] discovered caesium in the [[mineral water]] from [[Bad Dürkheim|Dürkheim]], Germany. Due to the bright-blue lines in its [[emission spectrum]], they chose a name derived from the Latin word ''caesius'', meaning sky-blue.<ref name="Weeks">{{cite journal |title = The discovery of the elements. XIII. Some spectroscopic discoveries |pages = 1413–1434|last = Weeks|first = Mary Elvira |authorlink=Mary Elvira Weeks|doi=10.1021/ed009p1413|journal = [[Journal of Chemical Education]] |volume =9 |issue =8 |year = 1932 |bibcode=1932JChEd...9.1413W}}</ref><ref group="note">Bunsen quotes [[Aulus Gellius]] [[Aulus Gellius|Noctes Atticae]] II, 26 by [[Nigidius Figulus]]: ''Nostris autem veteribus "caesia" dicta est, quae a Graecis glaukopis, ut Nigidius ait, "de colore caeli quasi caelia''.''</ref><ref>[[Oxford English Dictionary]], 2nd Edition</ref> Caesium was the first element to be discovered [[spectroscopy|spectroscopically]], only one year after the invention of the [[spectroscope]] by Bunsen and Kirchhoff.<ref name="caesium">{{cite web|url=http://pubs.acs.org/cen/80th/print/cesium.html |title=C&EN: It's Elemental: The Periodic Table – Cesium |publisher=American Chemical Society|accessdate=25 February 2010|author=Kaner, Richard|year = 2003}}</ref><br />
<br />
====Francium====<br />
<br />
There were at least four erroneous and incomplete discoveries<ref name="fontani">{{cite conference| first = Marco| last = Fontani| title = The Twilight of the Naturally-Occurring Elements: Moldavium (Ml), Sequanium (Sq) and Dor (Do)| booktitle = International Conference on the History of Chemistry| pages = 1–8| date = 10 September 2005| location = Lisbon|url = http://5ichc-portugal.ulusofona.pt/uploads/PaperLong-MarcoFontani.doc| archiveurl = http://web.archive.org/web/20060224090117/http://5ichc-portugal.ulusofona.pt/uploads/PaperLong-MarcoFontani.doc|archivedate=24 February 2006|accessdate =8 April 2007}}</ref><ref name="vanderkrogt-Fr"/><ref>{{cite news| title = Alabamine & Virginium|work=TIME| date = 15 February 1932|url = http://www.time.com/time/magazine/article/0,9171,743159,00.html| accessdate =1 April 2007}}</ref><ref>{{cite journal| last = MacPherson| first = H. G.| title = An Investigation of the Magneto-Optic Method of Chemical Analysis| journal = Physical Review| volume = 47| issue = 4| pages = 310–315| publisher = American Physical Society|year=1934|doi = 10.1103/PhysRev.47.310|bibcode = 1935PhRv...47..310M }}</ref> before [[Marguerite Perey]] of the [[Curie Institute (Paris)|Curie Institute]] in Paris, France discovered francium in 1939 by purifying a sample of [[isotopes of actinium|actinium-227]], which had been reported to have a decay energy of 220&nbsp;[[electronvolt|keV]]. However, Perey noticed decay particles with an energy level below 80&nbsp;keV. Perey thought this decay activity might have been caused by a previously unidentified decay product, one that was separated during purification, but emerged again out of the pure [[actinium]]-227. Various tests eliminated the possibility of the unknown element being [[thorium]], [[radium]], [[lead]], [[bismuth]], or [[thallium]]. The new product exhibited chemical properties of an alkali metal (such as coprecipitating with caesium salts), which led Perey to believe that it was element 87, caused by the [[alpha decay]] of actinium-227.<ref name="chemeducator">Adloff, Jean-Pierre; Kaufman, George B. (2005-09-25). [http://chemeducator.org/sbibs/s0010005/spapers/1050387gk.htm Francium (Atomic Number 87), the Last Discovered Natural Element]. ''The Chemical Educator'' '''10''' (5). Retrieved on 26 March 2007.</ref> Perey then attempted to determine the proportion of [[beta decay]] to alpha decay in actinium-227. Her first test put the alpha branching at 0.6%, a figure that she later revised to 1%.<ref name="mcgraw">{{Cite book| contribution = Francium| year = 2002| title = [[McGraw-Hill Encyclopedia of Science & Technology]]| volume = 7| pages = 493–494| publisher = McGraw-Hill Professional|isbn = 0-07-913665-6}}</ref> It was the last element discovered in [[nature]], rather than by synthesis.<ref group="note">Some synthetic elements, like [[technetium]] and [[plutonium]], have later been found in nature.</ref><br />
<br />
====Eka-francium====<br />
{{see also|Extended periodic table}}<br />
<br />
The next element below francium ([[Mendeleev's predicted elements|eka]]-francium) is very likely to be [[ununennium]] (Uue), element 119,<ref name="Uue"/>{{rp|1729–1730}} although this is not completely certain due to [[relativistic quantum chemistry|relativistic effects]].<ref name="tanm"/> The synthesis of ununennium was first attempted in 1985 by bombarding a target of [[einsteinium]]-254 with [[calcium]]-48 ions at the superHILAC accelerator at Berkeley, California. No atoms were identified, leading to a limiting yield of 300 [[barn (unit)|nb]].<ref name="link"/><ref name=vanderkrogt-uue>{{cite web|publisher = Elementymology & Elements Multidict|title = Ununennium| first = Peter|last =van der Krogt|url =http://elements.vanderkrogt.net/element.php?sym=Uue|accessdate =14 February 2011}}</ref><br />
<br />
:{{nuclide2|einsteinium|254|link=y}} + {{nuclide2|calcium|48|link=y}} → {{nuclide2|ununennium|302}}* → ''no atoms''<ref group="note">The [[asterisk]] denotes an [[excited state]].{{Citation needed|date=February 2012}}</ref><br />
<br />
It is highly unlikely<ref name="link" /> that this reaction will be able to create any atoms of ununennium in the near future, given the extremely difficult task of making sufficient amounts of [[Isotopes of einsteinium|<sup>254</sup>Es]], which is favoured for production of [[superheavy element|ultraheavy elements]] because of its large mass, relatively long half-life of 270 days, and availability in significant amounts of several micrograms,<ref>{{cite journal|last1=Schadel|first1=M.|last2=Brüchle|first2=W.|last3=Brügger|first3=M.|last4=Gäggeler|first4=H.|last5=Moody|first5=K.|last6=Schardt|first6=D.|last7=Sümmerer|first7=K.|last8=Hulet|first8=E.|last9=Dougan|first9=A.|last10=Dougan|displayauthors=9 |title=Heavy isotope production by multinucleon transfer reactions with <sup>254</sup>Es|journal=Journal of the Less Common Metals|volume=122|pages=411|year=1986|doi=10.1016/0022-5088(86)90435-2|first10=R.J.|last11=Landrum|first11=J.H.|last12=Lougheed|first12=R.W.|last13=Wild|first13=J.F.|last14=O'Kelley|first14=G.D.|last15=Hahn|first15=R.L.}}</ref> to make a large enough target to increase the sensitivity of the experiment to the required level; einsteinium has not been found in nature and has only been produced in laboratories. However, given that ununennium is only the first [[period 8 element]] on the [[extended periodic table]], it may well be discovered in the near future through other reactions; indeed, another attempt to synthesise ununennium by bombarding a [[berkelium]] target with [[titanium]] ions is under way at the [[GSI Helmholtz Centre for Heavy Ion Research]] in [[Darmstadt]], [[Germany]].<ref>{{cite news |title=Modern alchemy: Turning a line |author= |url=http://www.economist.com/node/21554502 |newspaper=[[The Economist]] |date=12 May 2012 |accessdate=5 October 2012}}</ref> Currently, none of the period 8 elements have been discovered yet, and it is also possible, due to [[nucleon drip line|drip instabilities]], that only the lower period 8 elements, up to around element 128, are physically possible.<ref name=EB>{{cite web|author=Seaborg, G. T.|url=http://www.britannica.com/EBchecked/topic/603220/transuranium-element|title=transuranium element (chemical element)|publisher=Encyclopædia Britannica|date=c. 2006|accessdate=16 March 2010}}</ref><ref name="emsley">{{cite book|last=Emsley|first=John|title=Nature's Building Blocks: An A-Z Guide to the Elements|edition=New|year=2011|publisher=Oxford University Press|location=New York, NY|isbn=978-0-19-960563-7|page=593}}</ref> No attempts at synthesis have been made for any heavier alkali metals, such as unhexpentium, due to their extremely high atomic number.<ref name="Uue"/>{{rp|1737–1739}}<br />
<br />
==Occurrence==<br />
<br />
===In the Solar System===<br />
[[Image:SolarSystemAbundances.png|thumb|right|800px|Estimated abundances of the chemical elements in the Solar system. Hydrogen and helium are most common, from the [[Big Bang]]. The next three elements (lithium, [[beryllium]], and [[boron]]) are rare because they are poorly synthesized in the Big Bang and also in stars. The two general trends in the remaining stellar-produced elements are: (1) an alternation of abundance in elements as they have even or odd atomic numbers, and (2) a general decrease in abundance, as elements become heavier. Iron is especially common because it represents the minimum energy nuclide that can be made by fusion of helium in supernovae.<ref name=lodders>{{cite journal | last1 = Lodders | first1 = Katharina | year = 2003 | title = Solar System Abundances and Condensation Temperatures of the Elements | url = | journal = The Astrophysical Journal | volume = 591 | issue = 2| pages = 1220–1247 |bibcode = 2003ApJ...591.1220L |doi = 10.1086/375492 }}</ref>]]<br />
The [[Oddo-Harkins rule]] holds that elements with even atomic numbers are more common that those with odd atomic numbers, with the exception of hydrogen. This rule argues that elements with odd atomic numbers have one unpaired proton and are more likely to capture another, thus increasing their atomic number. In elements with even atomic numbers, protons are paired, with each member of the pair offsetting the spin of the other, enhancing stability.<ref name=oddo>{{cite journal | doi = 10.1002/zaac.19140870118 | title = Die Molekularstruktur der radioaktiven Atome | year = 1914 | last1 = Oddo | first1 = Giuseppe | journal = Zeitschrift für anorganische Chemie | volume = 87 | pages = 253}}</ref><ref name=harkins>{{cite journal | doi = 10.1021/ja02250a002 | year = 1917 | last1 = Harkins | first1 = William D. | journal = Journal of the American Chemical Society | volume = 39 | issue = 5 | pages = 856 | title = The Evolution of the Elements and the Stability of Complex Atoms. I. A New Periodic System Which Shows a Relation Between the Abundance of the Elements and the Structure of the Nuclei of Atoms}}</ref><ref name=north>{{cite book|last=North|first=John|title=Cosmos an illustrated history of astronomy and cosmology|year=2008|publisher=Univ. of Chicago Press|isbn=978-0-226-59441-5|pages=602|url=http://books.google.com/?id=qq8Luhs7rTUC&lpg=PA602&dq=%22william%20draper%20harkins%22%20oddo&pg=PA602#v=onepage&q=%22william%20draper%20harkins%22%20oddo&f=false|edition=Rev. and updated}}</ref> All the alkali metals have odd atomic numbers and they are not as common as the elements with even atomic numbers adjacent to them (the [[noble gas]]es and the [[alkaline earth metal]]s) in the Solar System. The heavier alkali metals are also less abundant than the lighter ones as the alkali metals from rubidium onward can only be synthesized in [[supernova]]e and not in [[stellar nucleosynthesis]]. Lithium is also much less abundant than sodium and potassium as it is poorly synthesized in both [[Big Bang nucleosynthesis]] and in stars: the Big Bang could only produce trace quantities of lithium, [[beryllium]] and [[boron]] due to the absence of a stable nucleus with 5 or 8 [[nucleon]]s, and stellar nucleosynthesis could only pass this bottleneck by the [[triple-alpha process]], fusing three helium nuclei to form [[carbon]], and skipping over those three elements.<ref name=lodders/><br />
<br />
===On Earth===<br />
[[File:Spodumene-usa59abg.jpg|thumb|upright|[[Spodumene]], an important lithium mineral]]<br />
The [[Earth]] formed from the same cloud of matter that formed the Sun, but the planets acquired different compositions during the [[formation and evolution of the solar system]]. In turn, the [[history of Earth|natural history of the Earth]] caused parts of this planet to have differing concentrations of the elements. The mass of the Earth is approximately 5.98{{e|24}}&nbsp;kg. It is composed mostly of [[iron]] (32.1%), [[oxygen]] (30.1%), [[silicon]] (15.1%), [[magnesium]] (13.9%), [[sulfur]] (2.9%), [[nickel]] (1.8%), [[calcium]] (1.5%), and [[aluminium]] (1.4%); with the remaining 1.2% consisting of trace amounts of other elements. Due to [[mass segregation]], the core region is believed to be primarily composed of iron (88.8%), with smaller amounts of nickel (5.8%), sulfur (4.5%), and less than 1% trace elements.<ref name=pnas71_12_6973>{{cite journal | author=Morgan, J. W.; Anders, E. | title=Chemical composition of Earth, Venus, and Mercury | journal=Proceedings of the National Academy of Sciences | year=1980 | volume=77 | issue=12 | pages=6973–6977 | doi=10.1073/pnas.77.12.6973 | pmid=16592930 | pmc=350422 |bibcode = 1980PNAS...77.6973M }}</ref><br />
<br />
The alkali metals, due to their high reactivity, do not occur naturally in pure form in nature. They are [[Goldschmidt classification|lithophiles]] and therefore remain close to the Earth's surface because they combine readily with [[oxygen]] and so associate strongly with [[silica]], forming relatively low-density minerals that do not sink down into the Earth's core. Potassium, rubidium and caesium are also [[incompatible element]]s due to their low [[ionic radius|ionic radii]].<ref name="albarede">{{cite book | title = Geochemistry: an introduction | url = http://books.google.de/books?id=doVGzreGq14C&pg=PA17 | publisher = Cambridge University Press | year = 2003 | isbn = 978-0-521-89148-6 | first =Francis | last = Albarède }}</ref><br />
<br />
Sodium and potassium are very abundant in earth, both being among the ten [[abundance of elements in Earth's crust|most common elements in Earth's crust]];<ref name="webelements-occurrence">{{cite web|url = http://www.webelements.com/webelements/properties/text/image-flash/abund-crust.html|title = Abundance in Earth's Crust|publisher = WebElements.com|accessdate =14 April 2007}}</ref><ref name="IsraelScience&Technology">{{cite web|url = http://www.science.co.il/PTelements.asp?s=Earth|title = List of Periodic Table Elements Sorted by Abundance in Earth's crust|publisher = Israel Science and Technology Homepage|accessdate =15 April 2007}}</ref> sodium makes up approximately 2.6% of the [[Earth]]'s crust measured by weight, making it the [[Abundance of the chemical elements|sixth most abundant element]] overall<ref name="RubberBible86th">{{RubberBible86th}}</ref> and the most abundant alkali metal. Potassium makes up approximately 1.5% of the Earth's crust and is the seventh most abundant element.<ref name="RubberBible86th"/> Sodium is found in many different minerals, of which the most common is ordinary salt (sodium chloride), which occurs in vast quantities dissolved in seawater. Other solid deposits include [[halite]], [[amphibole]], [[cryolite]], [[nitratine]], and [[zeolite]].<ref name="RubberBible86th" /><br />
<br />
Lithium, due to its relatively low reactivity, can be found in seawater in large amounts; it is estimated that seawater is approximately 0.14 to 0.25 parts per million (ppm)<ref>{{cite web|url=http://www.ioes.saga-u.ac.jp/ioes-study/li/lithium/occurence.html |title=Lithium Occurrence|accessdate=13 March 2009|publisher=Institute of Ocean Energy, Saga University, Japan}}{{dead link|date=May 2012}}</ref><ref name=enc>{{cite web|url=http://www.enclabs.com/lithium.html|accessdate=15 October 2010|title=Some Facts about Lithium|publisher=ENC Labs}}</ref> or 25 [[micromolar]].<ref>{{cite journal|doi=10.1007/3-540-13534-0_3|title=Extraction of metals from sea water|volume =124/1984|pages= 91–133|author=Schwochau, Klaus|journal=Topics in Current Chemistry|year=1984|series=Topics in Current Chemistry|isbn=978-3-540-13534-0}}</ref><br />
<br />
Rubidium is approximately as abundant as [[zinc]] and more abundant than copper. It occurs naturally in the minerals [[leucite]], [[pollucite]], [[carnallite]], [[zinnwaldite]], and [[lepidolite]].<ref>{{Cite journal |title =Trace element chemistry of lithium-rich micas from rare-element granitic pegmatites |volume = 55<br />
| issue = 13 |year = 1995 |doi = 10.1007/BF01162588 |pages = 203–215 |journal = Mineralogy and Petrology |first = M. A. |last = Wise |bibcode = 1995MinPe..55..203W }}</ref> Caesium is more abundant than some commonly known elements, such as [[antimony]], [[cadmium]], [[tin]], and [[tungsten]], but is much less abundant than rubidium.<ref name="pubs.usgs"/><br />
<br />
[[Isotopes of francium|Francium-223]], the only naturally occurring isotope of francium,<ref name="atomicweights2007"/><ref name="atomicweights2009"/> is the [[decay product|product]] of the [[alpha decay]] of actinium-227 and can be found in trace amounts in [[uranium]] and [[thorium]] minerals.<ref name="CRC2006">{{Cite book |year =2006 |title = CRC Handbook of Chemistry and Physics |volume = 4|page= 12|publisher = CRC|isbn= 0-8493-0474-1}}</ref> In a given sample of uranium, there is estimated to be only one francium atom for every 10<sup>18</sup> uranium atoms.<ref name="nbb">{{cite book| last = Emsley|url=http://books.google.com/books?id=Yhi5X7OwuGkC&pg=PA151| first = John| title = Nature's Building Blocks| publisher = Oxford University Press| year = 2001| location = Oxford| pages = 151–153| isbn = 0-19-850341-5}}</ref><ref name="elemental">{{cite web| last = Gagnon| first = Steve| title = Francium| publisher = Jefferson Science Associates, LLC| url = http://education.jlab.org/itselemental/ele087.html| accessdate =1 April 2007| archiveurl= http://web.archive.org/web/20070331235139/http://education.jlab.org/itselemental/ele087.html| archivedate= 31 March 2007 <!--DASHBot-->| deadurl= no}}</ref> It has been calculated that there is at most 30&nbsp;g of francium in the [[Crust (geology)|earth's crust]] at any time, due to its extremely short [[half-life]] of 22 minutes.<ref name="Winter"/><ref name="itselemental">{{cite web|url = http://education.jlab.org/itselemental/index.html|title = It's Elemental&nbsp;— The Periodic Table of Elements|publisher = Jefferson Lab|accessdate =14 April 2007| archiveurl= http://web.archive.org/web/20070429032414/http://education.jlab.org/itselemental/index.html| archivedate= 29 April 2007 <!--DASHBot-->| deadurl= no}}</ref><br />
<br />
==Production and isolation==<br />
{{multiple image<br />
| footer = [[Salt flat]]s are rich in lithium, such as these in Salar del Hombre Muerto, Argentina (left) and [[Salar de Uyuni|Uyuni]], Bolivia (right). The lithium-rich brine is concentrated by pumping it into [[Salt evaporation pond|solar evaporation ponds]] (visible in Argentina image).<br />
| align = right<br />
| width1 = 160<br />
| width2 = 105<br />
| image1 = Lithium mine, Salar del Hombre Muerto, Argentina.jpg<br />
| alt1 = alt1<br />
| image2 = Uyuni landsat.JPG<br />
| alt2 = alt2<br />
}}<br />
The production of pure alkali metals is difficult due to their extreme reactivity with commonly used substances, such as water. The alkali metals are so reactive that they cannot be [[single displacement reaction|displaced]] by other elements and must be isolated through high-energy methods such as electrolysis.<ref name="rsc"/><ref name=generalchemistry/><br />
<br />
Lithium salts have to be extracted from the water of [[mineral spring]]s, [[brine]] pools, and brine deposits. The metal is produced electrolytically from a mixture of fused [[lithium chloride]] and [[potassium chloride]].<ref name="ober">{{cite web|url=http://minerals.usgs.gov/minerals/pubs/commodity/lithium/450798.pdf |title=Lithium|accessdate =19 August 2007|last=Ober |first=Joyce A |format=PDF |pages = 77–78| publisher=[[United States Geological Survey]]| archiveurl= http://web.archive.org/web/20070711062102/http://minerals.usgs.gov/minerals/pubs/commodity/lithium/450798.pdf| archivedate= 11 July 2007 <!--DASHBot-->| deadurl= no}}</ref><br />
<br />
Potassium occurs in many minerals, such as [[sylvite]] ([[potassium chloride]]).<ref name="rsc"/> It is occasionally produced through separating the potassium from the chlorine in potassium chloride, but is more often produced through electrolysis of [[potassium hydroxide]],<ref>{{cite web|publisher=Webelements|title=WebElements Periodic Table of the Elements {{pipe}} Potassium {{pipe}} Essential information |url=http://www.webelements.com/potassium/ |author=Winter, Mark|accessdate=27 November 2011}}</ref> found extensively in places such as [[Canada]], [[Russia]], [[Belarus]], [[Germany]], [[Israel]], [[United States]], and [[Jordan]], in a method similar to how sodium was produced in the late 1800s and early 1900s.<ref name=kirk>{{cite journal|doi=10.1002/0471238961.1915040912051311.a01.pub2|chapter=Sodium and Sodium Alloys|title=Kirk-Othmer Encyclopedia of Chemical Technology|year=2001|last1=Lemke|first1=Charles H.|last2=Markant|first2=Vernon H.|isbn=0471238961}}</ref> It can also be produced from [[seawater]]. Sodium occurs mostly in seawater and dried [[seabed]],<ref name="rsc"/> but is now produced through [[electrolysis]] of [[sodium chloride]] by lowering the melting point of the substance to below 700&nbsp;°C through the use of a [[Downs cell]].<ref name="pauling">{{Cite book|last=Pauling |first=Linus |title= General Chemistry |edition=1970 |publisher=Dover Publications}}</ref><ref name="losal">{{cite web|url=http://periodic.lanl.gov/11.shtml|title=Los Alamos National Laboratory – Sodium|accessdate=8 June 2007}}</ref> Extremely pure sodium can be produced through the thermal decomposition of [[sodium azide]].<ref>Merck Index, 9th ed., monograph 8325</ref><br />
<br />
[[File:Pichblende.jpg|thumb|This sample of [[uraninite]] contains about 100,000 atoms (3.3{{e|-20}}&nbsp;g) of francium-223 at any given time.<ref name="nbb" />|alt=A shiny gray 5-centimeter piece of matter with a rough surface.]]<br />
For several years in the 1950s and 1960s, a by-product of the potassium production called Alkarb was a main source for rubidium. Alkarb contained 21% rubidium while the rest was potassium and a small fraction of caesium.<ref>{{cite journal|title = Cesium and Rubidium Hit Market|journal = Chemical & Engineering News |volume = 37|issue = 22|pages = 50–56|year = 1959|doi = 10.1021/cen-v037n022.p050}}</ref> Today the largest producers of caesium, for example the [[Tanco Mine]], Manitoba, Canada, produce rubidium as by-product from [[pollucite]].<ref name=USGS/> Today, a common method for separating rubidium from potassium and caesium is the [[fractional crystallization (chemistry)|fractional crystallization]] of a rubidium and caesium [[alum]] ([[Caesium|Cs]], [[Rubidium|Rb]])[[Aluminium|Al]]([[Sulfate|SO<sub>4</sub>]])<sub>2</sub>·12[[Water|H<sub>2</sub>O]], which yields pure rubidium alum after approximately 30 different reactions.<ref name=USGS/><ref>{{cite book|url = http://books.google.com/?id=1ikjAQAAIAAJ&q=ferrocyanide+rubidium&dq=ferrocyanide+rubidium|publisher = United States. Bureau of Mines|title = bulletin 585|year = 1995}}</ref> The limited applications and the lack of a mineral rich in rubidium limits the production of rubidium compounds to 2 to 4 [[tonne]]s per year.<ref name=USGS>{{cite web|url = http://pubs.usgs.gov/of/2003/of03-045/of03-045.pdf |format = PDF|publisher = United States Geological Survey|accessdate =4 December 2010|title = Mineral Commodity Profile: Rubidium|first1 = William C.|last1 = Butterman|first2 = William E.|last2 = Brooks|first3 = Robert G.|last3 = Reese, Jr.|year=2003}}</ref> Caesium, however, is not produced from the above reaction. Instead, the mining of [[pollucite]] ore is the main method of obtaining pure caesium, extracted from the ore mainly by three methods: acid digestion, alkaline decomposition, and direct reduction.<ref name=USGS/><ref name=Burt>{{cite book|last = Burt|first = R. O.|year = 1993|chapter = Caesium and cesium compounds|title = Kirk-Othmer encyclopedia of chemical technology|edition = 4th|place = New York|publisher = John Wiley & Sons, Inc.|volume = 5|pages = 749–764|isbn = 978-0-471-48494-3}}</ref><br />
<!--<br />
Rubidium metal can be produced by [[redox|reducing]] rubidium chloride with [[calcium]], among other methods. In 1997, the cost of this metal in small quantities was about US$25/gram.{{citation needed|date=December 2010}}<br />
--><br />
<br />
[[Francium-223]], the only naturally occurring isotope of francium,<ref name="atomicweights2007"/><ref name="atomicweights2009"/> is produced naturally as the product of the [[alpha decay]] of [[actinium-227]]. Francium can be found in trace amounts in [[uranium]] and [[thorium]] minerals;<ref name="CRC2006" /> it has been calculated that at most there are 30&nbsp;g of francium in the [[Crust (geology)|earth's crust]] at any given time.<ref name="Winter">{{cite web<br />
|last = Winter|first = Mark<br />
|title = Geological information<br />
|work = Francium|publisher = The University of Sheffield<br />
|url = http://www.webelements.com/webelements/elements/text/Fr/geol.html<br />
|accessdate =26 March 2007}}</ref> As a result of its extreme rarity in nature, most francium is synthesized in the nuclear reaction <sup>197</sup>[[Gold|Au]] + <sup>18</sup>[[Oxygen|O]] → <sup>210</sup>[[Francium|Fr]] + 5 [[neutron|n]], yielding [[francium-209]], [[francium-210]], and [[francium-211]].<ref>{{cite journal |last1=Stancari |first1=G. |last2=Veronesi |first2=S. |last3=Corradi |first3=L. |last4=Atutov |first4=S. N. |last5=Calabrese |first5=R. |last6=Dainelli |first6=A. |last7=Mariotti |first7=E. |last8=Moi |first8=L. |last9=Sanguinetti |first9=S. |first10=L.|last10=Tomassetti|year=2006 |title=Production of Radioactive Beams of Francium |journal=Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment |volume=557 |issue=2 |pages=390–396 |doi=10.1016/j.nima.2005.11.193 |bibcode = 2006NIMPA.557..390S }}</ref> The greatest quantity of francium ever assembled to date is about 300,000 neutral atoms,<ref name="chemnews" /> which were synthesized using the nuclear reaction given above.<ref name="chemnews" /><br />
<br />
From their [[silicate]] ores, all the alkali metals may be obtained the same way: [[sulfuric acid]] is first used to dissolve the desired alkali metal ion and [[aluminium]](III) ions from the ore (leaching), whereupon basic precipitation removes aluminium ions from the mixture by precipitating it as the [[aluminium hydroxide|hydroxide]]. The remaining insoluble alkali metal [[carbonate]] is then precipitated selectively; the salt is then dissolved in [[hydrochloric acid]]. The result is then left to evaporate and the alkali metal can then be isolated through [[electrolysis]].<ref name=generalchemistry/><br />
<br />
Lithium and sodium are typically isolated through electrolysis from their liquid chlorides, with [[calcium chloride]] typically added to lower the melting point of the mixture. The heavier alkali metals, however, is more typically isolated in a different way, where a reducing agent (typically sodium for potassium and [[magnesium]] or [[calcium]] for the heaviest alkali metals) is used to reduce the alkali metal chloride. The liquid or gaseous product (the alkali metal) then undergoes [[fractional distillation]] for purification.<ref name=generalchemistry/><br />
<br />
==Applications==<br />
<br />
All of the discovered alkali metals excluding francium have many applications. Lithium is often used in [[lithium-ion battery|batteries]], and [[lithium oxide]] can help process silica. Lithium can also be used to make lubricating greases, air treatment, and aluminium production.<ref>{{Cite news|author=USGS |year=2011|title=Lithium|url= http://minerals.usgs.gov/minerals/pubs/commodity/lithium/mcs-2011-lithi.pdf|accessdate=4 December 2011|format=PDF}}</ref><br />
<br />
Pure sodium has many applications, including use in [[sodium-vapor lamp|sodium-vapour lamps]], which produce very efficient light compared to other types of lighting,<ref name="lamp1">{{cite book | url =http://books.google.com/books?id=0d7u9Nr33zIC&pg=PA112| page = 112 | title =Applied illumination engineering | isbn =978-0-88173-212-2 | author1 =Lindsey | first1 =Jack L | year =1997}}</ref><ref name="lamp2">{{cite book | url =http://books.google.com/books?id=AFNwNAFYtCAC&pg=PA241 | page = 241 | title =Revolution in lamps: A chronicle of 50 years of progress | isbn =978-0-88173-351-8 | author1 =Kane | first1 =Raymond | last2 =Sell | first2 =Heinz | year =2001}}</ref> and can help smooth the surface of other metals.<ref>{{cite book | url = http://books.google.com/?id=kyVWAAAAYAAJ&dq=sodium+descale+metal&q=METALLIC+SODIUM+++DESCALING+SEVERAL#search_anchor | title = Metal treatment and drop forging | author1 = Stampers | first1 = National Association of Drop Forgers and | year = 1957}}</ref><ref>{{cite book | url = http://books.google.de/books?id=LI4KmKqca78C&pg=PA76 | page = 76 | title = Metal cleaning bibliographical abstracts | author1 = Harris | first1 = Jay C | year = 1949}}</ref> Sodium compounds have many applications as well, the most well-known compound being [[table salt]].{{Citation needed|date=February 2012}} Sodium is also used in soap as salts of [[fatty acid]]s.{{Citation needed|date=February 2012}}<br />
<br />
Potassium compounds are often used as [[fertiliser]]s<ref name="Greenwood&Earnshaw">{{Greenwood&Earnshaw2nd}}</ref>{{rp|73}}<ref>{{cite book|author=Cordel, Oskar |title=Die Stassfurter Kalisalze in der Landwirthschalt: Eine Besprechung ...|url=http://books.google.com/books?id=EYpIAAAAYAAJ|accessdate=29 May 2011|year=1868|publisher=L. Schnock| language = German}}</ref> as potassium is an important element for plant nutrition. Other potassium ions are often used to hold [[anion]]s.{{Citation needed|date=January 2012}}{{Clarify|date=February 2012}} [[Potassium hydroxide]] is a very strong base, and is used to control the [[pH]] of various substances.<ref>{{cite book|publisher=Greenwood Publishing Group|url = http://books.google.com/books?id=UnjD4aBm9ZcC&pg=PA4|chapter = Personal Cleansing Products: Bar Soap|title = Chemical composition of everyday products|isbn = 978-0-313-32579-3|author1 = Toedt, John|author2 = Koza, Darrell|author3 = Cleef-Toedt, Kathleen Van|year = 2005}}</ref><ref>{{cite book|doi = 10.1002/14356007.a22_031.pub2|title = Ullmann's Encyclopedia of Industrial Chemistry|year = 2006|author= Schultz, H. ''et al.''|chapter = Potassium compounds|isbn = 3-527-30673-0|volume=A22|page=95}}</ref><br />
<br />
[[File:FOCS-1.jpg|thumb|right|alt=FOCS 1, a caesium atomic clock in Switzerland|FOCS 1, a caesium atomic clock in Switzerland]]<br />
Rubidium and caesium are often used in [[atomic clock]]s.<ref name="atomic-clocks">{{cite web|title = Cesium Atoms at Work|publisher=Time Service Department—U.S. Naval Observatory—Department of the Navy|url = http://tycho.usno.navy.mil/cesium.html|accessdate =20 December 2009}}</ref> Caesium atomic clocks are extraordinarily accurate; if a clock had been made at the time of the dinosaurs, it would be off by less than four seconds (after 80 million years).<ref name="pubs.usgs">{{cite web|url = http://pubs.usgs.gov/of/2004/1432/2004-1432.pdf|format = PDF|publisher = United States Geological Survey|accessdate =27 December 2009|title = Mineral Commodity Profile: Cesium|first1 = William C.|last1 = Butterman|first2 = William E.|last2 = Brooks|first3 = Robert G.|last3 = Reese, Jr.|year=2004| archiveurl= http://web.archive.org/web/20091122210358/http://pubs.usgs.gov/of/2004/1432/2004-1432.pdf| archivedate= 22 November 2009 <!--DASHBot-->| deadurl= no}}</ref> For that reason, caesium atoms are used as the definition of the second.<ref name="nist-second">{{cite web|title = The NIST reference on Constants, Units, and Uncertainty|publisher=National Institute of Standards and Technology|url=http://physics.nist.gov/cuu/Units/second.html}}</ref> Rubidium ions are often used in purple [[firework]]s,<ref>{{Cite journal |first = E.-C. |last = Koch |title = Special Materials in Pyrotechnics, Part II: Application of Caesium and Rubidium Compounds in Pyrotechnics |journal = Journal Pyrotechnics |year = 2002 |volume = 15 |pages = 9–24 |url=http://www.jpyro.com/wp/?p=179}}</ref> and caesium is often used in drilling fluids in the petroleum industry.<ref name="pubs.usgs" /><ref>{{cite book|title = Exploring Chemical Elements and their Compounds|author = Heiserman, David L.|publisher = McGraw-Hill|year = 1992|isbn = 0-8306-3015-5|pages = 201–203}}</ref><br />
<br />
Francium has no commercial applications,<ref name="nbb" /><ref name="elemental"/><ref>{{cite web| last = Winter| first = Mark| title = Uses| work = Francium| publisher = The University of Sheffield|url = http://www.webelements.com/webelements/elements/text/Fr/uses.html| accessdate =25 March 2007| archiveurl= http://web.archive.org/web/20070331031655/http://www.webelements.com/webelements/elements/text/Fr/uses.html| archivedate= 31 March 2007 <!--DASHBot-->| deadurl= no}}</ref> but because of francium's relatively simple [[atomic structure]], among other things, it has been used in [[spectroscopy]] experiments, leading to more information regarding [[energy level]]s and the [[coupling constant]]s between [[subatomic particle]]s.<ref>{{cite journal| last = Gomez| first = E| last2= Orozco|first2=L A|last3=Sprouse|first3=G D| title = Spectroscopy with trapped francium: advances and perspectives for weak interaction studies| journal = Rep. Prog. Phys.| volume = 69| issue = 1| pages = 79–118| date = 7 November 2005|doi = 10.1088/0034-4885/69/1/R02|bibcode = 2006RPPh...69...79G }}</ref> Studies on the light emitted by laser-trapped francium-210 ions have provided accurate data on transitions between atomic energy levels, similar to those predicted by [[quantum mechanics|quantum theory]].<ref>{{cite journal|last = Peterson|first = I|title = Creating, cooling, trapping francium atoms|page= 294|journal= Science News|date = 11 May 1996|url = http://www.sciencenews.org/pages/pdfs/data/1996/149-19/14919-06.pdf|accessdate =11 September 2009|volume=149|issue=19|doi = 10.2307/3979560}}</ref><br />
<br />
==Biological role and precautions==<br />
[[File:Lithium carbonate.jpg|thumb|right|[[Lithium carbonate]]]]<br />
<br />
Lithium naturally only occurs in traces in biological systems and has no known biological role, but does have effects on the body when ingested.<ref name="webelements-lithium"/> [[Lithium carbonate]] is used as a [[mood stabiliser]] in [[psychiatry]] to treat [[bipolar disorder]] ([[manic-depression]]) in daily doses of about 0.5 to 2&nbsp;grams, although there are side-effects.<ref name="webelements-lithium"/> Excessive ingestion of lithium causes drowsiness, slurred speech and vomiting, among other symptoms,<ref name="webelements-lithium"/> and [[poison]]s the [[central nervous system]],<ref name="webelements-lithium"/> which is dangerous as the required dosage of lithium to treat bipolar disorder is only slightly lower than the toxic dosage.<ref name="webelements-lithium">{{cite web|publisher=Webelements|title=WebElements Periodic Table of the Elements {{pipe}} Lithium {{pipe}} biological information |url=http://www.webelements.com/lithium/biology.html |author=Winter, Mark |accessdate=15 February 2011}}</ref><ref name="theodoregray-lithium">{{cite web |url=http://www.theodoregray.com/periodictable/Elements/003/index.s7.html |title=Facts, pictures, stories about the element Lithium in the Periodic Table |author=[[Theodore Gray|Gray, Theodore]] |work=theodoregray.com |accessdate=9 January 2012}}</ref> Its biochemistry, the way it is handled by the human body and studies using rats and goats suggest that it is an [[essential element|essential]] [[trace element]], although the natural biological function of lithium in humans has yet to be identified.<ref>{{cite journal |last1=Howland |first1=Robert H. |date=September 2007 |title=Lithium: Underappreciated and Underused? |journal=Psychiatric Annals |volume=37 |issue=9 |url=http://www.healio.com/journals/psycann/%7B19970467-072d-409e-8cda-9323edb2f73d%7D/lithium-underappreciated-and-underused |accessdate=6 November 2012}}</ref><ref>{{cite journal |last1=Zarse |first1=Kim |last2=Terao |first2=Takeshi |last3=Tian |first3=Jing |last4=Iwata |first4=Noboru |last5=Ishii |first5=Nobuyoshi |last6=Ristow |first6=Michael |date=August 2011 |title=Low-dose lithium uptake promotes longevity in humans and metazoans |journal=European Journal of Nutrition |volume=50 |issue=5 |pages=387–9 |publisher=Springer |doi=10.1007/s00394-011-0171-x |url=http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3151375/pdf/394_2011_Article_171.pdf |pmc=3151375 |accessdate=6 November 2012 |pmid=21301855}}</ref><br />
<br />
Sodium and potassium occur in all known biological systems, generally functioning as [[electrolytes]] inside and outside [[cell (biology)|cells]].<ref name="webelements-potassium"/><ref name="webelements-sodium"/> Sodium is an essential nutrient that regulates blood volume, blood pressure, osmotic equilibrium and [[pH]]; the minimum physiological requirement for sodium is 500&nbsp;milligrams per day.<ref name=r31>{{cite web|url=http://nuinfo-proto4.northwestern.edu/nutrition/factsheets/sodium.pdf|title=Sodium|publisher=Northewestern University|accessdate=21 November 2011}}{{dead link|date=August 2012}}</ref> [[Sodium chloride]] (also known as common salt) is the principal source of sodium in the diet, and is used as seasoning and preservative, such as for [[pickling]] and [[Jerky (food)|jerky]]; most of it comes from processed foods.<ref>{{cite web|url=http://health.ltgovernors.com/sodium-and-potassium-health-facts.html|title=Sodium and Potassium Quick Health Facts|accessdate=7 November 2011}}</ref> The [[Dietary Reference Intake|DRI]] for sodium is 1.5&nbsp;grams per day,<ref>{{cite web|title=Dietary Reference Intakes: Water, Potassium, Sodium, Chloride, and Sulfate|url=http://www.iom.edu/Reports/2004/Dietary-Reference-Intakes-Water-Potassium-Sodium-Chloride-and-Sulfate.aspx|publisher=Food and Nutrition Board, [[Institute of Medicine]], [[United States National Academies]]|date=11 February 2004|accessdate=23 November 2011}}</ref> but most people in the United States consume more than 2.3&nbsp;grams per day,<ref>{{cite book|author1=U.S. Department of Agriculture|author2=U.S. Department of Health and Human Services|authorlink1=United States Department of Agriculture|authorlink2=United States Department of Health and Human Services|title=Dietary Guidelines for Americans, 2010|page=22|edition=7th|date=December 2010|url=http://www.cnpp.usda.gov/Publications/DietaryGuidelines/2010/PolicyDoc/PolicyDoc.pdf|format=PDF|accessdate=23 November 2011|isbn=978-0-16-087941-8|oclc=738512922}}</ref> the minimum amount that promotes hypertension;<ref>{{cite journal|pmid=15369026|year=2004|last1=Geleijnse|first1=J. M.|last2=Kok|first2=F. J.|last3=Grobbee|first3=D. E.|title=Impact of dietary and lifestyle factors on the prevalence of hypertension in Western populations|volume=14|issue=3|pages=235–239|journal=European Journal of Public Health|doi=10.1093/eurpub/14.3.235}}</ref> this in turn causes 7.6 million premature deaths worldwide.<ref>{{cite journal|pmid=18456100|url=http://www.worldactiononsalt.com/evidence/docs/thelancet_hypertension_05.08.pdf|year=2008|last1=Lawes|first1=C. M.|last2=Vander Hoorn|first2=S.|last3=Rodgers|first3=A.|author4=International Society of Hypertension|title=Global burden of blood-pressure-related disease, 2001|volume=371|issue=9623|pages=1513–1518|doi=10.1016/S0140-6736(08)60655-8|journal=Lancet}}{{dead link|date=May 2012}}</ref><br />
<br />
Potassium is the major [[cation]] (positive ion) inside [[Cell (biology)|animal cells]],<ref name="webelements-potassium">{{cite web|url=http://www.webelements.com/potassium/biology.html|title=WebElements Periodic Table of the Elements {{pipe}} Potassium {{pipe}} biological information |publisher=WebElements |author= Winter, Mark |accessdate=13 January 2012}}</ref> while sodium is the major cation outside animal cells.<ref name="webelements-potassium" /><ref name="webelements-sodium">{{cite web|url=http://www.webelements.com/sodium/biology.html|title=WebElements Periodic Table of the Elements {{pipe}} Sodium {{pipe}} biological information |publisher=WebElements |author= Winter, Mark |accessdate=13 January 2012}}</ref> The [[concentration]] differences of these charged particles causes a difference in [[electric potential]] between the inside and outside of cells, known as the [[membrane potential]]. The balance between potassium and sodium is maintained by [[ion pumps]] in the [[cell membrane]].<ref name="pmid16253415">{{cite journal |author=Mikko Hellgren, Lars Sandberg, Olle Edholm |title=A comparison between two prokaryotic potassium channels (K<sub>ir</sub>Bac1.1 and KcsA) in a molecular dynamics (MD) simulation study |journal=Biophys. Chem. |volume=120 |issue=1 |pages=1–9 |year=2006 |pmid=16253415 |doi=10.1016/j.bpc.2005.10.002}}</ref> The cell membrane potential created by potassium and sodium ions allows the cell to generate an [[action potential]]—a "spike" of electrical discharge. The ability of cells to produce electrical discharge is critical for body functions such as [[neurotransmission]], muscle contraction, and heart function.<ref name="pmid16253415"/><br />
<br />
[[File:GoiâniaRadiationsource.gif|thumb|400px|right|A wheel type radiotherapy device which has a long [[collimator]] to focus the radiation into a narrow beam. The caesium-137 chloride radioactive source is the blue square, and gamma rays are represented by the beam emerging from the aperture. This was the radiation source involved in the Goiânia accident, containing about 93&nbsp;grams of caesium-137 chloride.]]<br />
Rubidium has no known biological role, but may help stimulate [[metabolism]],<ref name="webelements-rubidium">{{cite web|publisher=Webelements|title=WebElements Periodic Table of the Elements {{pipe}} Rubidium {{pipe}} biological information |url=http://www.webelements.com/rubidium/biology.html |author=Winter, Mark |accessdate=15 February 2011}}</ref><ref>{{cite journal |last1 = Relman |first1 = AS |title =The Physiological Behavior of Rubidium and Cesium in Relation to That of Potassium |journal = The Yale journal of biology and medicine |volume = 29 |issue = 3 |pages = 248–62 |year = 1956| pmid = 13409924|pmc = 2603856}}</ref><ref name="jcp.sagepub.com">{{cite journal | last1 = Meltzer | first1 = HL | title = A pharmacokinetic analysis of long-term administration of rubidium chloride | url = http://jcp.sagepub.com/content/31/2/179 | journal = Journal of clinical pharmacology | volume = 31 | issue = 2 | pages = 179–84 | year = 1991 | pmid = 2010564 | doi=10.1002/j.1552-4604.1991.tb03704.x}}</ref> and, similarly to caesium,<ref name="webelements-rubidium" /><ref name="webelements-caesium" /> replace potassium in the body causing [[hypokalemia|potassium deficiency]].<ref name="webelements-rubidium" /><ref name="jcp.sagepub.com"/> Caesium compounds are rarely encountered by most people, but most caesium compounds are mildly toxic because of chemical similarity of caesium to potassium, allowing the caesium to replace the potassium in the body, causing potassium deficiency.<ref name="webelements-caesium">{{cite web|url=http://www.webelements.com/caesium/biology.html|title=WebElements Periodic Table of the Elements {{pipe}} Caesium {{pipe}} biological information |publisher=WebElements |author= Winter, Mark |accessdate=13 January 2012}}</ref> Exposure to large amounts of caesium compounds can cause [[Irritability|hyperirritability]] and [[spasm]]s, but as such amounts would not ordinarily be encountered in natural sources, caesium is not a major chemical environmental pollutant.<ref>{{cite journal|doi = 10.1080/10934528109375003|title = Cesium in mammals: Acute toxicity, organ changes and tissue accumulation|year = 1981|last1 = Pinsky|first1 = Carl|first2 = Ranjan|first3 = J. R.|first4 = Jasper|first5 = Claude|first6 = James|journal = Journal of Environmental Science and Health, Part A|volume = 16|pages = 549– 567 |last2 = Bose|last3 = Taylor|last4 = McKee|last5 = Lapointe|last6 = Birchall|issue = 5}}</ref> The [[median lethal dose]] (LD<sub>50</sub>) value for [[caesium chloride]] in mice is 2.3&nbsp;g per kilogram, which is comparable to the LD<sub>50</sub> values of [[potassium chloride]] and [[sodium chloride]].<ref>{{cite journal|doi = 10.1016/0041-008X(75)90216-1|title = Acute toxicity of cesium and rubidium compounds|year = 1975|last1 = Johnson|first1 = Garland T.|journal = [[Toxicology and Applied Pharmacology]]|volume = 32|pages = 239–245|pmid = 1154391|first2 = Trent R.|first3 = D. Wagner|issue = 2|last2 = Lewis|last3 = Wagner}}</ref> Caesium chloride has been promoted as an alternative cancer therapy,<ref>{{cite journal | author = Sartori H. E. | year = 1984 | title = Cesium therapy in cancer patients | url = | journal = Pharmacol Biochem Behav | volume = 21 | issue = Suppl 1| pages = 11–13 | pmid = 6522427 | doi = 10.1016/0091-3057(84)90154-0 }}</ref> but has been linked to the deaths of over 50 patients, on whom it was used as part of a scientifically unvalidated cancer treatment.<ref>Wood, Leonie. {{cite web|url=http://www.smh.com.au/lifestyle/lifematters/cured-cancer-patients-died-court-told-20101119-180z9.html |title='Cured' cancer patients died, court told |work=The Sydney Morning Herald |date=20 November 2010}}</ref> [[Radioisotope]]s of caesium require special precautions: the improper handling of caesium-137 [[gamma ray]] sources can lead to release of this radioisotope and radiation injuries. Perhaps the best-known case is the Goiânia accident of 1987, in which an improperly-disposed-of radiation therapy system from an abandoned clinic in the city of [[Goiânia]], [[Brazil]], was scavenged from a junkyard, and the glowing [[caesium chloride|caesium salt]] sold to curious, uneducated buyers. This led to four deaths and serious injuries from radiation exposure. Together with [[caesium-134]], [[iodine-131]], and [[strontium-90]], caesium-137 was among the isotopes distributed by the [[Chernobyl disaster]] which constitute the greatest risk to health.<ref name="IAEA"/><br />
<br />
Francium has no biological role<ref name="webelements-francium">{{cite web|url=http://www.webelements.com/francium/biology.html|title=WebElements Periodic Table of the Elements {{pipe}} Francium {{pipe}} biological information |publisher=WebElements |author= Winter, Mark |accessdate=15 February 2011}}</ref> and is most likely to be toxic due to its extreme radioactivity, causing [[acute radiation syndrome|radiation poisoning]],<ref name="rsc-francium">{{cite web |url=http://www.rsc.org/periodic-table/element/87/Francium |title=Francium – Element information, properties and uses {{pipe}} Periodic Table |year=2012 |work=Visual Elements Periodic Table |publisher=[[Royal Society of Chemistry]] |accessdate=27 June 2012}}</ref> but since the greatest quantity of francium ever assembled to date is <!--a sphere of radius 1&nbsp;mm--> about 300,000 neutral atoms,<ref name="chemnews">{{cite journal|url=http://pubs.acs.org/cen/80th/francium.html|title=Francium|journal=Chemical and Engineering News|year=2003|author=Luis A. Orozco }}</ref> it is unlikely that most people will ever encounter francium.<br />
<br />
==Notes==<br />
{{reflist|group="note"|30em}}<br />
<br />
==References==<br />
{{reflist|30em}}<br />
<br />
==Further reading==<br />
{{refbegin|30em}}<br />
*{{Cite journal<br />
| author=Bauer, Brent A., Robert Houlihan, Michael J. Ackerman, Katya Johnson, and Himeshkumar Vyas<br />
| year=2006<br />
| title=Acquired Long QT Syndrome Secondary to Cesium Chloride Supplement<br />
| journal=[[Journal of Alternative and Complementary Medicine]]<br />
| volume=12 | pages=1011–1014<br />
| doi=10.1089/acm.2006.12.1011<br />
| pmid=17212573<br />
| issue=10<br />
}}<br />
*{{Cite journal<br />
| author=Campbell, Linda M., Aaron T. Fisk, Xianowa Wang, Gunter Kock, and Derek C. Muir<br />
| year= 2005<br />
| title=Evidence for Biomagnification of Rubidium in Freshwater and Marine Food Webs<br />
| journal=Canadian Journal of Fisheries and Aquatic Sciences<br />
| volume=62 | pages=1161–1167<br />
| doi=10.1139/f05-027<br />
| issue=5<br />
}}<br />
*{{Cite conference<br />
| author=Chang, Cheng-Hung, and Tian Y. Tsong<br />
| year=2005<br />
| title=Stochastic Resonance of Na, K-Ion Pumps on the Red Cell Membrane<br />
| booktitle=AIP Conference Proceedings: 18th International Conference on Noise and Fluctuations<br />
| volume=780 |pages=587–590<br />
| publisher=[[American Institute of Physics]]<br />
| doi=10.1063/1.2036821<br />
| isbn=0-7354-0267-1<br />
}}<br />
*{{Cite journal<br />
| author=Erermis, Serpil, Muge Tamar, Hatice Karasoy, Tezan Bildik, Eyup S. Ercan, and Ahmet Gockay<br />
| year=1997<br />
| title=Double-Blind Randomised Trial of Modest Salt Restriction in Older People<br />
| journal=[[The Lancet|Lancet]]<br />
| volume=350 | pages= 850–854<br />
| doi=10.1016/S0140-6736(97)02264-2<br />
| pmid = 9310603<br />
| issue=9081<br />
}}<br />
*{{Cite journal<br />
| author=Joffe, Russell T., Stephen T. Sokolov and Anthony J. Levitt<br />
| year=2006<br />
| title=Lithium and Triiodothyronine Augmentation of Antidepressants<br />
| journal=Canadian Journal of Psychiatry<br />
| volume=51 | pages= 791–3<br />
| pmid=17168254<br />
| issue=12<br />
}}<br />
*{{Cite journal<br />
| author=Krachler, M, and E Rossipal<br />
| year=1999<br />
| title=Trace Elements Transfer From Mother to the Newborn – Investigations on Triplets of Colostrum, Maternal and Umbilical Sera<br />
| journal=European Journal of Clinical Nutrition<br />
| volume=53 | pages=486–494<br />
| doi=10.1038/sj.ejcn.1600781<br />
| pmid=10403586<br />
| issue=6<br />
}}<br />
*{{Cite journal<br />
| author=Stein, Benjamin P., Stephen G. Benka, Phillip F. Schewe, and Bertram Schwarzhild<br />
| year=1996<br />
| title=Physics Update<br />
| journal=[[Physics Today]]<br />
| volume=49 | issue=6 | page=9<br />
| doi=10.1063/1.2807642<br />
|bibcode = 1996PhT....49f...9S }}<br />
{{refend}}<br />
<br />
==External links==<br />
*{{cite web|title=Group 1: The Alkali Metals|url=http://www.chemsoc.org/Viselements/pages/data/intro_groupi_data.html|work=Visual Elements|publisher=[[Royal Society of Chemistry]]|accessdate=8 December 2009}}<br />
*[http://www.chemguide.co.uk/inorganic/group1/properties.html Atomic and Physical Properties of the Group 1 Elements] An in-depth look at alkali metals<br />
*[http://www.theodoregray.com/periodictable/AlkaliBangs/index.html Alkali Metal Bangs] Filmed reactions of five-gram samples of the alkali metals with water<br />
{{PeriodicTablesFooter}}<br />
{{Compact periodic table}}<br />
{{Alkali metals}}<br />
{{Use dmy dates|date=August 2012}}<br />
<br />
{{DEFAULTSORT:Alkali Metal}}<br />
[[Category:Alkali metals| ]]<br />
[[Category:Groups in the periodic table]]<br />
[[Category:Periodic table]]</div>Adminhttps://en.formulasearchengine.com/index.php?title=Axiom&diff=218498Axiom2014-07-31T13:54:35Z<p>Admin: Reverted edits by MediaWiki spam cleanup (talk) to last revision by Paul August</p>
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<div>{{Use dmy dates|date=June 2013}}<br />
{{about|logical propositions}}<br />
{{redirect|Postulation|the term in algebraic geometry|Postulation (algebraic geometry)}}<br />
An '''axiom''', or '''postulate''', is a premise or starting point of reasoning. A self-evident principle or one that is accepted as true without proof as the basis for argument; a postulate. As classically conceived, an axiom is a premise so [[Self-evidence|evident]] as to be accepted as true without controversy.<ref><br />
"A proposition that commends itself to general acceptance; a well-established or universally conceded principle; a maxim, rule, law" axiom, n., definition 1a. ''Oxford English Dictionary'' Online, accessed 2012-04-28. Cf. Aristotle, ''[[Posterior Analytics]]'' I.2.72a18-b4.</ref><!-- HIDDEN UNTIL SOURCED —it is better known and more firmly believed than the conclusion.{{citation needed|date=May 2012}}--><br />
The word comes from the Greek ἀξίωμα (''āxīoma'') 'that which is thought worthy or fit' or 'that which commends itself as evident.'<ref>Cf. axiom, n., etymology. ''Oxford English Dictionary'', accessed 2012-04-28.</ref><ref>Oxford American College Dictionary: "n. a statement or proposition that is regarded as being established, accepted, or self-evidently true. ORIGIN: late 15th cent.: ultimately from Greek axiōma 'what is thought fitting,' from axios 'worthy.' http://www.highbeam.com/doc/1O997-axiom.html {{subscription}}</ref><br />
As used in modern [[logic]], an axiom is simply a premise or starting point for reasoning.<ref>"A proposition (whether true or false)" axiom, n., definition 2. ''Oxford English Dictionary'' Online, accessed 2012-04-28.</ref> Axioms define and delimit the realm of [[analysis]]; the relative truth of an axiom is taken for granted within the particular domain of analysis, and serves as a starting point for deducing and inferring other relative truths. No explicit view regarding the absolute truth of axioms is ever taken in the context of modern mathematics, as such a thing is considered to be an irrelevant and impossible contradiction in terms.<br />
<br />
In [[mathematics]], the term ''axiom'' is used in two related but distinguishable senses: [[#Logical axioms|"logical axioms"]] and [[#Non-logical axioms|"non-logical axioms"]]. Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms (e.g., {{nowrap|1= ''a'' + ''b'' = ''b'' + ''a''}}) are actually defining properties for the domain of a specific mathematical theory (such as [[arithmetic]]). When used in the latter sense, "axiom," "postulate", and "assumption" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. As modern mathematics admits multiple, equally "true" systems of logic, precisely the same thing must be said for logical axioms - they both define and are specific to the particular system of logic that is being invoked. To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain.<br />
<br />
In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Within the system they define, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by [[mathematical proof]]s, simply because they are starting points; there is nothing else from which they logically follow otherwise they would be classified as theorems. However, an axiom in one system may be a theorem in another, and vice versa.<br />
<br />
==Etymology==<br />
The word "axiom" comes from the [[Greek language|Greek]] word {{lang|grc|ἀξίωμα}} (''axioma''), a [[verbal noun]] from the verb {{lang|grc|ἀξιόειν}} (''axioein''), meaning "to deem worthy", but also "to require", which in turn comes from {{lang|grc|ἄξιος}} (''axios''), meaning "being in balance", and hence "having (the same) value (as)", "worthy", "proper". Among the [[ancient Greece|ancient Greek]] [[philosopher]]s an axiom was a claim which could be seen to be true without any need for proof.<br />
<br />
The root meaning of the word 'postulate' is to 'demand'; for instance, [[Euclid]] demands of us that we agree that some things can be done, e.g. any two points can be joined by a straight line, etc.<ref>Wolff, P. Breakthroughs in Mathematics, 1963, New York: New American Library, pp&nbsp;47–8</ref><br />
<br />
Ancient geometers maintained some distinction between axioms and postulates. While commenting Euclid's books [[Proclus]] remarks that "[[Geminus]] held that this [4th] Postulate should not be classed as a postulate but as an axiom, since it does not, like the first three Postulates, assert the possibility of some construction but expresses an essential property".<ref>[[T. L. Heath|Heath, T.]] 1956. The Thirteen Books of Euclid's Elements. New York: Dover. ''p200''</ref> [[Boethius]] translated 'postulate' as ''petitio'' and called the axioms ''notiones communes'' but in later manuscripts this usage was not always strictly kept.<br />
<br />
==Historical development==<br />
<br />
===Early Greeks===<br />
The logico-deductive method whereby conclusions (new knowledge) follow from premises (old knowledge) through the application of sound arguments ([[syllogisms]], rules of inference), was developed by the ancient Greeks, and has become the core principle of modern mathematics.{{Citation needed|date=August 2008}} [[tautology (logic)|Tautologies]] excluded, nothing can be deduced if nothing is assumed. Axioms and postulates are the basic assumptions underlying a given body of deductive knowledge. They are accepted without demonstration. All other assertions ([[theorem]]s, if we are talking about mathematics) must be proven with the aid of these basic assumptions. However, the interpretation of mathematical knowledge has changed from ancient times to the modern, and consequently the terms ''axiom'' and ''postulate'' hold a slightly different meaning for the present day mathematician, than they did for [[Aristotle]] and [[Euclid]].<br />
<br />
The ancient Greeks considered [[geometry]] as just one of several [[science]]s,{{Citation needed|date=August 2008}} and held the theorems of geometry on par with scientific facts. As such, they developed and used the logico-deductive method as a means of avoiding error, and for structuring and communicating knowledge. Aristotle's [[posterior analytics]] is a definitive exposition of the classical view.<br />
<br />
An "axiom", in classical terminology, referred to a self-evident assumption common to many branches of science. A good example would be the assertion that <blockquote>''When an equal amount is taken from equals, an equal amount results.''</blockquote><br />
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At the foundation of the various sciences lay certain additional hypotheses which were accepted without proof. Such a hypothesis was termed a ''postulate''. While the axioms were common to many sciences, the postulates of each particular science were different. Their validity had to be established by means of real-world experience. Indeed, Aristotle warns that the content of a science cannot be successfully communicated, if the learner is in doubt about the truth of the postulates.<ref>Aristotle, Metaphysics Bk IV, Chapter 3, 1005b "Physics also is a kind of Wisdom, but it is not the first kind. – And the attempts of some of those who discuss the terms on which truth should be accepted, are due to want of training in logic; for they should know these things already when they come to a special study, and not be inquiring into them while they are listening to lectures on it." W.D. Ross translation, in The Basic Works of Aristotle, ed. Richard McKeon, (Random House, New York, 1941)|date=June 2011</ref><br />
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The classical approach is well-illustrated by [[Euclid's Elements]], where a list of postulates is given (common-sensical geometric facts drawn from our experience), followed by a list of "common notions" (very basic, self-evident assertions).<br />
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:;Postulates<br />
:# It is possible to draw a [[straight line]] from any point to any other point.<br />
:# It is possible to extend a line segment continuously in both directions.<br />
:# It is possible to describe a [[circle]] with any center and any radius.<br />
:# It is true that all [[right angle]]s are equal to one another.<br />
:# ("[[Parallel postulate]]") It is true that, if a straight line falling on two straight lines make the [[polygon|interior angles]] on the same side less than two right angles, the two straight lines, if produced indefinitely, [[Line-line intersection|intersect]] on that side on which are the [[angles]] less than the two right angles.<br />
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:;Common notions:<br />
:# Things which are equal to the same thing are also equal to one another.<br />
:# If equals are added to equals, the wholes are equal.<br />
:# If equals are subtracted from equals, the remainders are equal.<br />
:# Things which coincide with one another are equal to one another.<br />
:# The whole is greater than the part.<br />
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===Modern development===<br />
A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions (axioms, postulates, [[propositional logic|propositions]], theorems) and definitions. One must concede the need for [[primitive notion]]s, or undefined terms or concepts, in any study. Such abstraction or formalization makes mathematical knowledge more general, capable of multiple different meanings, and therefore useful in multiple contexts. [[Alessandro Padoa]], [[Mario Pieri]], and [[Giuseppe Peano]] were pioneers in this movement.<br />
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Structuralist mathematics goes further, and develops theories and axioms (e.g. [[Field theory (mathematics)|field theory]], [[group (mathematics)|group theory]], [[topological space|topology]], [[linear space|vector spaces]]) without ''any'' particular application in mind. The distinction between an "axiom" and a "postulate" disappears. The postulates of Euclid are profitably motivated by saying that they lead to a great wealth of geometric facts. The truth of these complicated facts rests on the acceptance of the basic hypotheses. However, by throwing out Euclid's fifth postulate we get theories that have meaning in wider contexts, [[hyperbolic geometry]] for example. We must simply be prepared to use labels like "line" and "parallel" with greater flexibility. The development of hyperbolic geometry taught mathematicians that postulates should be regarded as purely formal statements, and not as facts based on experience.<br />
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When mathematicians employ the [[Field (mathematics)|field]] axioms, the intentions are even more abstract. The propositions of field theory do not concern any one particular application; the mathematician now works in complete abstraction. There are many examples of fields; field theory gives correct knowledge about them all.<br />
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It is not correct to say that the axioms of field theory are "propositions that are regarded as true without proof." Rather, the field axioms are a set of constraints. If any given system of addition and multiplication satisfies these constraints, then one is in a position to instantly know a great deal of extra information about this system.<br />
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Modern mathematics formalizes its foundations to such an extent that mathematical theories can be regarded as mathematical objects, and mathematics itself can be regarded as a branch of [[logic]]. [[Gottlob Frege|Frege]], [[Bertrand Russell|Russell]], [[Henri Poincaré|Poincaré]], [[David Hilbert|Hilbert]], and [[Kurt Gödel|Gödel]] are some of the key figures in this development.<br />
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In the modern understanding, a set of axioms is any [[Class (set theory)|collection]] of formally stated assertions from which other formally stated assertions follow by the application of certain well-defined rules. In this view, logic becomes just another formal system. A set of axioms should be [[consistent]]; it should be impossible to derive a contradiction from the axiom. A set of axioms should also be non-redundant; an assertion that can be deduced from other axioms need not be regarded as an axiom.<br />
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It was the early hope of modern logicians that various branches of mathematics, perhaps all of mathematics, could be derived from a consistent collection of basic axioms. An early success of the formalist program was Hilbert's formalization of [[Euclidean geometry]], and the related demonstration of the consistency of those axioms.<br />
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In a wider context, there was an attempt to base all of mathematics on [[Georg Cantor|Cantor's]] [[set theory]]. Here the emergence of [[Russell's paradox]], and similar antinomies of [[naïve set theory]] raised the possibility that any such system could turn out to be inconsistent.<br />
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The formalist project suffered a decisive setback, when in 1931 Gödel showed that it is possible, for any sufficiently large set of axioms ([[peano arithmetic|Peano's axioms]], for example) to construct a statement whose truth is independent of that set of axioms. As a [[corollary]], Gödel proved that the consistency of a theory like [[Peano arithmetic]] is an unprovable assertion within the scope of that theory.<br />
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It is reasonable to believe in the consistency of Peano arithmetic because it is satisfied by the system of [[natural number]]s, an [[Infinity|infinite]] but intuitively accessible formal system. However, at present, there is no known way of demonstrating the consistency of the modern [[Zermelo–Fraenkel axioms]] for set theory. The [[axiom of choice]], a key hypothesis of this theory, remains a very controversial assumption. Furthermore, using techniques of [[forcing (mathematics)|forcing]] ([[Paul Cohen (mathematician)|Cohen]]) one can show that the [[continuum hypothesis]] (Cantor) is independent of the Zermelo–Fraenkel axioms. Thus, even this very general set of axioms cannot be regarded as the definitive foundation for mathematics.<br />
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===Other sciences===<br />
Axioms play a key role not only in mathematics, but also in other sciences, notably in [[theoretical physics]]. In particular, the monumental work of [[Isaac Newton]] is essentially based on [[Euclid]]'s axioms, augmented by a postulate on the non-relation of [[spacetime]] and the physics taking place in it at any moment.<br />
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In 1905, Newton's axioms were replaced by those of [[Albert Einstein]]'s [[special relativity]], and later on by those of [[general relativity]].<br />
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Another paper of Albert Einstein and coworkers (see [[EPR paradox]]), almost immediately contradicted by [[Niels Bohr]], concerned the interpretation of [[quantum mechanics]]. This was in 1935. According to Bohr, this new theory should be [[Probability theory|probabilistic]], whereas according to Einstein it should be [[deterministic]]. Notably, the underlying quantum mechanical theory, i.e. the set of "theorems" derived by it, seemed to be identical. Einstein even assumed that it would be sufficient to add to quantum mechanics "hidden variables" to enforce determinism. However, thirty years later, in 1964, [[John Stewart Bell|John Bell]] found a theorem, involving complicated optical correlations (see [[Bell inequalities]]), which yielded measurably different results using Einstein's axioms compared to using Bohr's axioms. And it took roughly another twenty years until an experiment of [[Alain Aspect]] got results in favour of Bohr's axioms, not Einstein's. (Bohr's axioms are simply: The theory should be probabilistic in the sense of the [[Copenhagen interpretation]].)<br />
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As a consequence, it is not necessary to explicitly cite Einstein's axioms, the more so since they concern subtle points on the "reality" and "locality" of experiments.<br />
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Regardless, the role of axioms in mathematics and in the above-mentioned sciences is different. In mathematics one neither "proves" nor "disproves" an axiom for a set of theorems; the point is simply that in the conceptual realm identified by the axioms, the theorems logically follow. In contrast, in physics a comparison with experiments always makes sense, since a [[Falsifiability|falsified]] physical theory needs modification.<br />
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==Mathematical logic==<br />
In the field of [[mathematical logic]], a clear distinction is made between two notions of axioms: ''logical'' and ''non-logical'' (somewhat similar to the ancient distinction between "axioms" and "postulates" respectively).<br />
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===Logical axioms===<br />
These are certain [[Formula (mathematical logic)|formulas]] in a [[formal language]] that are universally [[validity|valid]], that is, formulas that are [[satisfiability|satisfied]] by every [[Assignment (mathematical logic)|assignment]] of values. Usually one takes as logical axioms ''at least'' some minimal set of tautologies that is sufficient for proving all [[tautology (logic)|tautologies]] in the language; in the case of [[predicate logic]] more logical axioms than that are required, in order to prove [[logical truth]]s that are not tautologies in the strict sense.<br />
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====Examples====<br />
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=====Propositional logic=====<br />
In [[propositional logic]] it is common to take as logical axioms all formulae of the following forms, where <math>\phi</math>, <math>\chi</math>, and <math>\psi</math> can be any formulae of the language and where the included [[Logical connective|primitive connectives]] are only "<math>\neg</math>" for [[negation]] of the immediately following proposition and "<math>\to\,</math>" for [[Entailment|implication]] from antecedent to consequent propositions:<br />
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#<math>\phi \to (\psi \to \phi)</math><br />
#<math>(\phi \to (\psi \to \chi)) \to ((\phi \to \psi) \to (\phi \to \chi))</math><br />
#<math>(\lnot \phi \to \lnot \psi) \to (\psi \to \phi).</math><br />
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Each of these patterns is an ''[[axiom schema]]'', a rule for generating an infinite number of axioms. For example, if <math>A</math>, <math>B</math>, and <math>C</math> are [[propositional variable]]s, then <math>A \to (B \to A)</math> and <math>(A \to \lnot B) \to (C \to (A \to \lnot B))</math> are both instances of axiom schema 1, and hence are axioms. It can be shown that with only these three axiom schemata and ''[[modus ponens]]'', one can prove all tautologies of the propositional calculus. It can also be shown that no pair of these schemata is sufficient for proving all tautologies with ''modus ponens''.<br />
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Other axiom schemas involving the same or different sets of primitive connectives can be alternatively constructed.<ref>Mendelson, "6. Other Axiomatizations" of Ch. 1</ref><br />
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These axiom schemata are also used in the [[predicate calculus]], but additional logical axioms are needed to include a quantifier in the calculus.<ref>Mendelson, "3. First-Order Theories" of Ch. 2</ref><br />
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=====Mathematical logic=====<br />
<div style="border: 1px solid #CCCCCC; padding-left: 5px; "><br />
'''Axiom of Equality.''' Let <math>\mathfrak{L}\,</math> be a [[first-order language]]. For each variable <math>x\,</math>, the formula<br />
<br />
<center><br />
<math>x = x\,</math><br />
</center><br />
<br />
is universally valid.<br />
</div><br />
<br />
This means that, for any [[Free variables and bound variables|variable symbol]] <math>x\,,</math> the formula <math>x = x\,</math> can be regarded as an axiom. Also, in this example, for this not to fall into vagueness and a never-ending series of "primitive notions", either a precise notion of what we mean by <math>x = x\,</math> (or, for that matter, "to be equal") has to be well established first, or a purely formal and syntactical usage of the symbol <math>=\,</math> has to be enforced, only regarding it as a string and only a string of symbols, and mathematical logic does indeed do that.<br />
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Another, more interesting example [[axiom scheme]], is that which provides us with what is known as '''Universal Instantiation''':<br />
<br />
<div style="border: 1px solid #CCCCCC; padding-left: 5px; "><br />
'''Axiom scheme for Universal Instantiation.''' Given a formula <math>\phi\,</math> in a first-order language <math>\mathfrak{L}\,</math>, a variable <math>x\,</math> and a [[First order logic#Terms|term]] <math>t\,\!</math> that is [[First-order logic#Rules of inference|substitutable]] for <math>x\,</math> in <math>\phi\,</math>, the formula<br />
<br />
<center><br />
<math>\forall x \, \phi \to \phi^x_t</math><br />
</center><br />
<br />
is universally valid.<br />
</div><br />
<br />
Where the symbol <math>\phi^x_t</math> stands for the formula <math>\phi\,</math> with the term <math>t\,\!</math> substituted for <math>x\,</math>. (See [[Substitution of variables]].) In informal terms, this example allows us to state that, if we know that a certain property <math>P\,</math> holds for every <math>x\,</math> and that <math>t\,\!</math> stands for a particular object in our structure, then we should be able to claim <math>P(t)\,</math>. Again, ''we are claiming that the formula'' <math>\forall x \phi \to \phi^x_t</math> ''is valid'', that is, we must be able to give a "proof" of this fact, or more properly speaking, a ''metaproof''. Actually, these examples are ''metatheorems'' of our theory of mathematical logic since we are dealing with the very concept of ''proof'' itself. Aside from this, we can also have '''Existential Generalization''':<br />
<br />
<div style="border: 1px solid #CCCCCC; padding-left: 5px; "><br />
'''Axiom scheme for Existential Generalization.''' Given a formula <math>\phi\,</math> in a first-order language <math>\mathfrak{L}\,</math>, a variable <math>x\,</math> and a term <math>t\,\!</math> that is substitutable for <math>x\,</math> in <math>\phi\,</math>, the formula<br />
<br />
<center><br />
<math>\phi^x_t \to \exists x \, \phi</math><br />
</center><br />
<br />
is universally valid.<br />
</div><br />
<br />
===Non-logical axioms===<br />
'''Non-logical axioms''' are formulas that play the role of theory-specific assumptions. Reasoning about two different structures, for example the [[natural number]]s and the [[integer]]s, may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such as [[group (algebra)|groups]]). Thus non-logical axioms, unlike logical axioms, are not ''[[Tautology (logic)|tautologies]]''. Another name for a non-logical axiom is ''postulate''.<ref>Mendelson, "3. First-Order Theories: Proper Axioms" of Ch. 2</ref><br />
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Almost every modern [[mathematical theory]] starts from a given set of non-logical axioms, and it was thought{{Citation needed|date=July 2011}} that in principle every theory could be axiomatized in this way and formalized down to the bare language of logical formulas.<!-- This turned out to be impossible{{Citation needed|date=March 2010}} and proved to be quite a story (''[[#role|see below]]''); however recently this approach has been resurrected in the form of [[neo-logicism]].--><br />
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Non-logical axioms are often simply referred to as ''axioms'' in mathematical [[discourse]]. This does not mean that it is claimed that they are true in some absolute sense. For example, in some groups, the group operation is [[commutative]], and this can be asserted with the introduction of an additional axiom, but without this axiom we can do quite well developing (the more general) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.<br />
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Thus, an ''axiom'' is an elementary basis for a [[Formal_system#Logical_system|formal logic system]] that together with the [[rules of inference]] define a '''[[deductive system]]'''.<br />
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====Examples====<br />
This section gives examples of mathematical theories that are developed entirely from a set of non-logical axioms (axioms, henceforth). A rigorous treatment of any of these topics begins with a specification of these axioms.<br />
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Basic theories, such as [[arithmetic]], [[real analysis]] and [[complex analysis]] are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of [[Zermelo–Fraenkel set theory]] with choice, abbreviated ZFC, or some very similar system of [[axiomatic set theory]] like [[Von Neumann–Bernays–Gödel set theory]], a [[conservative extension]] of ZFC. Sometimes slightly stronger theories such as [[Morse-Kelley set theory]] or set theory with a [[strongly inaccessible cardinal]] allowing the use of a [[Grothendieck universe]] are used, but in fact most mathematicians can actually prove all they need in systems weaker than ZFC, such as [[second-order arithmetic]].<br />
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The study of topology in mathematics extends all over through [[point set topology]], [[algebraic topology]], [[differential topology]], and all the related paraphernalia, such as [[homology theory]], [[homotopy theory]]. The development of ''abstract algebra'' brought with itself [[group theory]], [[ring (mathematics)|rings]] and [[field (mathematics)|fields]], [[Galois theory]].<br />
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This list could be expanded to include most fields of mathematics, including [[measure theory]], [[ergodic theory]], [[probability]], [[representation theory]], and [[differential geometry]].<br />
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[[Combinatorics]] is an example of a field of mathematics which does not, in general, follow the axiomatic method.<br />
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=====Arithmetic=====<br />
The [[Peano axioms]] are the most widely used ''axiomatization'' of [[first-order arithmetic]]. They are a set of axioms strong enough to prove many important facts about [[number theory]] and they allowed Gödel to establish his famous [[Gödel's second incompleteness theorem|second incompleteness theorem]].<ref>Mendelson, "5. The Fixed Point Theorem. Gödel's Incompleteness Theorem" of Ch. 2</ref><br />
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We have a language <math>\mathfrak{L}_{NT} = \{0, S\}\,</math> where <math>0\,</math> is a constant symbol and <math>S\,</math> is a [[unary function]] and the following axioms:<br />
<br />
# <math>\forall x. \lnot (Sx = 0) </math><br />
# <math>\forall x. \forall y. (Sx = Sy \to x = y) </math><br />
# <math>((\phi(0) \land \forall x.\,(\phi(x) \to \phi(Sx))) \to \forall x.\phi(x)</math> for any <math>\mathfrak{L}_{NT}\,</math> formula <math>\phi\ </math> with one free variable.<br />
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The standard structure is <math>\mathfrak{N} = \langle\N, 0, S\rangle\,</math> where <math>\N\,</math> is the set of natural numbers, <math>S\,</math> is the [[successor function]] and <math>0\,</math> is naturally interpreted as the number 0.<br />
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=====Euclidean geometry=====<br />
Probably the oldest, and most famous, list of axioms are the 4 + 1 [[Euclid's postulates]] of [[Euclidean geometry|plane geometry]]. The axioms are referred to as "4 + 1" because for nearly two millennia the [[parallel postulate|fifth (parallel) postulate]] ("through a point outside a line there is exactly one parallel") was suspected of being derivable from the first four. Ultimately, the fifth postulate was found to be independent of the first four. Indeed, one can assume that exactly one parallel through a point outside a line exists, or that infinitely many exist. This choice gives us two alternative forms of geometry in which the interior [[angle]]s of a [[triangle]] add up to exactly 180 degrees or less, respectively, and are known as Euclidean and [[hyperbolic geometry|hyperbolic]] geometries. If one also removes the second postulate ("a line can be extended indefinitely") then [[elliptic geometry|elliptic]] geometry arises, where there is no parallel through a point outside a line, and in which the interior angles of a triangle add up to more than 180 degrees.<br />
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=====Real analysis=====<br />
The object of study is the [[real numbers]]. The real numbers are uniquely picked out (up to [[isomorphism]]) by the properties of a ''Dedekind complete ordered field'', meaning that any nonempty set of real numbers with an upper bound has a least upper bound. However, expressing these properties as axioms requires use of [[second-order logic]]. The [[Löwenheim-Skolem theorem]]s tell us that if we restrict ourselves to [[first-order logic]], any axiom system for the reals admits other models, including both models that are smaller than the reals and models that are larger. Some of the latter are studied in [[non-standard analysis]].<br />
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===<span id="role">Role in mathematical logic</span>===<br />
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====Deductive systems and completeness====<br />
A '''[[deductive]] system''' consists of a set <math>\Lambda\,</math> of logical axioms, a set <math>\Sigma\,</math> of non-logical axioms, and a set <math>\{(\Gamma, \phi)\}\,</math> of ''rules of inference''. A desirable property of a deductive system is that it be '''complete'''. A system is said to be complete if, for all formulas <math>\phi</math>,<br />
<center><br />
<math>\text{if }\Sigma \models \phi\text{ then }\Sigma \vdash \phi</math><br />
</center><br />
<br />
that is, for any statement that is a ''logical consequence'' of <math>\Sigma\,</math> there actually exists a ''deduction'' of the statement from <math>\Sigma\,</math>. This is sometimes expressed as "everything that is true is provable", but it must be understood that "true" here means "made true by the set of axioms", and not, for example, "true in the intended interpretation". [[Gödel's completeness theorem]] establishes the completeness of a certain commonly used type of deductive system.<br />
<br />
Note that "completeness" has a different meaning here than it does in the context of [[Gödel's first incompleteness theorem]], which states that no ''recursive'', ''consistent'' set of non-logical axioms <math>\Sigma\,</math> of the Theory of Arithmetic is ''complete'', in the sense that there will always exist an arithmetic statement <math>\phi\,</math> such that neither <math>\phi\,</math> nor <math>\lnot\phi\,</math> can be proved from the given set of axioms.<br />
<br />
There is thus, on the one hand, the notion of ''completeness of a deductive system'' and on the other hand that of ''completeness of a set of non-logical axioms''. The completeness theorem and the incompleteness theorem, despite their names, do not contradict one another.<br />
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===Further discussion===<br />
Early [[mathematician]]s regarded [[axiomatic geometry]] as a model of [[physical space]], and obviously there could only be one such model. The idea that alternative mathematical systems might exist was very troubling to mathematicians of the 19th century and the developers of systems such as [[Boolean algebra (logic)|Boolean algebra]] made elaborate efforts to derive them from traditional arithmetic. [[Évariste Galois|Galois]] showed just before his untimely death that these efforts were largely wasted. Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details and [[abstract algebra|modern algebra]] was born. In the modern view axioms may be any set of formulas, as long as they are not known to be inconsistent.<br />
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==See also==<br />
{{Portal|Mathematics|Logic}}<br />
* [[Axiomatic system]]<br />
* [[Dogma]]<br />
* [[List of axioms]]<br />
* [[Model theory]]<br />
* [[Regulæ Juris]]<br />
* [[Theorem]]<br />
<br />
==References==<br />
<references/><br />
* Mendelson, Elliot (1987). ''Introduction to mathematical logic.'' Belmont, California: Wadsworth & Brooks. ISBN 0-534-06624-0<br />
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==External links==<br />
{{Wiktionary|axiom|given}}<br />
{{Wikisource1911Enc}}<br />
* {{PhilPapers|search|axiom}}<br />
* {{planetmath|urlname=Axiom|title=Axiom}}<br />
* [http://us.metamath.org/mpegif/mmset.html#axioms ''Metamath'' axioms page]<br />
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{{Logic}}<br />
<br />
[[Category:Mathematical axioms|*]]<br />
[[Category:Mathematical terminology]]<br />
[[Category:Formal systems]]<br />
[[Category:Concepts in logic]]</div>Adminhttps://en.formulasearchengine.com/index.php?title=Actinium&diff=218485Actinium2014-07-31T13:53:47Z<p>Admin: Reverted edits by MediaWiki spam cleanup (talk) to last revision by Materialscientist</p>
<hr />
<div>{{good article}}<br />
{{Infobox actinium}}<br />
'''Actinium''' is a radioactive [[chemical element]] with symbol&nbsp;'''Ac''' (not to be confused with the abbreviation for an [[Acetyl|acetyl group]]) and [[atomic number]]&nbsp;89, which was discovered in 1899. It was the first [[Primordial element|non-primordial radioactive element]] to be isolated. [[Polonium]], [[radium]] and [[radon]] were observed before actinium, but they were not isolated until 1902. Actinium gave the name to the [[actinide]] series, a group of 15 similar elements between actinium and [[lawrencium]] in the [[periodic table]].<br />
<br />
A soft, silvery-white [[radioactive]] metal, actinium reacts rapidly with oxygen and moisture in air forming a white coating of actinium oxide that prevents further oxidation. As with most [[lanthanide]]s and actinides, actinium assumes [[oxidation state]] +3 in nearly all its chemical compounds. Actinium is found only in traces in [[uranium]] ores as the [[isotope]] <sup>227</sup>Ac, which decays with a [[half-life]] of 21.772 years, predominantly emitting [[beta particle]]s. One [[tonne]] of [[uranium]] ore contains about 0.2 milligrams of actinium. The close similarity of physical and chemical properties of actinium and [[lanthanum]] makes separation of actinium from the ore impractical. Instead, the element is prepared, in milligram amounts, by the neutron irradiation of <sup>226</sup>{{radium}} in a [[nuclear reactor]]. Owing to its scarcity, high price and radioactivity, actinium has no significant industrial use. Its current applications include a neutron source and an agent for [[radiation therapy]] targeting cancer cells in the body.<br />
<br />
==History==<br />
[[André-Louis Debierne]], a French chemist, announced the discovery of a new element in 1899. He separated it from [[uraninite|pitchblende]] residues left by [[Marie Curie|Marie]] and [[Pierre Curie]] after they had extracted [[radium]]. In 1899, Debierne described the substance as similar to [[titanium]]<ref>{{cite journal |title = Sur un nouvelle matière radio-active |first = André-Louis |last = Debierne |journal = Comptes rendus |volume = 129 |pages = 593–595 |year = 1899 |url = http://gallica.bnf.fr/ark:/12148/bpt6k3085b/f593.table |language=French}}</ref> and (in 1900) as similar to [[thorium]].<ref>{{cite journal |title = Sur un nouvelle matière radio-actif – l'actinium |first = André-Louis |last = Debierne |journal = Comptes rendus |volume = 130 |pages = 906–908 |year = 1900–1901 |url = http://gallica.bnf.fr/ark:/12148/bpt6k3086n/f906.table |language=French}}</ref> [[Friedrich Oskar Giesel]] independently discovered actinium in 1902<ref>{{cite journal |title = Ueber Radium und radioactive Stoffe |first = Friedrich Oskar |last = Giesel |journal = Berichte der Deutschen Chemische Geselschaft |volume = 35 |issue = 3 |pages = 3608–3611 |year = 1902 |doi = 10.1002/cber.190203503187 |language=German}}</ref> as a substance being similar to [[lanthanum]] and called it "emanium" in 1904.<ref>{{cite journal |title = Ueber den Emanationskörper (Emanium) |first = Friedrich Oskar |last = Giesel |journal = Berichte der Deutschen Chemische Geselschaft |volume = 37 |issue = 2 |pages = 1696–1699 |year = 1904 |doi = 10.1002/cber.19040370280 |language=German}}</ref> After a comparison of the substances half-lives determined by Debierne,<ref>{{cite journal |title = Sur l'actinium |first = André-Louis |last = Debierne |journal = Comptes rendus |volume = 139 |pages = 538–540 |year = 1904 |language=French}}</ref> [[Hariett Brooks]] in 1904, and [[Otto Hahn]] and [[Otto Sackur]] in 1905, Debierne's chosen name for the new element was retained because it had seniority.<ref>{{cite journal |title = Ueber Emanium |first = Friedrich Oskar |last = Giesel |journal = Berichte der Deutschen Chemische Geselschaft |volume = 37 |issue = 2 |pages = 1696–1699 |year = 1904 |doi = 10.1002/cber.19040370280 |language=German}}</ref><ref>{{cite journal |title = Ueber Emanium |first = Friedrich Oskar |last = Giesel |journal = Berichte der Deutschen Chemische Geselschaft |volume = 38 |issue = 1 |pages = 775–778 |year = 1905 |doi = 10.1002/cber.190503801130 |language=German}}</ref><br />
<br />
Articles published in the 1970s<ref>{{cite journal |title = The Discovery of Actinium |first = Harold W. |last = Kirby |journal = Isis |volume = 62 |issue = 3 |pages = 290–308<br />
|year = 1971 |jstor=229943 |doi =10.1086/350760}}</ref> and later<ref name="Adloff">{{cite journal |title = The centenary of a controversial discovery: actinium |first = J. P. |last = Adloff |journal = Radiochim. Acta |volume = 88 |pages = 123–128 |year = 2000 |doi = 10.1524/ract.2000.88.3-4.123 |issue = 3–4_2000}}</ref> suggest that Debierne's results published in 1904 conflict with those reported in 1899 and 1900. This has led some authors to advocate that Giesel alone should be credited with the discovery.<ref>{{cite journal |last1 = Kirby |first1 = Harold W. |last2 = Morss |first2 = Lester R. |title = The Chemistry of the Actinide and Transactinide Elements |pages = 18 |year = 2006 |doi = 10.1007/1-4020-3598-5_2 |chapter = Actinium |isbn = 978-1-4020-3555-5}}</ref> A less confrontational vision of scientific discovery is proposed by Adloff.<ref name="Adloff" /> He suggests that hindsight criticism of the early publications should be mitigated by the nascent state of radiochemistry: highlighting the prudence of Debierne's claims in the original papers, he notes that nobody can contend that Debierne's substance did not contain actinium. Debierne, who is now considered by the vast majority of historians as the discoverer, lost interest in the element and left the topic. Giesel, on the other hand, can rightfully be credited with the first preparation of radiochemically pure actinium and with the identification of its atomic number 89.<br />
<br />
The name actinium originates from the [[Ancient Greek]] ''aktis, aktinos'' (ακτίς, ακτίνος), meaning beam or ray.<ref name=CRC/> Its symbol Ac is also used in abbreviations of other compounds that have nothing to do with actinium, such as [[acetyl]], [[acetate]]<ref>{{cite book |author1=Gilley, Cynthia Brooke |author2=University of California, San Diego. Chemistry |title=New convertible isocyanides for the Ugi reaction; application to the stereoselective synthesis of omuralide |url=http://books.google.com/books?id=vJQPInUTy3QC&pg=PR11 |accessdate=12 August 2011 |year=2008 |publisher=ProQuest |isbn=978-0-549-79554-4 |page=11}}</ref> and sometimes [[acetaldehyde]].<ref>{{cite book |author=Reimers, Jeffrey R. |title=Computational Methods for Large Systems: Electronic Structure Approaches for Biotechnology and Nanotechnology |url=http://books.google.com/books?id=Ca9z4_cH-W8C&pg=PA575 |accessdate=12 August 2011 |date=20 July 2011 |publisher=John Wiley and Sons |isbn=978-0-470-48788-4 |page=575}}</ref><br />
<br />
==Properties==<br />
Actinium is a soft, silvery-white,<ref name="blueglow"/><ref name=brit>''Actinium'', in Encyclopædia Britannica, 15th edition, 1995, p. 70</ref> [[radioactive]], metallic element. Its estimated [[shear modulus]] is similar to that of [[lead]].<ref>Frederick Seitz, David Turnbull [http://books.google.com/books?id=F9V3a-0V3r8C&pg=PA289 Solid state physics: advances in research and applications], Academic Press, 1964 ISBN 0-12-607716-9 pp. 289–291</ref> Owing to its strong radioactivity, actinium glows in the dark with a pale blue light, which originates from the surrounding air ionized by the emitted energetic particles.<ref>{{cite book |author=Richard A. Muller |title=Physics and Technology for Future Presidents: An Introduction to the Essential Physics Every World Leader Needs to Know |url=http://books.google.com/books?id=jMWCDsJesbcC&pg=PA136 |accessdate=12 August 2011 |date=12 April 2010 |publisher=Princeton University Press |isbn=978-0-691-13504-5 |pages=136–}}</ref> Actinium has similar chemical properties as [[lanthanum]] and other lanthanides, and therefore these elements are difficult to separate when extracting from uranium ores. Solvent extraction and [[ion chromatography]] are commonly used for the separation.<ref>{{cite journal |title = Chemistry of the Actinide Elements Annual Review of Nuclear Science |volume = 1 |pages = 245–262 |year = 1952 |first = J. J. |last = Katz |doi = 10.1146/annurev.ns.01.120152.001333 |journal = Annual Review of Nuclear Science |last2 = Manning |first2 = W M |bibcode = 1952ARNPS...1..245K }}</ref><br />
<br />
The first element of the [[actinide]]s, actinium gave the group its name, much as [[lanthanum]] had done for the [[lanthanide]]s. The group of elements is more diverse than the lanthanides and therefore it was not until 1945 that [[Glenn T. Seaborg]] proposed the most significant change to [[Dmitri Mendeleev]]'s [[periodic table]], by introducing the actinides.<ref>{{cite journal |title = The Transuranium Elements |first = Glenn T. |last = Seaborg |journal = Science |volume = 104 |issue = 2704 |year = 1946 |pages = 379–386 |jstor=1675046 |doi = 10.1126/science.104.2704.379 |pmid = 17842184 |bibcode = 1946Sci...104..379S }}</ref><br />
<br />
Actinium reacts rapidly with oxygen and moisture in air forming a white coating of actinium oxide that prevents further oxidation.<ref name="blueglow">{{cite journal |title = Preparation of Actinium Metal |first = Joseph G. |last = Stites |journal = J. Am. Chem. Soc. |year = 1955 |volume = 77 |issue = 1 |pages = 237–240 |doi = 10.1021/ja01606a085 |last2 = Salutsky |first2 = Murrell L. |last3 = Stone |first3 = Bob D.}}</ref> As with most lanthanides and actinides, actinium exists in the [[oxidation state]] +3, and the Ac<sup>3+</sup> ions are colorless in solutions.<ref name=bse/> The oxidation state +3 originates from the 6d<sup>1</sup>7s<sup>2</sup> electronic configuration of actinium, that is it easily donates 3 electrons assuming a stable closed-shell structure of the [[noble gas]] [[radon]].<ref name=brit/> The rare oxidation state +2 is only known for actinium dihydride (AcH<sub>2</sub>).<ref name=ach/><br />
<br />
==Chemical compounds==<br />
Only a limited number of actinium compounds are known including AcF<sub>3</sub>, AcCl<sub>3</sub>, AcBr<sub>3</sub>, AcOF, AcOCl, AcOBr, Ac<sub>2</sub>S<sub>3</sub>, Ac<sub>2</sub>O<sub>3</sub> and AcPO<sub>4</sub>. Except for AcPO<sub>4</sub>, they are all similar to the corresponding lanthanum compounds and contain actinium in the oxidation state +3.<ref name=bse/><ref>{{cite journal |title = The Preparation and Identification of Some Pure Actinium Compounds |journal = Journal of the American Chemical Society |last = Sherman |first = Fried |pages = 771–775 |doi = 10.1021/ja01158a034 |year =1950 |volume = 72 |last2 = Hagemann |first2 = French |last3 = Zachariasen |first3 = W. H. |issue = 2}}</ref> In particular, the lattice constants of the analogous lanthanum and actinium compounds differ by only a few percent.<ref name=j2/><br />
<br />
{| Class = "wikitable collapsible collapsed" style = "text-align: center"<br />
! Formula<br />
! color<br />
! symmetry<br />
! [[space group]]<br />
! No<br />
! [[Pearson symbol]]<br />
! ''a'' (pm)<br />
! ''b'' (pm)<br />
! ''c'' (pm)<br />
! ''Z''<br />
! density, <br />g/cm<sup>3</sup><br />
|-<br />
| Ac<br />
| silvery<br />
| ''[[Cubic crystal system|fcc]]''<ref name=ach>{{cite journal |doi=10.1016/0022-1902(61)80369-2 |last1=Farr |year=1961 |first1=J |pages=42 |volume=18 |journal=Journal of Inorganic and Nuclear Chemistry |title=The crystal structure of actinium metal and actinium hydride |last2=Giorgi |first2=A.L. |last3=Bowman |first3=M.G. |last4=Money |first4=R.K.}}</ref><br />
| Fm{{overline|3}}m<br />
| 225<br />
| cF4<br />
| 531.1<br />
| 531.1<br />
| 531.1<br />
| 4<br />
| 10.07<br />
|-<br />
| AcH<sub>2</sub><br />
|<br />
| cubic<ref name=ach/><br />
| Fm{{overline|3}}m<br />
| 225<br />
| cF12<br />
| 567<br />
| 567<br />
| 567<br />
| 4<br />
| 8.35<br />
|-<br />
| Ac<sub>2</sub>O<sub>3</sub><br />
| white<ref name="blueglow"/><br />
| [[Trigonal crystal system|trigonal]]<ref name=aco>{{cite journal |doi=10.1107/S0365110X49001016 |last1=Zachariasen |year=1949 |first1=W. H. |pages=388 |volume=2 |journal=Acta Crystallographica |title=Crystal chemical studies of the 5f-series of elements. XII. New compounds representing known structure types |issue=6}}</ref><br />
| P{{overline|3}}m1<br />
| 164<br />
| hP5<br />
| 408<br />
| 408<br />
| 630<br />
| 1<br />
| 9.18<br />
|-<br />
| Ac<sub>2</sub>S<sub>3</sub><br />
|<br />
| cubic<ref name=acs>{{cite journal |doi=10.1107/S0365110X49000126 |last1=Zachariasen |year=1949 |first1=W. H. |pages=57 |volume=2 |journal=Acta Crystallographica |title=Crystal chemical studies of the 5f-series of elements. VI. The Ce2S3-Ce3S4 type of structure}}</ref><br />
| I{{overline|4}}3d<br />
| 220<br />
| cI28<br />
| 778.56<br />
| 778.56<br />
| 778.56<br />
| 4<br />
| 6.71<br />
|-<br />
| AcF<sub>3</sub><br />
| white<ref name=m71>Meyer, p. 71</ref><br />
| [[Hexagonal crystal system|hexagonal]]<ref name=j2/><ref name=aco/><br />
| P{{overline|3}}c1<br />
| 165<br />
| hP24<br />
| 741<br />
| 741<br />
| 755<br />
| 6<br />
| 7.88<br />
|-<br />
| AcCl<sub>3</sub><br />
|<br />
| hexagonal<ref name=j2/><ref name=accl>{{cite journal |doi=10.1107/S0365110X48000703 |last1=Zachariasen |year=1948 |first1=W. H. |pages=265 |volume=1 |journal=Acta Crystallographica |title=Crystal chemical studies of the 5f-series of elements. I. New structure types |issue=5}}</ref><br />
| P6<sub>3</sub>/m<br />
| 165<br />
| hP8<br />
| 764<br />
| 764<br />
| 456<br />
| 2<br />
| 4.8<br />
|-<br />
| AcBr<sub>3</sub><br />
| white<ref name=j2/><br />
| hexagonal<ref name=accl/><br />
| P6<sub>3</sub>/m<br />
| 165<br />
| hP8<br />
| 764<br />
| 764<br />
| 456<br />
| 2<br />
| 5.85<br />
|-<br />
| AcOF<br />
| white<ref name=m87/><br />
| cubic<ref name=j2/><br />
| Fm{{overline|3}}m<br />
|<br />
|<br />
| 593.1<br />
|<br />
|<br />
|<br />
| 8.28<br />
|-<br />
| AcOCl<br />
|<br />
| [[Tetragonal crystal system|tetragonal]]<ref name=j2/><br />
|<br />
|<br />
|<br />
| 424<br />
| 424<br />
| 707<br />
|<br />
| 7.23<br />
|-<br />
| AcOBr<br />
|<br />
| tetragonal<ref name=j2/><br />
|<br />
|<br />
|<br />
| 427<br />
| 427<br />
| 740<br />
|<br />
| 7.89<br />
|-<br />
| AcPO<sub>4</sub>·0.5H<sub>2</sub>O<br />
|<br />
| hexagonal<ref name=j2/><br />
|<br />
|<br />
|<br />
| 721<br />
| 721<br />
| 664<br />
|<br />
| 5.48<br />
|}<br />
<br />
Here ''a'', ''b'' and ''c'' are lattice constants, No is space group number and ''Z'' is the number of [[formula unit]]s per [[unit cell]]. Density was not measured directly but calculated from the lattice parameters.<br />
<br />
===Oxides===<br />
Actinium oxide (Ac<sub>2</sub>O<sub>3</sub>) can be obtained by heating the hydroxide at 500 °C or the [[oxalate]] at 1100 °C, in vacuum. It crystal lattice is [[Isomorphism (crystallography)|isotypic]] with the oxides of most trivalent rare-earth metals.<ref name=j2/><br />
<br />
===Halides===<br />
Actinium trifluoride can be produced either in solution or in solid reaction. The former reaction is carried out at room temperature, by adding [[hydrofluoric acid]] to a solution containing actinium ions. In the latter method, actinium metal is treated with hydrogen fluoride vapors at 700 °C in an all-platinum setup. Treating actinium trifluoride with [[ammonium hydroxide]] at 900–1000 °C yields [[oxyfluoride]] AcOF. Whereas lanthanum oxyfluoride can be easily obtained by burning lanthanum trifluoride in air at 800 °C for an hour, similar treatment of actinium trifluoride yields no AcOF and only results in melting of the initial product.<ref name=j2/><ref name=m87>Meyer, pp. 87–88</ref><br />
<br />
:AcF<sub>3</sub> + 2 NH<sub>3</sub> + H<sub>2</sub>O → AcOF + 2 NH<sub>4</sub>F<br />
<br />
Actinium trichloride is obtained by reacting actinium hydroxide or [[oxalate]] with [[carbon tetrachloride]] vapors at temperatures above 960 °C. Similar to oxyfluoride, actinium [[oxychloride]] can be prepared by hydrolyzing actinium trichloride with [[ammonium hydroxide]] at 1000 °C. However, in contrast to the oxyfluoride, the oxychloride could well be synthesized by igniting a solution of actinium trichloride in [[hydrochloric acid]] with [[ammonia]].<ref name=j2/><br />
<br />
Reaction of [[aluminium bromide]] and actinium oxide yields actinium tribromide:<br />
:Ac<sub>2</sub>O<sub>3</sub> + 2 AlBr<sub>3</sub> → 2 AcBr<sub>3</sub> + Al<sub>2</sub>O<sub>3</sub><br />
<br />
and treating it with ammonium hydroxide at 500 °C results in the oxybromide AcOBr.<ref name=j2/><br />
<br />
===Other compounds===<br />
Actinium hydride was obtained by reduction of actinium trichloride with potassium at 300 °C, and its structure was deduced by analogy with the corresponding LaH<sub>2</sub> hydride. The source of hydrogen in the reaction was uncertain.<ref>Meyer, p. 43</ref><br />
<br />
Mixing [[monosodium phosphate]] (NaH<sub>2</sub>PO<sub>4</sub>) with a solution of actinium in hydrochloric acid yields white-colored actinium phosphate hemihydrate (AcPO<sub>4</sub>·0.5H<sub>2</sub>O), and heating actinium oxalate with [[hydrogen sulfide]] vapors at 1400 °C for a few minutes results in a black actinium sulfide Ac<sub>2</sub>S<sub>3</sub>. It may possibly be produced by acting with a mixture of [[hydrogen sulfide]] and [[carbon disulfide]] on [[actinium oxide]] at 1000 °C.<ref name=j2/><br />
<br />
==Isotopes==<br />
{{main|Isotopes of actinium}}<br />
Naturally occurring actinium is composed of one radioactive [[isotope]]; {{chem|227|Ac}}. Thirty-six [[radioisotope]]s have been identified, the most stable being {{chem|227|Ac}} with a [[half-life]] of 21.772 years, {{chem|225|Ac}} with a half-life of 10.0 days and {{chem|226|Ac}} with a half-life of 29.37 hours. All remaining [[radioactive decay|radioactive]] isotopes have half-lives that are less than 10 hours and the majority of them have half-lives shorter than one minute. The shortest-lived known isotope of actinium is {{chem|217|Ac}} (half-life of 69 nanoseconds) which decays through [[alpha decay]] and [[electron capture]]. Actinium also has two [[meta state]]s.<ref name ="nubas">{{cite journal |last = Audi |first = Georges |title = The NUBASE Evaluation of Nuclear and Decay Properties |journal = Nuclear Physics A |volume = 729 |pages = 3–128 |publisher = Atomic Mass Data Center |year = 2003 |doi=10.1016/j.nuclphysa.2003.11.001 |bibcode=2003NuPhA.729....3A |last2 = Bersillon |first2 = O. |last3 = Blachot |first3 = J. |last4 = Wapstra |first4 = A.H.}}</ref><br />
<br />
Purified {{chem|227|Ac}} comes into equilibrium with its decay products at the end of 185 days. It decays according to its 21.772-year half-life emitting mostly beta (98.8%) and some alpha particles (1.2%);<ref name=bse>[http://bse.sci-lib.com/article008169.html Actinium], [[Great Soviet Encyclopedia]] (in Russian)</ref> the successive decay products are part of the [[actinium series]]. Owing to the low available amounts, low energy of its beta particles (46 keV) and low intensity of alpha radiation, {{chem|227|Ac}} is difficult to detect directly by its emission and it is therefore traced via its decay products.<ref name=bse/> The isotopes of actinium range in [[atomic weight]] from 206&nbsp;[[atomic mass unit|u]] ({{chem|206|Ac}}) to 236&nbsp;u ({{chem|236|Ac}}).<ref name ="nubas"/><br />
<br />
{| class="wikitable" style="text-align:center"<br />
!Isotope<br />
!Production<br />
!Decay<br />
!Half-life<br />
|-<br />
|<sup>221</sup>Ac<br />
|<sup>232</sup>Th(d,9n)<sup>225</sup>Pa(α)→<sup>221</sup>Ac<br />
|α<br />
|52 ms<br />
|-<br />
|<sup>222</sup>Ac<br />
|<sup>232</sup>Th(d,8n)<sup>226</sup>Pa(α)→<sup>222</sup>Ac<br />
|α<br />
|5.0 s<br />
|-<br />
|<sup>223</sup>Ac<br />
|<sup>232</sup>Th(d,7n)<sup>227</sup>Pa(α)→<sup>223</sup>Ac<br />
|α<br />
|2.1 min<br />
|-<br />
|<sup>224</sup>Ac<br />
|<sup>232</sup>Th(d,6n)<sup>228</sup>Pa(α)→<sup>224</sup>Ac<br />
|α<br />
|2.78 hours<br />
|-<br />
|<sup>225</sup>Ac<br />
|<sup>232</sup>Th(n,γ)<sup>233</sup>Th(β<sup>−</sup>)→<sup>233</sup>Pa(β<sup>−</sup>)→<sup>233</sup>U(α)→<sup>229</sup>Th(α)→<sup>225</sup>Ra(β<sup>−</sup>)<sup>225</sup>Ac<br />
|α<br />
|10 days<br />
|-<br />
|<sup>226</sup>Ac<br />
|<sup>226</sup>Ra(d,2n)<sup>226</sup>Ac<br />
|α, β<sup>−</sup> <br />electron capture<br />
|29.37 hours<br />
|-<br />
|<sup>227</sup>Ac<br />
|<sup>235</sup>U(α)→<sup>231</sup>Th(β<sup>−</sup>)→<sup>231</sup>Pa(α)→<sup>227</sup>Ac<br />
|α, β<sup>−</sup><br />
|21.77 years<br />
|-<br />
|<sup>228</sup>Ac<br />
|<sup>232</sup>Th(α)→<sup>228</sup>Ra(β<sup>−</sup>)→<sup>228</sup>Ac<br />
|β<sup>−</sup><br />
|6.15 hours<br />
|-<br />
|<sup>229</sup>Ac<br />
|<sup>228</sup>Ra(n,γ)<sup>229</sup>Ra(β<sup>−</sup>)→<sup>229</sup>Ac<br />
|β<sup>−</sup><br />
|62.7 min<br />
|-<br />
|<sup>230</sup>Ac<br />
|<sup>232</sup>Th(d,α)<sup>230</sup>Ac<br />
|β<sup>−</sup><br />
|122 s<br />
|-<br />
|<sup>231</sup>Ac<br />
|<sup>232</sup>Th(γ,p)<sup>231</sup>Ac<br />
|β<sup>−</sup><br />
|7.5 min<br />
|-<br />
|<sup>232</sup>Ac<br />
|<sup>232</sup>Th(n,p)<sup>232</sup>Ac<br />
|β<sup>−</sup><br />
|119 s<br />
|}<br />
<br />
==Occurrence and synthesis==<br />
[[File:Uraninite-39029.jpg|150px|thumb|[[Uraninite]] ores have elevated concentrations of actinium.]]<br />
Actinium is found only in traces in [[uranium]] ores as <sup>227</sup>Ac – one tonne of ore contains about 0.2 milligrams of actinium.<ref name=j1>{{cite journal |doi=10.1021/ja01158a033 |last1=Hagemann |year=1950 |first1=French |pages=768 |volume=72 |journal=Journal of the American Chemical Society |title=The Isolation of Actinium |issue=2}}</ref><ref name=g946>{{Greenwood&Earnshaw2nd|page=946}}</ref> The actinium [[isotope]] <sup>227</sup>Ac is a transient member of the [[Decay chain#Actinium series|actinium series]] [[decay chain]], which begins with the parent isotope [[Uranium-235|<sup>235</sup>U]] (or [[Plutonium-239|<sup>239</sup>Pu]]) and ends with the stable lead isotope [[Isotopes of lead|<sup>207</sup>Pb]]. Another actinium isotope (<sup>225</sup>Ac) is transiently present in the [[Decay chain#Neptunium series|neptunium series]] [[decay chain]], beginning with [[Neptunium|<sup>237</sup>Np]] (or [[Uranium-233|<sup>233</sup>U)]] and ending with [[thallium]] (<sup>205</sup>Tl) and near-stable [[bismuth]] (<sup>209</sup>Bi).<br />
<br />
The low natural concentration, and the close similarity of physical and chemical properties to those of lanthanum and other lanthanides, which are always abundant in actinium-bearing ores, render separation of actinium from the ore impractical, and complete separation was never achieved.<ref name=j2>{{cite journal |doi=10.1021/ja01158a034 |last1=Fried |year=1950 |first1=Sherman |pages=771 |volume=72 |journal=Journal of the American Chemical Society |last2=Hagemann |first2=French |last3=Zachariasen |first3=W. H. |title=The Preparation and Identification of Some Pure Actinium Compounds |issue=2}}</ref> Instead, actinium is prepared, in milligram amounts, by the neutron irradiation of <sup>226</sup>{{radium}} in a [[nuclear reactor]].<ref name=g946/><ref>{{cite book |author=Emeleus, H. J. |title=Advances in inorganic chemistry and radiochemistry |url=http://books.google.com/books?id=K5_LSQqeZ_IC&pg=PA16 |accessdate=12 August 2011 |date=July 1987 |publisher=Academic Press |isbn=978-0-12-023631-2 |pages=16–}}</ref><br />
<br />
:<math>\mathrm{^{226}_{\ 88}Ra\ +\ ^{1}_{0}n\ \longrightarrow \ ^{227}_{\ 88}Ra\ \xrightarrow[42.2 \ min]{\beta^-} \ ^{227}_{\ 89}Ac}</math><br />
<br />
The reaction yield is about 2% of the radium weight. <sup>227</sup>Ac can further capture neutrons resulting in small amounts of <sup>228</sup>Ac. After the synthesis, actinium is separated from radium and from the products of decay and nuclear fusion, such as thorium, polonium, lead and bismuth. The extraction can be performed with thenoyltrifluoroacetone-[[benzene]] solution from an aqueous solution of the radiation products, and the selectivity to a certain element is achieved by adjusting the [[pH]] (to about 6.0 for actinium).<ref name=j1/> An alternative procedure is anion exchange with an appropriate [[resin]] in [[nitric acid]], which can result in a separation factor of 1,000,000 for radium and actinium vs. thorium in a two-stage process. Actinium can then be separated from radium, with a ratio of about 100, using a low cross-linking cation exchange resin and nitric acid as [[eluant]].<ref name=sep/><br />
<br />
<sup>225</sup>Ac was first produced artificially at the [[Institute for Transuranium Elements]] (ITU) in Germany using a [[cyclotron]] and at [[St George Hospital, Sydney|St George Hospital]] in Sydney using a [[Linear particle accelerator|linac]] in 2000.<ref>{{cite journal |doi = 10.1016/j.apradiso.2008.11.012 |year = 2009 |author = Melville, G; Allen, Bj |title = Cyclotron and linac production of Ac-225 |volume = 67 |issue = 4 |pages = 549–55 |pmid = 19135381 |journal = Applied radiation and isotopes}}</ref> This rare isotope has potential applications in radiation therapy and is most efficiently produced by bombarding a radium-226 target with 20–30 MeV [[deuterium]] ions. This reaction also yields <sup>226</sup>Ac which however decays with a half-life of 29 hours and thus does not contaminate <sup>225</sup>Ac.<ref>Russell, Pamela J.; Jackson, Paul and Kingsley, Elizabeth Anne [http://books.google.com/books?id=K1y6k5bdlWkC&pg=PA336 Prostate cancer methods and protocols], Humana Press, 2003, ISBN 0-89603-978-1, p. 336</ref><br />
<br />
Actinium metal has been prepared by the reduction of actinium fluoride with [[lithium]] vapor in vacuum at a temperature between 1100 and 1300 °C. Higher temperatures resulted in evaporation of the product and lower ones lead to an incomplete transformation. Lithium was chosen among other alkali metals because its fluoride is most volatile.<ref name=CRC>Hammond, C. R. ''The Elements'' in {{RubberBible86th}}</ref><ref name="blueglow"/><br />
<br />
==Applications==<br />
Owing to its scarcity, high price and radioactivity, actinium currently has no significant industrial use.<ref name=CRC/><!--http://www.osti.gov/energycitations/product.biblio.jsp?osti_id=4066566--><br />
<br />
<sup>227</sup>Ac is highly radioactive and was therefore studied for use as an active element of [[radioisotope thermoelectric generator]]s, for example in spacecraft. The oxide of <sup>227</sup>Ac pressed with [[beryllium]] is also an efficient [[neutron source]] with the activity exceeding that of the standard americium-beryllium and radium-beryllium pairs.<ref name=b1>Russell, Alan M. and Lee, Kok Loong [http://books.google.com/books?id=fIu58uZTE-gC&pg=PA470 Structure-property relations in nonferrous metals], Wiley, 2005, ISBN 0-471-64952-X, pp. 470–471</ref> In all those applications, <sup>227</sup>Ac (a beta source) is merely a progenitor which generates alpha-emitting isotopes upon its decay. Beryllium captures alpha particles and emits neutrons owing to its large cross-section for the (α,n) nuclear reaction:<br />
<br />
: <math>\mathrm{^{9}_{4}Be\ +\ ^{4}_{2}He\ \longrightarrow \ ^{12}_{\ 6}C\ +\ ^{1}_{0}n\ +\ \gamma}</math><br />
<br />
The <sup>227</sup>AcBe neutron sources can be applied in a [[neutron probe]] – a standard device for measuring the quantity of water present in soil, as well as moisture/density for quality control in highway construction.<ref>Majumdar, D. K. [http://books.google.com/books?id=hf1j9v4v3OEC&pg=PA108 Irrigation Water Management: Principles and Practice], 2004 ISBN 81-203-1729-7 p. 108</ref><ref>Chandrasekharan, H. and Gupta, Navindu [http://books.google.com/books?id=45IDh4Lt8xsC&pg=PA203 Fundamentals of Nuclear Science – Application in Agriculture], 2006 ISBN 81-7211-200-9 pp. 202 ff</ref> Such probes are also used in well logging applications, in [[neutron radiography]], tomography and other radiochemical investigations.<ref>{{cite journal |title = Neutron Spectrum of an Actinium–Beryllium Source |first = W.R. |last = Dixon |journal = Can. J. Phys./Rev. Can. Phys. |volume = 35 |issue = 6 |pages = 699–702 |year = 1957 |url = http://pubs.nrc-cnrc.gc.ca/cgi-bin/rp/rp2_abst_e?cjp_p57-075_35_ns_nf_cjp |doi = 10.1139/p57-075 |last2 = Bielesch |first2 = Alice |last3 = Geiger |first3 = K. W.|bibcode = 1957CaJPh..35..699D }}</ref><br />
<br />
[[File:DOTA polyaminocarboxylic acid.png|thumb|150px|Chemical structure of the [[DOTA (chelator)|DOTA]] carrier for <sup>225</sup>Ac in radiation therapy.]]<br />
<sup>225</sup>Ac is applied in medicine to produce <sup>213</sup>{{bismuth}} in a reusable generator<ref name=sep>{{cite journal |doi = 10.1016/j.apradiso.2004.12.003 |year = 2005 |volume = 62 |issue = 5 |pages =667–679 |title = Production of actinium-225 for alpha particle mediated radioimmunotherapy |last = Bolla |first = Rose A. |journal = Applied Radiation and Isotopes |pmid = 15763472 |last2 = Malkemus |first2 = D |last3 = Mirzadeh |first3 = S}}</ref> or can be used alone as an agent for [[radiation therapy]], in particular targeted alpha therapy (TAT). This isotope has a half-life of 10 days that makes it much more suitable for radiation therapy than <sup>213</sup>Bi (half-life 46 minutes). Not only <sup>225</sup>Ac itself, but also its decay products emit alpha particles which kill cancer cells in the body. The major difficulty with application of <sup>225</sup>Ac was that intravenous injection of simple actinium complexes resulted in their accumulation in the bones and liver for a period of tens of years. As a result, after the cancer cells were quickly killed by alpha particles from <sup>225</sup>Ac, the radiation from the actinium and its decay products might induce new mutations. To solve this problem, <sup>225</sup>Ac was bound to a [[Chelation|chelating]] agent, such as [[citrate]], [[ethylenediaminetetraacetic acid]] (EDTA) or [[pentetic acid|diethylene triamine pentaacetic acid]] (DTPA). This reduced actinium accumulation in the bones, but the excretion from the body remained slow. Much better results were obtained with such chelating agents as HEHA(1,4,7,10,13,16-hexaazacyclohexadecane-N,N`,N``,N```,N````,N`````-hexaacetic acid)<ref>{{cite journal |title=Improved in Vivo Stability of Actinium-225 Macrocyclic Complexes}}</ref> or [[DOTA (chelator)|DOTA]] (1,4,7,10-tetraazacyclododecane-1,4,7,10-tetraacetic acid) coupled to [[trastuzumab]], a [[monoclonal antibody]] that interferes with the [[HER2/neu]] [[Receptor (biochemistry)|receptor]]. The latter delivery combination was tested on mice and proved to be effective against [[leukemia]], [[lymphoma]], [[breast cancer|breast]], [[Ovarian cancer|ovarian]], [[neuroblastoma]] and [[prostate cancer]]s.<ref>{{cite journal|last1=McDevitt|first1=Michael R.|last2=Ma|first2=Dangshe|last3=Lai|first3=Lawrence T.|last4=Simon|first4=Jim|last5=Borchardt|first5=Paul|last6=Frank|first6=R. Keith|last7=Wu|first7=Karen|last8=Pellegrini|first8=Virginia|last9=Curcio|first9=Michael J.|last10=Miederer|first10=Matthias|last11=Bander|first11=Neil H.|last12=Scheinberg|first12=David A.|displayauthors=3|title=Tumor Therapy with Targeted Atomic Nanogenerators|year=2001|journal=Science|volume=294|issue=5546|pages=1537–1540|doi=10.1126/science.1064126|bibcode=2001Sci...294.1537M|pmid=11711678|url=http://www.studybusiness.com/HTML/Bio/10021/10021-04-2003-BIO-04-E.pdf}}</ref><ref>{{cite journal |url=http://cancerres.aacrjournals.org/content/63/16/5084.full.pdf |title=Targeted Actinium-225 in Vivo Generators for Therapy of Ovarian Cancer |author=Borchardt, Paul E. et al. |journal=Cancer Research |volume=63 |issue=16 |pages= 5084–5090 |year=2003 |pmid=12941838}}</ref><ref>{{cite journal |author=Ballangrud, A. M. ''et al.'' |title=Alpha-particle emitting atomic generator (Actinium-225)-labeled trastuzumab (herceptin) targeting of breast cancer spheroids: efficacy versus HER2/neu expression |journal=Clinical cancer research : an official journal of the American Association for Cancer Research |volume=10 |issue=13 |pages=4489–97 |year=2004 |pmid=15240541 |doi=10.1158/1078-0432.CCR-03-0800}}</ref><br />
<br />
The medium half-life of <sup>227</sup>Ac (21.77 years) makes it very convenient radioactive isotope in modeling the slow vertical mixing of oceanic waters. The associated processes cannot be studied with the required accuracy by direct measurements of current velocities (of the order 50 meters per year). However, evaluation of the concentration depth-profiles for different isotopes allows estimating the mixing rates. The physics behind this method is as follows: oceanic waters contain homogeneously dispersed <sup>235</sup>U. Its decay product, <sup>231</sup>Pa, gradually precipitates to the bottom, so that its concentration first increases with depth and then stays nearly constant. <sup>231</sup>Pa decays to <sup>227</sup>Ac; however, the concentration of the latter isotope does not follow the <sup>231</sup>Pa depth profile, but instead increases toward the sea bottom. This occurs because of the mixing processes which raise some additional <sup>227</sup>Ac from the sea bottom. Thus analysis of both <sup>231</sup>Pa and <sup>227</sup>Ac depth profiles allows to model the mixing behavior.<ref>{{cite journal |last1=Nozaki |first1=Yoshiyuki |title=Excess 227Ac in deep ocean water |journal=Nature |volume=310 |pages=486 |year=1984 |doi=10.1038/310486a0 | issue=5977 | bibcode = 1984Natur.310..486N}}</ref><ref>{{cite journal |last1=Geibert |first1=W. |last2=Rutgers Van Der Loeff |first2=M.M. |last3=Hanfland |first3=C. |last4=Dauelsberg |first4=H.-J. |title=Actinium-227 as a deep-sea tracer: sources, distribution and applications |journal=Earth and Planetary Science Letters |volume=198 |pages=147 |year=2002 |doi=10.1016/S0012-821X(02)00512-5 |bibcode=2002E&PSL.198..147G}}</ref><br />
<br />
==Precautions==<br />
<sup>227</sup>Ac is highly radioactive and experiments with it are carried out in a specially designed laboratory equipped with a [[glove box]]. When actinium trichloride is administered intravenously to rats, about 33% of actinium is deposited into the bones and 50% into the liver. Its toxicity is comparable to, but slightly lower than that of americium and plutonium.<ref>{{cite journal |doi = 10.2172/4406766 |title = Toxicology of Actinium Equilibrium Mixture |first2 = J. |last = Langham |last2 = Storer |first = W. |year = 1952 | journal = Los Alamos Scientific Lab.: Technical Report}}</ref><br />
<br />
==See also==<br />
* [[Decay chain#Actinium series|Actinium series]]<br />
{{Subject bar<br />
|portal=Chemistry<br />
|book1=Actinium<br />
|book2=Actinides<br />
|book3=Period 7 elements<br />
|book4=Group 3 elements<br />
|book5=Chemical elements (sorted&nbsp;alphabetically)<br />
|book6=Chemical elements (sorted by number)<br />
|commons=y<br />
|wikt=y<br />
|wikt-search=actinium<br />
}}<br />
<br />
==References==<br />
{{Reflist|30em}}<br />
<br />
==Bibliography==<br />
* Meyer, Gerd and Morss, Lester R. [http://books.google.com/books?id=bnS5elHL2w8C&pg=PA87 Synthesis of lanthanide and actinide compounds], Springer, 1991, ISBN 0-7923-1018-7<br />
<br />
==External links==<br />
* [http://www.periodicvideos.com/videos/089.htm Actinium] at ''[[The Periodic Table of Videos]]'' (University of Nottingham)<br />
* [http://toxnet.nlm.nih.gov/cgi-bin/sis/search/r?dbs+hsdb:@term+@na+@rel+actinium,+radioactive NLM Hazardous Substances Databank – Actinium, Radioactive]<br />
* [http://radchem.nevada.edu/classes/rdch710/files/actinium.pdf Actinium] in {{cite book<br />
| title = The Chemistry of the Actinide and Transactinide Elements<br />
| editor1-last = Morss |editor2-first = Norman M.<br />
| editor2-last = Edelstein<br />
| editor3-last = Fuger |editor3-first = Jean<br />
| last = Haire |first = Richard G.<br />
| publisher = Springer<br />
| year = 2006<br />
| isbn = 1-4020-3555-1<br />
| location = Dordrecht, The Netherlands<br />
| edition = 3rd<br />
}}<br />
<br />
{{Compact periodic table}}<br />
{{Use dmy dates|date=December 2011}}<br />
<br />
[[Category:Chemical elements]]<br />
[[Category:Actinides]]<br />
[[Category:Actinium]]<br />
<br />
{{Link GA|es}}<br />
{{Link FA|ro}}<br />
{{Link GA|zh}}</div>Adminhttps://en.formulasearchengine.com/index.php?title=Algorithms_for_calculating_variance&diff=218510Algorithms for calculating variance2014-07-31T13:52:48Z<p>Admin: Reverted edits by MediaWiki spam cleanup (talk) to last revision by Bilorv</p>
<hr />
<div>{{merge from|Standard deviation#Rapid calculation methods|date=February 2013}}<br />
<br />
'''Algorithms for calculating variance''' play a major role in [[statistics|statistical]] computing. A key problem in the design of good [[algorithm]]s for this problem is that formulas for the [[variance]] may involve sums of squares, which can lead to [[numerical instability]] as well as to [[arithmetic overflow]] when dealing with large values.<br />
<br />
==Naïve algorithm==<br />
A formula for calculating the variance of an entire [[statistical population|population]] of size ''N'' is:<br />
<br />
:<math>\sigma^2 = \displaystyle\frac {\sum_{i=1}^N x_i^2 - (\sum_{i=1}^N x_i)^2/N}{N}. \!</math><br />
<br />
A formula for calculating an [[estimator bias|unbiased]] estimate of the population variance from a finite [[statistical sample|sample]] of ''n'' observations is:<br />
<br />
:<math>s^2 = \displaystyle\frac {\sum_{i=1}^n x_i^2 - (\sum_{i=1}^n x_i)^2/n}{n-1}. \!</math><br />
<br />
Therefore a naive algorithm to calculate the estimated variance is given by the following:<br />
<br />
<source lang="python"><br />
def naive_variance(data):<br />
n = 0<br />
Sum = 0<br />
Sum_sqr = 0<br />
<br />
for x in data:<br />
n = n + 1<br />
Sum = Sum + x<br />
Sum_sqr = Sum_sqr + x*x<br />
<br />
variance = (Sum_sqr - (Sum*Sum)/n)/(n - 1)<br />
return variance<br />
</source><br />
<br />
This algorithm can easily be adapted to compute the variance of a finite population: simply divide by ''N'' instead of ''n''&nbsp;−&nbsp;1 on the last line.<br />
<br />
Because <code>Sum_sqr</code> and <code>(Sum*Sum)/n</code> can be very similar numbers, [[Loss of significance|cancellation]] can lead to the [[precision (arithmetic)|precision]] of the result to be much less than the inherent precision of the [[floating-point]] arithmetic used to perform the computation. Thus this algorithm should not be used in practice.<ref name="Einarsson2005">{{cite book|author=Bo Einarsson|title=Accuracy and Reliability in Scientific Computing|url=http://books.google.com/books?id=8hrDV5EbrEsC|accessdate=17 February 2013|date=1 August 2005|publisher=SIAM|isbn=978-0-89871-584-2|page=47}}</ref><ref name="Chan1983">{{cite journal|url=http://www.cs.yale.edu/publications/techreports/tr222.pdf|author=T.F.Chan, G.H. Golub and R.J. LeVeque|title="Algorithms for computing the sample variance: Analysis and recommendations", The American Statistician, 37|pages=242–247|year=1983}}</ref> This is particularly bad if the standard deviation is small relative to the mean. However, the algorithm can be improved by adopting the method of the [[assumed mean]].<br />
<br />
==Two-pass algorithm==<br />
An alternative approach, using a different formula for the variance, first computes the sample mean,<br />
:<math>\bar x = \displaystyle \frac {\sum_{j=1}^n x_j}{n}</math>,<br />
and then computes the sum of the squares of the differences from the mean,<br />
:<math>\mathrm{variance} = s^2 = \displaystyle\frac {\sum_{i=1}^n (x_i - \bar x)^2}{n-1} \!</math>,<br />
where s is the standard deviation. This is given by the following pseudocode:<br />
<br />
<source lang="python"><br />
def two_pass_variance(data):<br />
n = 0<br />
sum1 = 0<br />
sum2 = 0<br />
<br />
for x in data:<br />
n = n + 1<br />
sum1 = sum1 + x<br />
<br />
mean = sum1/n<br />
<br />
for x in data:<br />
sum2 = sum2 + (x - mean)*(x - mean)<br />
<br />
variance = sum2/(n - 1)<br />
return variance<br />
</source><br />
<br />
This algorithm is always numerically stable, unless n is large.<ref name="Einarsson2005"/><ref>{{cite book|first=Nicholas | last=Higham |title=Accuracy and Stability of Numerical Algorithms (2 ed) (Problem 1.10)| publisher=SIAM|year=2002}}</ref> Although it can be worse if much of the data is very close to but not precisely equal to the mean and some are quite far away from it{{Citation needed|date=November 2011}}.<!-- The first algorithm has less subtractions, that are a common form of losing precision in algorithms implemented in finite precision computers.--><br />
<br />
The results of both of these simple algorithms (I and II) can depend inordinately on the ordering of the data and can give poor results for very large data sets due to repeated roundoff error in the accumulation of the sums. Techniques such as [[compensated summation]] can be used to combat this error to a degree.<br />
<br />
===Compensated variant===<br />
The compensated-summation version of the algorithm above reads:<ref name=":0" /><!--Where did this algorithm come from? It is not the normal form for a Kahan summation.--><br />
<br />
<source lang="python"><br />
def compensated_variance(data):<br />
n = 0<br />
sum1 = 0<br />
for x in data:<br />
n = n + 1<br />
sum1 = sum1 + x<br />
mean = sum1/n<br />
<br />
sum2 = 0<br />
sum3 = 0<br />
for x in data:<br />
sum2 = sum2 + (x - mean)**2<br />
sum3 = sum3 + (x - mean)<br />
variance = (sum2 - sum3**2/n)/(n - 1)<br />
return variance<br />
</source><br />
<br />
==Online algorithm==<br />
It is often useful to be able to compute the variance in a single pass, inspecting each value <math>x_i</math> only once; for example, when the data are being collected without enough storage to keep all the values, or when costs of memory access dominate those of computation. For such an [[online algorithm]], a [[recurrence relation]] is required between quantities from which the required statistics can be calculated in a numerically stable fashion.<br />
<br />
The following formulas can be used to update the [[mean]] and (estimated) variance of the sequence, for an additional element <math>x_{\mathrm{new}}</math>. Here, ''{{overline|x}}<sub>n</sub>'' denotes the sample mean of the first ''n'' samples (''x''<sub>1</sub>, ..., ''x<sub>n</sub>''), ''s''<sup>2</sup><sub>''n''</sub> their sample variance, and ''σ''<sup>2</sup><sub>''N''</sub> their population variance.<br />
<br />
:<math>\bar x_n = \frac{(n-1) \, \bar x_{n-1} + x_n}{n} = \bar x_{n-1} + \frac{x_n - \bar x_{n-1}}{n} \!</math><br />
<br />
:<math>s^2_n = \frac{(n-2)}{(n-1)} \, s^2_{n-1} + \frac{(x_n - \bar x_{n-1})^2}{n}, \quad n>1 </math><br />
<br />
:<math>\sigma^2_N = \frac{(N-1) \, \sigma^2_{N-1} + (x_N - \bar x_{N-1})(x_N - \bar x_{N})}{N}.</math><br />
<br />
It turns out that a more suitable quantity for updating is the sum of squares of differences from the (current) mean, <math>\textstyle\sum_{i=1}^n (x_i - \bar x_n)^2</math>, here denoted <math>M_{2,n}</math>:<br />
<br />
:<math>M_{2,n}\! = M_{2,n-1} + (x_n - \bar x_{n-1})(x_n - \bar x_n)</math><br />
:<math>s^2_n = \frac{M_{2,n}}{n-1}</math><br />
:<math>\sigma^2_N = \frac{M_{2,N}}{N}</math><br />
<br />
A numerically stable algorithm is given below. It also computes the mean.<br />
This algorithm is due to Knuth,<ref>[[Donald E. Knuth]] (1998). ''[[The Art of Computer Programming]]'', volume 2: ''Seminumerical Algorithms'', 3rd edn., p. 232. Boston: Addison-Wesley.</ref> who cites Welford,<ref>B. P. Welford (1962).[http://www.jstor.org/stable/1266577 "Note on a method for calculating corrected sums of squares and products"]. ''[[Technometrics]]'' 4(3):419–420.</ref> and it has been thoroughly analyzed.<ref>Chan, Tony F.; Golub, Gene H.; LeVeque, Randall J. (1983). Algorithms for Computing the Sample Variance: Analysis and Recommendations. The American Statistician 37, 242-247. http://www.jstor.org/stable/2683386</ref><ref>Ling, Robert F. (1974). Comparison of Several Algorithms for Computing Sample Means and Variances. Journal of the American Statistical Association, Vol. 69, No. 348, 859-866. {{doi|10.2307/2286154}}</ref> It is also common to denote <math>M_k = \bar x_k</math> and <math>S_k = M_{2,k}</math>.<ref>http://www.johndcook.com/standard_deviation.html</ref><br />
<br />
<source lang="python"><br />
def online_variance(data):<br />
n = 0<br />
mean = 0<br />
M2 = 0<br />
<br />
for x in data:<br />
n = n + 1<br />
delta = x - mean<br />
mean = mean + delta/n<br />
M2 = M2 + delta*(x - mean)<br />
<br />
variance = M2/(n - 1)<br />
return variance<br />
</source><br />
<br />
This algorithm is much less prone to loss of precision due to [[Catastrophic cancellation|massive cancellation]], but might not be as efficient because of the division operation inside the loop. For a particularly robust two-pass algorithm for computing the variance, first compute and subtract an estimate of the mean, and then use this algorithm on the residuals.<br />
<br />
The [[Algorithms for calculating variance#Parallel algorithm|parallel algorithm]] below illustrates how to merge multiple sets of statistics calculated online.<br />
<br />
==Weighted incremental algorithm==<br />
The algorithm can be extended to handle unequal sample weights, replacing the simple counter ''n'' with the sum of weights seen so far. West (1979)<ref>D. H. D. West (1979). ''[[Communications of the ACM]]'', 22, 9, 532-535: ''Updating Mean and Variance Estimates: An Improved Method''</ref> suggests this incremental algorithm:<br />
<br />
<source lang="python"><br />
def weighted_incremental_variance(dataWeightPairs):<br />
sumweight = 0<br />
mean = 0<br />
M2 = 0<br />
<br />
for x, weight in dataWeightPairs: # Alternatively "for x, weight in zip(data, weights):"<br />
temp = weight + sumweight<br />
delta = x - mean<br />
R = delta * weight / temp<br />
mean = mean + R<br />
M2 = M2 + sumweight * delta * R # Alternatively, "M2 = M2 + weight * delta * (x−mean)"<br />
sumweight = temp<br />
<br />
variance_n = M2/sumweight<br />
variance = variance_n * len(dataWeightPairs)/(len(dataWeightPairs) - 1)<br />
</source><br />
<br />
==Parallel algorithm==<br />
Chan et al.<ref name=":0">{{Citation<br />
| last1 = Chan | first1 = Tony F. | author1-link = Tony F. Chan<br />
| last2 = Golub | first2 = Gene H. | author2-link = Gene H. Golub<br />
| last3 = LeVeque | first3 = Randall J.<br />
| contribution = Updating Formulae and a Pairwise Algorithm for Computing Sample Variances.<br />
| title = Technical Report STAN-CS-79-773<br />
| publisher = Department of Computer Science, Stanford University<br />
| year = 1979<br />
| contribution-url = ftp://reports.stanford.edu/pub/cstr/reports/cs/tr/79/773/CS-TR-79-773.pdf }}.</ref> note that the above online algorithm III is a special case of an algorithm that works for any partition of the sample <math>X</math> into sets <math>X_A</math>, <math>X_B</math>:<br />
:<math>\delta\! = \bar x_B - \bar x_A</math><br />
:<math>\bar x_X = \bar x_A + \delta\cdot\frac{n_B}{n_X}</math><br />
:<math>M_{2,X} = M_{2,A} + M_{2,B} + \delta^2\cdot\frac{n_A n_B}{n_X}</math>.<br />
This may be useful when, for example, multiple processing units may be assigned to discrete parts of the input.<br />
<br />
Chan's method for estimating the mean is numerically unstable when <math>n_A \approx n_B</math> and both are large, because the numerical error in <math>\bar x_B - \bar x_A</math> is not scaled down in the way that it is in the <math>n_B = 1</math> case. In such cases, prefer <math>\bar x_X = \frac{n_A \bar x_A + n_B \bar x_B}{n_A + n_B}</math>.<br />
<br />
==Example==<br />
Assume that all floating point operations use the standard [[IEEE 754#Double-precision 64 bit|IEEE 754 double-precision]] arithmetic. Consider the sample (4, 7, 13, 16) from an infinite population. Based on this sample, the estimated population mean is 10, and the unbiased estimate of population variance is 30. Both Algorithm I and Algorithm II compute these values correctly. Next consider the sample (10<sup>8</sup>&nbsp;+&nbsp;4, 10<sup>8</sup>&nbsp;+&nbsp;7, 10<sup>8</sup>&nbsp;+&nbsp;13, 10<sup>8</sup>&nbsp;+&nbsp;16), which gives rise to the same estimated variance as the first sample. Algorithm II computes this variance estimate correctly, but Algorithm I returns 29.333333333333332 instead of 30. While this loss of precision may be tolerable and viewed as a minor flaw of Algorithm I, it is easy to find data that reveal a major flaw in the naive algorithm: Take the sample to be (10<sup>9</sup>&nbsp;+&nbsp;4, 10<sup>9</sup>&nbsp;+&nbsp;7, 10<sup>9</sup>&nbsp;+&nbsp;13, 10<sup>9</sup>&nbsp;+&nbsp;16). Again the estimated population variance of 30 is computed correctly by Algorithm II, but the naive algorithm now computes it as −170.66666666666666. This is a serious problem with Algorithm I and is due to [[catastrophic cancellation]] in the subtraction of two similar numbers at the final stage of the algorithm.<br />
<br />
==Higher-order statistics==<br />
Terriberry<ref>{{Citation<br />
| last=Terriberry<br />
| first=Timothy B.<br />
| year=2007<br />
| title=Computing Higher-Order Moments Online<br />
| url=http://people.xiph.org/~tterribe/notes/homs.html<br />
}}</ref> extends Chan's formulae to calculating the third and fourth [[central moment]]s, needed for example when estimating [[skewness]] and [[kurtosis]]:<br />
:<math>M_{3,X} = M_{3,A} + M_{3,B} + \delta^3\frac{n_A n_B (n_A - n_B)}{n_X^2} + 3\delta\frac{n_AM_{2,B} - n_BM_{2,A}}{n_X}</math><br />
:<math>\begin{align}<br />
M_{4,X} = M_{4,A} + M_{4,B} & + \delta^4\frac{n_A n_B \left(n_A^2 - n_A n_B + n_B^2\right)}{n_X^3} \\<br />
& + 6\delta^2\frac{n_A^2 M_{2,B} + n_B^2 M_{2,A}}{n_X^2} + 4\delta\frac{n_AM_{3,B} - n_BM_{3,A}}{n_X} \\<br />
\end{align}</math><br />
<br />
Here the <math>M_k</math> are again the sums of powers of differences from the mean <math>\Sigma(x - \overline{x})^k</math>, giving<br />
:skewness: <math>g_1 = \frac{\sqrt{n} M_3}{M_2^{3/2}},</math><br />
:kurtosis: <math>g_2 = \frac{n M_4}{M_2^2}-3.</math><br />
<br />
For the incremental case (i.e., <math>B = \{x\}</math>), this simplifies to:<br />
:<math>\delta\! = x - m</math><br />
:<math>m' = m + \frac{\delta}{n}</math><br />
:<math>M_2' = M_2 + \delta^2 \frac{ n-1}{n}</math><br />
:<math>M_3' = M_3 + \delta^3 \frac{ (n - 1) (n - 2)}{n^2} - \frac{3\delta M_2}{n}</math><br />
:<math>M_4' = M_4 + \frac{\delta^4 (n - 1) (n^2 - 3n + 3)}{n^3} + \frac{6\delta^2 M_2}{n^2} - \frac{4\delta M_3}{n}</math><br />
<br />
By preserving the value <math>\delta / n</math>, only one division operation is needed and the higher-order statistics can thus be calculated for little incremental cost.<br />
<br />
An example of the online algorithm for kurtosis implemented as described is:<br />
<source lang="python"><br />
def online_kurtosis(data):<br />
n = 0<br />
mean = 0<br />
M2 = 0<br />
M3 = 0<br />
M4 = 0<br />
<br />
for x in data:<br />
n1 = n<br />
n = n + 1<br />
delta = x - mean<br />
delta_n = delta / n<br />
delta_n2 = delta_n * delta_n<br />
term1 = delta * delta_n * n1<br />
mean = mean + delta_n<br />
M4 = M4 + term1 * delta_n2 * (n*n - 3*n + 3) + 6 * delta_n2 * M2 - 4 * delta_n * M3<br />
M3 = M3 + term1 * delta_n * (n - 2) - 3 * delta_n * M2<br />
M2 = M2 + term1<br />
<br />
kurtosis = (n*M4) / (M2*M2) - 3<br />
return kurtosis<br />
</source><br />
<br />
Pébay<ref>{{Citation<br />
| last=Pébay<br />
| first=Philippe<br />
| year=2008<br />
| contribution=Formulas for Robust, One-Pass Parallel Computation of Covariances and Arbitrary-Order Statistical Moments<br />
| title=Technical Report SAND2008-6212<br />
| publisher=Sandia National Laboratories<br />
| contribution-url=http://infoserve.sandia.gov/sand_doc/2008/086212.pdf<br />
}}</ref><br />
further extends these results to arbitrary-order [[central moment]]s, for the incremental and the pairwise cases. One can also find there similar formulas for [[covariance]].<br />
<br />
Choi and Sweetman<br />
<ref name="Choi2010">{{Citation<br />
| last1 = Choi | first1 = Muenkeun<br />
| last2 = Sweetman | first2 = Bert<br />
| year=2010<br />
| title=Efficient Calculation of Statistical Moments for Structural Health Monitoring<br />
| url=http://www.rms-group.org/RMS_Papers/TAMUG_Papers/MK/Efficient_Moments_2010.pdf<br />
}}</ref><br />
offer two alternative methods to compute the skewness and kurtosis, each of which can save substantial computer memory requirements and CPU time in certain applications. The first approach is to compute the statistical moments by separating the data into bins and then computing the moments from the geometry of the resulting histogram, which effectively becomes a [[one-pass algorithm]] for higher moments. One benefit is that the statistical moment calculations can be carried out to arbitrary accuracy such that the computations can be tuned to the precision of, e.g., the data storage format or the original measurement hardware. A relative histogram of a random variable can be constructed in<br />
the conventional way: the range of potential values is<br />
divided into bins and the number of occurrences within each bin are<br />
counted and plotted such that the area of each rectangle equals<br />
the portion of the sample values within that bin:<br />
<br />
: <math> H(x_k)=\frac{h(x_k)}{A}</math><br />
<br />
where <math>h(x_k)</math> and <math>H(x_k)</math> represent the frequency and<br />
the relative frequency at bin <math>x_k</math> and <math>A= \sum_{k=1}^{K} h(x_k)<br />
\,\Delta x_k</math> is the total area of the histogram. After this<br />
normalization, the <math>n</math> raw moments and central moments of <math>x(t)</math><br />
can be calculated from the relative histogram:<br />
<br />
: <math><br />
m_n^{(h)} = \sum_{k=1}^{K} x_k^n \, H(x_k) \Delta x_k<br />
= \frac{1}{A} \sum_{k=1}^{K} x_k^n \, h(x_k) \Delta x_k<br />
</math><br />
<br />
: <math><br />
\theta_n^{(h)}= \sum_{k=1}^{K} \Big(x_k-m_1^{(h)}\Big)^n \, H(x_k)\Delta x_k<br />
= \frac{1}{A} \sum_{k=1}^{K} \Big(x_k-m_1^{(h)}\Big)^n \, h(x_k) \Delta x_k<br />
</math><br />
<br />
where the superscript <math>^{(h)}</math> indicates the moments are<br />
calculated from the histogram. For constant bin width <math>\Delta<br />
x_k=\Delta x</math> these two expressions can be simplified using <math>I= A/\Delta x</math>:<br />
<br />
: <math><br />
m_n^{(h)}= \frac{1}{I} {\sum_{k=1}^{K} x_k^n \, h(x_k)}<br />
</math><br />
<br />
: <math><br />
\theta_n^{(h)}= \frac{1}{I}{\sum_{k=1}^{K} \Big(x_k-m_1^{(h)}\Big)^n \, h(x_k)}<br />
</math><br />
<br />
The second approach from Choi and Sweetman<br />
<ref name="Choi2010" /><br />
is an analytical methodology to combine statistical moments from individual segments of a time-history such that the resulting overall moments are those of the complete time-history. This methodology could be used for parallel computation of statistical moments with subsequent combination of those moments, or for combination of statistical moments computed at sequential times.<br />
<br />
If <math>Q</math> sets of statistical moments are known:<br />
<math>(\gamma_{0,q},\mu_{q},\sigma^2_{q},\alpha_{3,q},\alpha_{4,q})<br />
\quad </math> for <math>q=1,2,...,Q </math>, then each <math>\gamma_n</math> can<br />
be expressed in terms of the equivalent <math>n</math> raw moments:<br />
<br />
: <math><br />
\gamma_{n,q}= m_{n,q} \gamma_{0,q} \qquad \quad \textrm{for} \quad n=1,2,3,4 \quad \text{ and } \quad q = 1,2, \dots ,Q<br />
</math><br />
<br />
where <math>\gamma_{0,q}</math> is generally taken to be the duration of the <math>q^{th}</math> time-history, or the number of points if <math>\Delta t</math> is constant.<br />
<br />
The benefit of expressing the statistical moments in<br />
terms of <math>\gamma</math> is that the <math>Q</math> sets can be combined by<br />
addition, and there is no upper limit on the value of <math>Q</math>.<br />
<br />
: <math><br />
\gamma_{n,c}= \sum_{q=1}^{Q}\gamma_{n,q} \quad \quad \textrm{for} \quad n=0,1,2,3,4<br />
</math><br />
<br />
where the subscript <math>_c</math> represents the concatenated<br />
time-history or combined <math>\gamma</math>. These combined values of<br />
<math>\gamma</math> can then be inversely transformed into raw moments<br />
representing the complete concatenated time-history<br />
<br />
: <math><br />
m_{n,c}=\frac{\gamma_{n,c}}{\gamma_{0,c}} \quad \textrm{for} \quad n=1,2,3,4<br />
</math><br />
<br />
Known relationships between the raw moments (<math>m_n</math>) and the central moments (<math> \theta_n = E[(x-\mu)^n])</math>)<br />
are then used to compute the central moments of the concatenated time-history. Finally, the statistical moments of the concatenated history are computed from the central moments:<br />
<br />
: <math><br />
\mu_c=m_{1,c}<br />
\ \ \ \ \ \sigma^2_c=\theta_{2,c}<br />
\ \ \ \ \ \alpha_{3,c}=\frac{\theta_{3,c}}{\sigma_c^3}<br />
\ \ \ \ \ \alpha_{4,c}={\frac{\theta_{4,c}}{\sigma_c^4}}-3<br />
</math><br />
<br />
==Covariance==<br />
Very similar algorithms can be used to compute the [[covariance]]. The naive algorithm is:<br />
:<math>\operatorname{Cov}(X,Y) = \displaystyle\frac {\sum_{i=1}^n x_i y_i - (\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)/n}{n}. \!</math><br />
<br />
For the algorithm above, one could use the following pseudocode:<br />
<source lang="python"><br />
def naive_covariance(data1, data2):<br />
n = len(data1)<br />
sum12 = 0<br />
sum1 = sum(data1)<br />
sum2 = sum(data2)<br />
<br />
for i in range(n):<br />
sum12 += data1[i]*data2[i]<br />
<br />
covariance = (sum12 - sum1*sum2 / n) / n<br />
return covariance<br />
</source><br />
<br />
A more numerically stable two-pass algorithm first computes the sample means, and then the covariance:<br />
:<math>\bar x = \displaystyle \sum_{i=1}^n x_i/n</math><br />
:<math>\bar y = \displaystyle \sum_{i=1}^n y_i/n</math><br />
:<math>\operatorname{Cov}(X,Y) = \displaystyle\frac {\sum_{i=1}^n (x_i - \bar x)(y_i - \bar y)}{n}. \!</math><br />
<br />
The two-pass algorithm may be written as:<br />
<source lang="python"><br />
def two_pass_covariance(data1, data2):<br />
n = len(data1)<br />
<br />
mean1 = sum(data1) / n<br />
mean2 = sum(data2) / n <br />
<br />
covariance = 0<br />
<br />
for i in range(n):<br />
a = data1[i] - mean1 <br />
b = data2[i] - mean2<br />
covariance += a*b / n<br />
<br />
return covariance<br />
</source><br />
<br />
A slightly more accurate compensated version performs the full naive algorithm on the residuals. The final sums <math>\textstyle\sum x_i</math> and <math>\textstyle\sum y_i</math> ''should'' be zero, but the second pass compensates for any small error.<br />
<br />
A stable one-pass algorithm exists, similar to the one above, that computes co-moment <math>\textstyle C_n = \sum_{i=1}^n (x_i - \bar x_n)(y_i - \bar y_n)</math>:<br />
:<math>\bar x_n = \bar x_{n-1} + \frac{x_n - \bar x_{n-1}}{n} \!</math><br />
:<math>\bar y_n = \bar y_{n-1} + \frac{y_n - \bar y_{n-1}}{n} \!</math><br />
:<math>C_n = C_{n-1} + (x_n - \bar x_n)(y_n - \bar y_{n-1}) = C_{n-1} + (y_n - \bar y_n)(x_n - \bar x_{n-1})</math><br />
The apparent asymmetry in that last equation is due to the fact that <math>\textstyle (x_n - \bar x_n) = \frac{n-1}{n}(x_n - \bar x_{n-1})</math>, so both update terms are equal to <math>\textstyle \frac{n-1}{n}(x_n - \bar x_{n-1})(y_n - \bar y_{n-1})</math>. Even greater accuracy can be achieved by first computing the means, then using the stable one-pass algorithm on the residuals.<br />
<br />
Thus we can compute the covariance as<br />
:<math>\begin{align}<br />
\operatorname{Cov}_N(X,Y) = \frac{C_N}{N} &= \frac{\operatorname{Cov}_{N-1}(X,Y)\cdot(N-1) + (x_n - \bar x_n)(y_n - \bar y_{n-1})}{N}\\<br />
&= \frac{\operatorname{Cov}_{N-1}(X,Y)\cdot(N-1) + (y_n - \bar y_n)(x_n - \bar x_{n-1})}{N}\\<br />
&= \frac{\operatorname{Cov}_{N-1}(X,Y)\cdot(N-1) + \frac{N-1}{N}(x_n - \bar x_{n-1})(y_n - \bar y_{n-1})}{N}.<br />
\end{align}</math><br />
<br />
Likewise, there is a formula for combining the covariances of two sets that can be used to parallelize the computation:<br />
:<math>C_X = C_A + C_B + (\bar x_A - \bar x_B)(\bar y_A - \bar y_B)\cdot\frac{n_A n_B}{n_X}</math>.<br />
<br />
==See also==<br />
*[[Computational formula for the variance]]<br />
<br />
==References==<br />
<references /><br />
<br />
==External links==<br />
* {{MathWorld|title=Sample Variance Computation|urlname=SampleVarianceComputation}}<br />
<br />
{{DEFAULTSORT:Algorithms For Calculating Variance}}<br />
[[Category:Statistical algorithms]]<br />
[[Category:Statistical deviation and dispersion]]<br />
[[Category:Articles with example pseudocode]]</div>Adminhttps://en.formulasearchengine.com/index.php?title=Angle&diff=218528Angle2014-07-31T13:51:04Z<p>Admin: Reverted edits by MediaWiki spam cleanup (talk) to last revision by Dbfirs</p>
<hr />
<div>{{About|angles in geometry}}<br />
{{Redirect|Oblique angle|the cinematographic technique|Dutch angle}}<br />
[[Image:Angle Symbol.svg|thumb|right|∠, the angle symbol in [[Unicode]] is [[List of XML and HTML character entity references|U+2220]]|228x228px]]<br />
<br />
In [[geometry]], an '''angle''' is the figure formed by two [[Ray (geometry)|rays]], called the ''sides'' of the angle, sharing a common endpoint, called the ''[[vertex (geometry)|vertex]]'' of the angle.<ref>{{SpringerEOM|id=Angle&oldid=13323|title=Angle|year=2001|last=Sidorov|first=L.A.}}</ref><br />
Angles are usually presumed to be in a [[Euclidean plane]] or in the [[Euclidean space]], but are also defined in [[non-Euclidean geometry|non-Euclidean geometries]]. In particular, in [[spherical geometry]], the [[spherical angle]]s are defined, using arcs of [[great circle]]s instead of rays.<br />
<br />
''Angle'' is also used to designate the [[measure (mathematics)|measure]] of an angle or of a [[Rotation (mathematics)|rotation]]. This measure is the ratio of the length of a [[arc (geometry)|circular arc]] to its [[radius]]. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation.<br />
<br />
The word ''angle'' comes from the [[Latin]] word ''angulus'', meaning "a corner". The word ''angulus'' is a diminutive, of which the primitive form, ''angus'', does not occur in Latin. [[Cognate]] words are the [[Greek language|Greek]] {{lang|grc|ἀγκύλος}} ''(ankylοs)'', meaning "crooked, curved," and the [[English language|English]] word "[[ankle]]". Both are connected with the [[Proto-Indo-European language|Proto-Indo-European]] root ''*ank-'', meaning "to bend" or "bow".<ref>{{citation|last=Slocum|first=Jonathan|year=2007<br />
|url=http://www.utexas.edu/cola/centers/lrc/ielex/X/P0089.html<br />
|title=Preliminary Indo-European lexicon&nbsp;— Pokorny PIE data<br />
|accessdate=2 Feb 2010|publisher=[[Linguistics Research Center at UT Austin|University of Texas research department: linguistics research center]]}}</ref><br />
<!--Note: ἀγκύλος rather than ἀνκύλος is correct, the γκ is a digraph pronounced [ŋk].--><br />
<br />
[[Euclid]] defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to [[Proclus]] an angle must be either a quality or a quantity, or a relationship. The first concept was used by [[Eudemus of Rhodes|Eudemus]], who regarded an angle as a deviation from a [[straight line]]; the second by [[Carpus of Antioch]], who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles are certainly quantitative.<ref>{{harvnb|Chisholm|1911}}; {{harvnb|Heiberg|1908|pp=177–178}}</ref><!-- This paragraph is quoted from EB1911, but its source seems to be Heath.--><br />
<br />
==Identifying angles==<br />
In mathematical expressions, it is common to use [[Greek letter]]s (<var>α</var>, <var>β</var>, <var>γ</var>, <var>θ</var>, <var>φ</var>, ...) to serve as [[Variable (mathematics)|variables]] standing for the size of some angle. (To avoid confusion with its other meaning, the symbol [[Pi|π]] is typically not used for this purpose.) Lower case roman letters (a, b, c, ...) are also used. See the figures in this article for examples.<br />
<br />
In geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB and AC (i.e. the lines from point A to point B and point A to point C) is denoted ∠BAC or <math>\widehat{\rm BAC}.</math> Sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex ("angle A").<br />
<br />
Potentially, an angle denoted, say, ∠BAC might refer to any of four angles: the clockwise angle from B to C, the anticlockwise angle from B to C, the clockwise angle from C to B, or the anticlockwise angle from C to B, where the direction in which the angle is measured determines its sign (see [[#Positive and negative angles|Positive and negative angles]]). However, in many geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant, and no ambiguity arises. Otherwise, a convention may be adopted so that ∠BAC always refers to the anticlockwise (positive) angle from B to C, and ∠CAB to the anticlockwise (positive) angle from C to B.<br />
<br />
==Types of angles==<br />
<!-- old images<br />
{|<br />
|- style="vertical-align: top;"<br />
|[[Image:Right angle.svg|thumb|134px|[[Right angle]].]]<br />
|[[Image:Complement angle.svg|thumb|134px|The [[complementary angles]] <var>a</var> and <var>b</var> (<var>b<var> is the complement of <var>a</var>, and <var>a</var> is the complement of <var>b</var>).]]<br />
|}<br />
{|<br />
|[[Image:Angle obtuse acute straight.svg|thumb|241px|Acute (<var>a</var>), obtuse (<var>b</var>), and straight (<var>c</var>) angles. Here, <var>a</var> and <var>b</var> are [[supplementary angles]].]]<br />
|[[Image:Reflex angle.svg|thumb|96px|[[Reflex angle]]. Here the sum of the reflex angle and the acute angle makes an [[explementary angle]].]]<br />
|}<br />
<br />
| [[Image:angle acute.png|thumb|150px|Acute angle]]<br />
| [[Image:angle obtuse.png|thumb|200px|Obtuse angle]]<br />
| [[Image:angle straight.png|thumb|200px|Straight angle]]<br />
--><br />
<br />
=== Individual angles ===<br />
[[Image:Right angle.svg|thumb|left|150px|[[Right angle]].]]<br />
[[Image:Reflex angle.svg|thumb|right|110px|[[Reflex angle]].]]<br />
[[Image:Angle obtuse acute straight.svg|thumb|center|250px|Acute (<var>a</var>), obtuse (<var>b</var>), and straight (<var>c</var>) angles. The acute and obtuse angles are also known as oblique angles]]<br />
<br />
*Angles smaller than a right angle (less than 90°) are called ''acute angles'' ("acute" meaning "sharp").<br />
*An angle equal to 1/4 turn (90° or ''[[π]] / ''2 radians) is called a ''[[right angle]]''. Two lines that form a right angle are said to be ''[[normal (geometry)|normal]]'', ''[[orthogonality|orthogonal]]'', or ''[[perpendicular]]''.<br />
*Angles larger than a right angle and smaller than a straight angle (between 90° and 180°) are called ''obtuse angles'' ("obtuse" meaning "blunt").<br />
*An angle equal to 1/2 turn (180° or ''[[pi|π]]'' radians) is called a ''straight angle''.<br />
*Angles larger than a straight angle but less than 1 turn (between 180° and 360°) are called ''reflex angles''.<br />
*An angle equal to 1 turn (360° or 2''[[pi|π]]'' radians) is called a ''full angle'', ''complete angle'', or a ''perigon''.<br />
*Angles that are not right angles or a multiple of a right angle are called ''oblique angles''.<br />
<br />
The names, intervals, and measured units are shown in a table below:<br />
<br />
{|class = wikitable style="text-align:center;"<br />
|style = "background:#f2f2f2; text-align:center;" | '''Name'''&nbsp;&nbsp; <br />
|style = "width:3em;" | acute<br />
|style = "width:3em;" | right angle<br />
|style = "width:3em;" | obtuse<br />
|style = "width:3em;" | straight<br />
|style = "width:3em;" | reflex<br />
|style = "width:3em;" | perigon<br />
|-<br />
! Units !! colspan=10 | Interval<br />
|-<br />
|style = "background:#f2f2f2; text-align:center;" | '''[[Turn (geometry)|Turns]]'''&nbsp;&nbsp; <br />
|style = "width:3em;" | <math>(0,\tfrac{1}{4})</math><br />
|style = "width:3em;" | <math>\tfrac{1}{4}</math><br />
|style = "width:3em;" | <math>(\tfrac{1}{4},\tfrac{1}{2})</math><br />
|style = "width:3em;" | <math>\tfrac{1}{2}</math><br />
|style = "width:3em;" | <math>(\tfrac{1}{2},1)</math><br />
|style = "width:3em;" | <math>1</math><br />
|-<br />
|style = "background:#f2f2f2; text-align:center;" | '''[[Radian]]s'''<br />
| <math>(0,\tfrac{1}{2}\pi)</math><br />
| <math>\tfrac{1}{2}\pi</math><br />
| <math>(\tfrac{1}{2}\pi,\pi)</math><br />
| <math>\pi</math><br />
| <math>(\pi,2\pi)</math><br />
| <math>2\pi\,</math><br />
|-<br />
|style = "background:#f2f2f2; text-align:center;" | '''[[Degree (angle)|Degrees]]'''&nbsp;&nbsp; <br />
|style = "width:3em;" | (0,90)°<br />
|style = "width:3em;" | 90°<br />
|style = "width:3em;" | (90,180)°<br />
|style = "width:3em;" | 180°<br />
|style = "width:3em;" | (180,360)°<br />
|style = "width:3em;" | 360°<br />
|-<br />
|}<br />
<br />
=== Equivalence angle pairs ===<br />
*Angles that have the same measure (i.e. the same magnitude) are said to be ''equal'' or ''[[Congruence (geometry)|congruent]]''. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g. all ''right angles'' are equal in measure).<br />
*Two angles which share terminal sides, but differ in size by an integer multiple of a turn, are called ''coterminal angles''.<br />
*A ''reference angle'' is the acute version of any angle determined by repeatedly subtracting or adding straight angle (1/2 turn, 180°, or π radians), to the results as necessary, until the magnitude of result is an acute angle, a value between 0 and 1/4 turn, 90°, or π/2 radians. For example, an angle of 30 degrees has a reference angle of 30 degrees, and an angle of 150 degrees also has a reference angle of 30 degrees (180-150). An angle of 750 degrees has a reference angle of 30 degrees (750-720).<ref>http://www.mathwords.com/r/reference_angle.htm</ref><br />
<br />
=== Intersecting angle pairs ===<br />
[[File:Vertical Angles.svg|thumb|150px|left|Angles A and B are a pair of vertical angles; angles C and D are a pair of vertical angles.]]<br />
[[Image:Adjacentangles.svg|right|thumb|225px|Angles ''A'' and ''B'' are adjacent.]]<br />
<br />
When two straight lines intersect at a point, four angles are formed. Pairwise these angles are named according to their location relative to each other.<br />
*A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called ''vertical angles'' or ''opposite angles'' or ''vertically opposite angles''. They are abbreviated as ''vert. opp. ∠s''.<ref name="tb">{{cite book|first1=TW|last1=Wong|first2=MS|last2=Wong|title=New Century Mathematics|location=Hong Kong|publisher=Oxford University Press|edition=1|volume=1B|pages=161–163|chapter=Angles in Intersecting and Parallel Lines|isbn=978-0-19-800176-8}}</ref><br />
<br />
:The equality of vertically opposite angles is called the ''vertical angle theorem''. [[Eudemus of Rhodes]] attributed the proof to [[Thales|Thales of Miletus]].<ref>{{cite book|author=[[Euclid]]|title=[[Euclid's Elements|The Elements]]|year=c. 300 BC}} Proposition I:13.</ref><ref name="William G. Shute 1960 pp. 25-27">William G. Shute, William W. Shirk, George F. Porter, ''Plane and Solid Geometry'', American Book Company (1960) pp. 25-27</ref> The proposition showed that since both of a pair of vertical angles is supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical Note,<ref name="William G. Shute 1960 pp. 25-27"/> when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as: all straight angles are equal, equals added to equals are equal, and equals subtracted from equals are equal.<br />
<br />
:In the figure, assume the measure of Angle ''A'' = ''x''. When two adjacent angles form a straight line, they are supplementary. Therefore, the measure of Angle ''C'' = 180 − ''x''. Similarly, the measure of Angle ''D'' = 180 − ''x''. Both Angle ''C'' and Angle ''D'' have measures equal to 180 - ''x'' and are congruent. Since Angle ''B'' is supplementary to both Angles ''C'' and ''D'', either of these angle measures may be used to determine the measure of Angle ''B''. Using the measure of either Angle ''C'' or Angle ''D'' we find the measure of Angle ''B'' = 180 - (180 - ''x'') = 180 - 180 + ''x'' = ''x''. Therefore, both Angle ''A'' and Angle ''B'' have measures equal to ''x'' and are equal in measure.<br />
<br />
*''Adjacent angles'', often abbreviated as ''adj. ∠s'', are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles that are side by side, or adjacent. Adjacent angles which sum to a right angle, straight angle or full angle are special and are respectively called ''complementary'', ''supplementary'' and ''explementary'' angles (see "Combine angle pairs" below).<br />
<br />
A [[Transversal (geometry)|transversal]] is a line that intersects a pair of (often parallel) lines and is associated with ''alternate angles'', ''corresponding angle'', ''interior angles'', and ''exterior angles''.{{citation needed|date=September 2013}}<br />
<br />
=== Combine angle pairs ===<br />
<br />
There are three special angle pairs which involves the summation of angles:<br />
<br />
*''Complementary angles'' are angle pairs whose measures sum to one right angle (1/4 turn, 90°, or π / 2 radians). If the two complementary angles are adjacent their non-shared sides form a right angle. In Euclidean geometry, the two acute angles in a right triangle are complementary, because the sum of internal angles of a triangle is 180 degrees, and the right angle itself accounts for ninety degrees.<br />
:The adjective complementary is from Latin ''complementum'', associated with the verb ''complere'', "to fill up". An acute angle is "filled up" by its complement to form a right angle.<br />
:The difference between an angle and a right angle is termed the ''complement'' of the angle.<br />
:If angles ''A'' and ''B'' are complementary, the following relationships hold:<br />
:<math>\begin{align} \sin^2A + \sin^2B &= 1\\ \cos^2A + \cos^2B &= 1\\ \tan A &= \cot B \\ \sec A &= \csc B \end{align}.</math><br />
:(The [[tangent]] of an angle equals the [[cotangent]] of its complement and its secant equals the [[cosecant]] of its complement.)<br />
:The [[prefix]] "co-" in the names of some trigonometric ratios refers to the word "complementary".<br />
<br />
*Two angles that sum to a straight angle (1/2 turn, 180°, or π radians) are called ''supplementary angles''.<br />
:If the two supplementary angles are [[adjacent angles|adjacent]] (i.e. have a common [[vertex (geometry)|vertex]] and share just one side), their non-shared sides form a [[line (geometry)|straight line]]. However, supplementary angles do not have to be on the same line, and can be separated in space. For example, adjacent angles of a [[parallelogram]] are supplementary, and opposite angles of a [[cyclic quadrilateral]] (one whose vertices all fall on a single circle) are supplementary.<br />
:If a point P is exterior to a circle with center O, and if the [[tangent lines to circles|tangent lines]] from P touch the circle at points T and Q, then ∠TPQ and ∠TOQ are supplementary.<br />
:The sines of supplementary angles are equal. Their cosines and tangents (unless undefined) are equal in magnitude but have opposite signs.<br />
:In Euclidean geometry, any sum of two angles in a triangle are supplementary to the third, because the sum of internal angles of a triangle is a straight angle.<br />
<br />
*Two angles that sum to a complete angle (1 turn, 360°, or 2π radians) are called ''explementary angles'' or ''conjugate angles''.<br />
*:The difference between an angle and a complete angle is termed the ''explement'' of the angle or ''conjugate'' of an angle.<br />
{{clr}}<br />
{|<br />
|- style="vertical-align: top;"<br />
[[Image:Complement angle.svg|thumb|left|150px|The ''complementary'' angles <var>a</var> and <var>b</var> (<var>b<var> is the ''complement'' of <var>a</var>, and <var>a</var> is the complement of <var>b</var>).]]<br />
[[Image:Reflex angle.svg|thumb|right|125px|The sum of the reflex angle and the acute angle makes an ''explementary'' angle.]]<br />
[[Image:Angle obtuse acute straight.svg|thumb|center|300px|The angles <var>a</var> and <var>b</var> are ''supplementary'' angles.]]<br />
|}<br />
{{clr}}<br />
<br />
=== Polygon related angles ===<br />
*An angle that is part of a [[simple polygon]] is called an ''[[interior angle]]'' if it lies on the inside of that simple polygon. A [[Convex and concave polygons|concave]] simple polygon has at least one interior angle that is a reflex angle.<br />
*:In [[Euclidean geometry]], the measures of the interior angles of a [[triangle]] add up to ''π'' radians, 180°, or 1/2 turn; the measures of the interior angles of a simple [[quadrilateral]] add up to 2''π'' radians, 360°, or 1 turn. In general, the measures of the interior angles of a [[polygon|simple polygon]] with ''n'' sides add up to [(''n''&nbsp;−&nbsp;2)&nbsp;×&nbsp;''π''] radians, or [(''n''&nbsp;−&nbsp;2)&nbsp;×&nbsp;180]°, (''2n''&nbsp;−&nbsp;4) right angles, or (''n/2''&nbsp;−&nbsp;1) turn.<br />
*The supplement of the interior angle is called the ''[[exterior angle]]''. It measures the amount of rotation one has to make at this vertex to trace out the polygon. If the corresponding interior angle is a reflex angle, the exterior angle should be considered [[Negative number|negative]]. Even in a non-simple polygon it may be possible to define the exterior angle, but one will have to pick an [[orientation (mathematics)|orientation]] of the [[plane (mathematics)|plane]] (or [[surface]]) to decide the sign of the exterior angle measure.<br />
*:In Euclidean geometry, the sum of the exterior angles of a simple polygon will be one full turn (360°). The exterior angle here could be called a ''supplementary exterior angle''. Because each side of an angle has a supplementary angle, there are two exterior angles per vertex. It is commonly used in [[Logo (programming language)|Logo Turtle Geometry]] when drawing regular polygons. <br />
*Some authors use the name ''exterior angle'' of a simple polygon to simply mean the ''explement exterior angle'' (''not'' supplement!) of the interior angle.<ref>{{MathWorld |urlname=ExteriorAngle |title=Exterior Angle}}</ref> This conflicts with the above usage.<br />
<br />
=== Plane related angles ===<br />
*The angle between two [[Plane (mathematics)|planes]] (such as two adjacent faces of a [[polyhedron]]) is called a ''[[dihedral angle]]''. It may be defined as the acute angle between two lines [[Normal (geometry)|normal]] to the planes.<br />
*The angle between a plane and an intersecting straight line is equal to ninety degrees minus the angle between the intersecting line and the line that goes through the point of intersection and is normal to the plane.<br />
<br />
==Measuring angles==<!-- linked from [[Degree (angle)]] --><br />
The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other. Angles that have the same size are sometimes called ''congruent angles''.<br />
<br />
In some contexts, such as identifying a point on a circle or describing the ''orientation'' of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full [[Turn (geometry)|turn]] are effectively equivalent. In other contexts, such as identifying a point on a [[spiral]] curve or describing the ''cumulative rotation'' of an object in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of a full turn are not equivalent.<br />
<br />
[[Image:Angle measure.svg|right|thumb|The measure of angle <var>θ</var> is the quotient of <var>s</var> and <var>r</var>.]]<br />
<br />
In order to measure an angle <var>[[theta|θ]]</var>, a [[circular arc]] centered at the vertex of the angle is drawn, e.g. with a pair of [[Compasses (drafting)|compasses]]. The length of the arc <var>s</var> is then divided by the radius of the arc <var>r</var>, and possibly multiplied by a scaling constant <var>k</var> (which depends on the units of measurement that are chosen):<br />
<br />
:<math> \theta = k \frac{s}{r}. </math><br />
<br />
The value of <var>θ</var> thus defined is independent of the size of the circle: if the length of the radius is changed then the arc length changes in the same proportion, so the ratio ''s''/''r'' is unaltered. (Proof. The formula above can be rewritten as {{nowrap|1= ''k'' = θ''r''/''s''.}} One turn, for which {{nowrap|1= θ = ''n''}} units, corresponds to an arc equal in length to the circle's [[circumference]], which is 2π''r'', so {{nowrap|1= ''s'' = 2π''r''}}. Substituting ''n'' for θ and 2π''r'' for ''s'' in the formula, results in {{nowrap|1= ''k'' = ''nr''/(2π''r'') = ''n''/(2π).}}) {{refn|group=Note|This approach requires however an additional proof that the measure of the angle does not change with changing radius <var>r</var>, in addition to the issue of "measurement units chosen." A smoother approach is to measure the angle by the length of the corresponding unit circle arc. Here "unit" can be chosen to be dimensionless in the sense that it is the real number 1 associated with the unit segment on the real line. See R. Dimitric for instance.<ref>R. Dimitric: [http://elib.mi.sanu.ac.rs/files/journals/tm/29/tm1525.pdf On angles and angle measurements]</ref>}}<br />
<br />
===Units===<br />
Units used to represent angles are listed below in descending magnitude order. Of these units, the ''[[degree (angle)|degree]]'' and the ''[[radian]]'' are by far the most commonly used. Angles expressed in radians are dimensionless for the purposes of [[dimensional analysis]].<br />
<br />
Most units of angular measurement are defined such that one ''[[Turn (geometry)|turn]]'' (i.e. one full circle) is equal to ''n'' units, for some whole number ''n''. The two exceptions are the radian and the diameter part.<br />
<br />
;Turn (''n''&nbsp;=&nbsp;1): The ''turn'', also ''cycle'', ''full circle'', ''revolution'', and ''rotation'', is complete circular movement or measure (as to return to the same point) with circle or ellipse. A turn is abbreviated {{tau}},''cyc'', ''rev'', or ''rot'' depending on the application, but in the acronym ''[[Revolutions per minute|rpm]]'' (revolutions per minute), just ''r'' is used. A ''turn'' of ''n'' units is obtained by setting {{nowrap|1= ''k'' = 1/(2{{pi}})}} in the formula above. The equivalence of 1 ''turn'' is 360°, 2{{pi}} rad, 400 grad, and 4 right angles. The symbol {{tau}} can also be used as a [[mathematical constant]] to represent 2{{pi}} radians. Used in this way ({{nowrap|1= ''k'' = {{tau}}/(2{{pi}})}}) allows for radians to be expressed as a fraction of a turn. For example, half a turn is {{nowrap|1= {{tau}}/2 = {{pi}}}}.<br />
;Quadrant (''n''&nbsp;=&nbsp;4): The ''quadrant'' is 1/4 of a turn, i.e. a ''[[right angle]]''. It is the unit used in [[Euclid's Elements]]. 1 quad. = 90° = {{pi}}/2&nbsp;rad = 1/4 turn = 100&nbsp;grad. In German the symbol <sup>∟</sup> has been used to denote a quadrant.<br />
;Sextant (''n''&nbsp;=&nbsp;6): The ''sextant'' (''angle of the [[equilateral triangle]]'') is 1/6 of a turn. It was the unit used by the [[Babylonians]],<ref>[[J.H. Jeans]] (1947), ''The Growth of Physical Science'', [http://books.google.co.uk/books?hl=en&lr=&id=JX49AAAAIAAJ&oi=fnd&pg=PA7 p.7]; [[Francis Dominic Murnaghan (mathematician)|Francis Dominic Murnaghan]] (1946), ''Analytic Geometry'', p.2</ref> and is especially easy to construct with ruler and compasses. The degree, minute of arc and second of arc are [[sexagesimal]] subunits of the Babylonian unit. 1 Babylonian unit = 60° = {{pi}}/3&nbsp;rad ≈ 1.047197551&nbsp;rad.<br />
[[Image:Angle radian.svg|right|thumb|<var>θ</var> = <var>s</var>/<var>r</var> rad = 1 rad.]]<br />
;Radian (''n''&nbsp;=&nbsp;6.283...): The ''[[radian]]'' is the angle subtended by an arc of a circle that has the same length as the circle's radius. The case of radian for the formula given earlier, a ''radian'' of ''n'' = 2{{pi}} units is obtained by setting ''k'' = 2{{pi}} /(2{{pi}}) = 1. One turn is 2{{pi}} radians, and one radian is 180/{{pi}} degrees, or about 57.2958 degrees. The radian is abbreviated ''rad'', though this symbol is often omitted in mathematical texts, where radians are assumed unless specified otherwise. When radians are used angles are considered as dimensionless. The radian is used in virtually all mathematical work beyond simple practical geometry, due, for example, to the pleasing and "natural" properties that the [[trigonometric function]]s display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the [[SI]] system. <br />
;Hour angle (''n''&nbsp;=&nbsp;24): The astronomical ''[[hour angle]]'' is 1/24 of a turn. Since this system is amenable to measuring objects that cycle once per day (such as the relative position of stars), the sexagesimal subunits are called ''minute of time'' and ''second of time''. Note that these are distinct from, and 15 times larger than, minutes and seconds of arc. 1&nbsp;hour = 15° = {{pi}}/12&nbsp;rad = 1/6&nbsp;quad. = 1/24 ''turn'' ≈ 16.667&nbsp;grad.<br />
;Point (''n''&nbsp;=&nbsp;32): The ''[[compass point|point]]'', used in [[navigation]], is 1/32 of a turn. 1&nbsp;point = 1/8 of a right angle = 11.25° = 12.5&nbsp;grad. Each point is subdivided in four quarter-points so that 1 turn equals 128 quarter-points.<br />
;Hexacontade (''n''&nbsp;=&nbsp;60): The ''hexacontade'' is a unit of 6° that [[Eratosthenes]] used, so that a whole turn was divided into 60 units.<br />
;Pechus (''n''&nbsp;=&nbsp;144–180): The ''pechus'' was a [[Babylonian mathematics|Babylonian]] unit equal to about 2° or 2½°.<br />
;Binary degree (''n''&nbsp;=&nbsp;256): The ''binary degree'', also known as the ''[[binary radian]]'' (or ''brad''), is 1/256 of a turn.<ref>[http://web.archive.org/web/20080628051746/http://www.oopic.com/pgchap15.htm ooPIC Programmer's Guide (archived)] ''www.oopic.com''</ref> The binary degree is used in computing so that an angle can be efficiently represented in a single [[byte]] (albeit to limited precision). Other measures of angle used in computing may be based on dividing one whole turn into 2<sup>''n''</sup> equal parts for other values of ''n''.<ref>[http://blogs.msdn.com/shawnhar/archive/2010/01/04/angles-integers-and-modulo-arithmetic.aspx Angles, integers, and modulo arithmetic] Shawn Hargreaves ''blogs.msdn.com''</ref><br />
;Degree (''n''&nbsp;=&nbsp;360): The ''[[degree (angle)|degree]]'', denoted by a small superscript circle (°), is 1/360 of a turn, so one ''turn'' is 360°. The case of degrees for the formula given earlier, a ''degree'' of ''n'' = 360° units is obtained by setting ''k'' = 360°/(2{{pi}}). One advantage of this old [[sexagesimal]] subunit is that many angles common in simple geometry are measured as a whole number of degrees. Fractions of a degree may be written in normal decimal notation (e.g. 3.5° for three and a half degrees), but the "minute" and "second" sexagesimal subunits of the "degree-minute-second" system are also in use, especially for [[Geographic coordinate system|geographical coordinates]] and in [[astronomy]] and [[ballistics]]:<br />
;Diameter part (''n''&nbsp;=&nbsp;376.99...): The ''diameter part'' (occasionally used in Islamic mathematics) is 1/60 radian. One "diameter part" is approximately 0.95493°. There are about 376.991 diameter parts per turn.<br />
;Grad (''n''&nbsp;=&nbsp;400): The ''[[grad (angle)|grad]]'', also called ''grade'', ''gradian'', or ''gon'', is 1/400 of a turn, so a right angle is 100 grads. It is a decimal subunit of the quadrant. A [[kilometre]] was historically defined as a [[centi]]-grad of arc along a great circle of the Earth, so the kilometer is the decimal analog to the [[sexagesimal]] nautical mile. The grad is used mostly in [[triangulation]].<br />
;Mil (''n''&nbsp;=&nbsp;6000–6400): The ''[[angular mil|mil]]'' is any of several units that are ''approximately'' equal to a [[milliradian]]. There are several definitions ranging from 0.05625 to 0.06 degrees (3.375 to 3.6 minutes), with the milliradian being approximately 0.05729578 degrees (3.43775 minutes). In [[NATO]] countries, it is defined as 1/6400th of a circle. Its value is approximately equal to the angle subtended by a width of 1 metre as seen from 1&nbsp;km away (2{{pi}} / 6400 = 0.0009817… ≒ 1/1000).<br />
;Minute of arc (''n''&nbsp;=&nbsp;21,600): The ''[[minute of arc]]'' (or ''MOA'', ''arcminute'', or just ''minute'') is 1/60 of a degree = 1/21600 turn. It is denoted by a single prime (&nbsp;′&nbsp;). For example, 3°&nbsp;30′ is equal to 3&nbsp;+&nbsp;30/60 degrees, or 3.5 degrees. A mixed format with decimal fractions is also sometimes used, e.g. 3°&nbsp;5.72′ = 3&nbsp;+&nbsp;5.72/60 degrees. A [[nautical mile]] was historically defined as a minute of arc along a [[great circle]] of the Earth.<br />
;Second of arc (''n''&nbsp;=&nbsp;1,296,000): The ''[[second of arc]]'' (or ''arcsecond'', or just ''second'') is 1/60 of a minute of arc and 1/3600 of a degree. It is denoted by a double prime (&nbsp;″&nbsp;). For example, 3°&nbsp;7′&nbsp;30″ is equal to 3 + 7/60 + 30/3600 degrees, or 3.125&nbsp;degrees.<br />
<br />
===Positive and negative angles===<br />
Although the definition of the measurement of an angle does not support the concept of a negative angle, it is frequently useful to impose a convention that allows positive and negative angular values to represent orientations and/or rotations in opposite directions relative to some reference.<br />
<br />
In a two-dimensional [[Cartesian coordinate system]], an angle is typically defined by its two sides, with its vertex at the origin. The ''initial side'' is on the positive [[x-axis]], while the other side or ''terminal side'' is defined by the measure from the initial side in radians, degrees, or turns. With ''positive angles'' representing rotations toward the positive [[y-axis]] and ''negative angles'' representing rotations toward the negative y-axis. When Cartesian coordinates are represented by ''standard position'', defined by the x-axis rightward and the y-axis upward, positive rotations are [[anticlockwise]] and negative rotations are [[clockwise]].<br />
<br />
In many contexts, an angle of −''θ'' is effectively equivalent to an angle of "one full turn minus ''θ''". For example, an orientation represented as &nbsp;−&nbsp;45° is effectively equivalent to an orientation represented as 360°&nbsp;−&nbsp;45° or 315°. However, a rotation of &nbsp;−&nbsp;45° would not be the same as a rotation of 315°.<br />
<br />
In three-dimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined relative to some reference, which is typically a [[Vector (geometric)|vector]] passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.<br />
<br />
In [[navigation]], [[bearing (navigation)|bearings]] are measured relative to north. By convention, viewed from above, bearing angle are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so a north-west orientation corresponds to a bearing of 315°.<br />
.<br />
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===Alternative ways of measuring the size of an angle===<br />
There are several alternatives to measuring the size of an angle by the corresponding angle of rotation.<br />
The ''[[grade (slope)|grade of a slope]]'', or ''gradient'' is equal to the [[tangent (trigonometric function)|tangent]] of the angle, or sometimes (rarely) the [[sine]]. Gradients are often expressed as a percentage. For very small values (less than 5%), the grade of a slope is approximately the measure of an angle in radians.<br />
<br />
In [[rational geometry]] the ''spread'' between two lines is defined at the square of sine of the angle between the lines. Since the sine of an angle and the sine of its supplementary angle are the same any angle of rotation that maps one of the lines into the other leads to the same value of the spread between the lines.<br />
<br />
===Astronomical approximations===<br />
Astronomers measure angular separation of objects in degrees from their point of observation.<br />
* 0.5° is approximately the width of the sun or moon.<br />
* 1° is approximately the width of a little finger at arm's length.<br />
* 10° is approximately the width of a closed fist at arm's length.<br />
* 20° is approximately the width of a handspan at arm's length.<br />
<br />
These measurements clearly depend on the individual subject, and the above should be treated as rough [[rule of thumb]] approximations only.<br />
<br />
==Angles between curves==<br />
[[Image:Curve angles.svg|thumb|right|The angle between the two curves at P is defined as the angle between the tangents <var>A</var> and <var>B</var> at <var>P</var>]]<br />
The angle between a line and a [[curve]] (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the [[tangent]]s at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—''amphicyrtic'' (Gr. ''{{lang|grc|ἀμφί}}'', on both sides, ''κυρτός'', convex) or ''cissoidal'' (Gr. ''κισσός'', ivy), biconvex; ''xystroidal'' or ''sistroidal'' (Gr. ''ξυστρίς'', a tool for scraping), concavo-convex; ''amphicoelic'' (Gr. ''κοίλη'', a hollow) or ''angulus lunularis'', biconcave.<ref>{{harvnb|Chisholm|1911}}; {{harvnb|Heiberg|1908|p=178}}</ref><!-- Again, most of this paragraph is from EB1911 with Heath used as its source. --><br />
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==Dot product and generalisation==<br />
In the [[Euclidean space|Euclidean plane]], the angle θ between two [[Vector (geometric)|vectors]] '''u''' and '''v''' is related to their [[dot product]] and their lengths by the formula<br />
<br />
:<math>\mathbf{u} \cdot \mathbf{v} = \cos(\theta)\ \|\mathbf{u}\|\ \|\mathbf{v}\|.</math><br />
<br />
This formula supplies an easy method to find the angle between two planes (or curved surfaces) from their [[normal vector]]s and between [[skew lines]] from their vector equations.<br />
<br />
==Inner product==<br />
To define angles in an abstract real [[inner product space]], we replace the Euclidean dot product ( '''·''' ) by the inner product <math>\langle\cdot,\cdot\rangle</math>, i.e.<br />
<br />
:<math>\langle\mathbf{u},\mathbf{v}\rangle = \cos(\theta)\ \|\mathbf{u}\|\ \|\mathbf{v}\|.</math><br />
<br />
In a complex [[inner product space]], the expression for the cosine above may give non-real values, so it is replaced with<br />
<br />
:<math>\operatorname{Re}(\langle\mathbf{u},\mathbf{v}\rangle) = \cos(\theta)\ \|\mathbf{u}\|\ \|\mathbf{v}\|.</math><br />
<br />
or, more commonly, using the absolute value, with<br />
<br />
:<math> |\langle\mathbf{u},\mathbf{v}\rangle| = \cos(\theta)\ \|\mathbf{u}\|\ \|\mathbf{v}\|.</math><br />
<br />
The latter definition ignores the direction of the vectors and thus describes the angle between one-dimensional subspaces <math>\operatorname{span}(\mathbf{u})</math> and <math>\operatorname{span}(\mathbf{v})</math> spanned by the vectors <math>\mathbf{u}</math> and <math>\mathbf{v}</math> correspondingly.<br />
<br />
==Angles between subspaces==<br />
The definition of the angle between one-dimensional subspaces <math>\operatorname{span}(\mathbf{u})</math> and <math>\operatorname{span}(\mathbf{v})</math> given by<br />
<br />
:<math> |\langle\mathbf{u},\mathbf{v}\rangle| = \cos(\theta)\ \|\mathbf{u}\|\ \|\mathbf{v}\|</math><br />
<br />
in a [[Hilbert space]] can be extended to subspaces of any finite dimensions. Given two subspaces <math>\mathcal{U},\mathcal{W}</math> with <math>\operatorname{dim}(\mathcal{U}):=k\leq \operatorname{dim}(\mathcal{W}):=l</math>, this leads to a definition of <math>k</math> angles called canonical or [[principal angles]] between subspaces.<br />
<br />
==Angles in Riemannian geometry==<br />
In [[Riemannian geometry]], the [[metric tensor]] is used to define the angle between two [[tangent]]s. Where ''U'' and ''V'' are tangent vectors and ''g''<sub>''ij''</sub> are the components of the metric tensor ''G'',<br />
<br />
:<math><br />
\cos \theta = \frac{g_{ij}U^iV^j}<br />
{\sqrt{ \left| g_{ij}U^iU^j \right| \left| g_{ij}V^iV^j \right|}}.<br />
</math><br />
<br />
==Angles in geography and astronomy==<br />
In [[geography]], the location of any point on the Earth can be identified using a ''[[geographic coordinate system]]''. This system specifies the [[latitude]] and [[longitude]] of any location in terms of angles subtended at the centre of the Earth, using the [[equator]] and (usually) the [[Greenwich meridian]] as references.<br />
<br />
In [[astronomy]], a given point on the [[celestial sphere]] (that is, the apparent position of an astronomical object) can be identified using any of several ''[[astronomical coordinate systems]]'', where the references vary according to the particular system. Astronomers measure the ''[[angular separation]]'' of two [[star]]s by imagining two lines through the centre of the [[Earth]], each intersecting one of the stars. The angle between those lines can be measured, and is the angular separation between the two stars.<br />
<br />
Astronomers also measure the ''apparent size'' of objects as an [[angular diameter]]. For example, the [[full moon]] has an angular diameter of approximately 0.5°, when viewed from Earth. One could say, "The Moon's diameter subtends an angle of half a degree." The [[small-angle formula]] can be used to convert such an angular measurement into a distance/size ratio.<br />
<br />
==See also==<br />
*[[Bisection#Angle bisector|Angle bisector]]<br />
*[[Angular velocity]]<br />
*[[Argument (complex analysis)]]<br />
*[[Astrological aspect]]<br />
*[[Central angle]]<br />
*[[Clock angle problem]]<br />
*[[Great circle distance]]<br />
*[[Hyperbolic angle]]<br />
*[[Inscribed angle]]<br />
*[[Irrational angle]]<br />
*[[Protractor]]<br />
*[[Solid angle]] for a concept of angle in three dimensions.<br />
*[[Spherical angle]]<br />
<br />
==Notes==<br />
{{reflist|group=Note}}<br />
<br />
==References==<br />
{{Reflist}}<br />
<br />
==Sources==<br />
*{{cite book |ref=harv|first=Johan Ludvig |last=Heiberg |year=1908 |title=Euclid |editor-first=T. L. |editor-last=Heath |editor-link=T. L. Heath |series=The thirteen books of Euclid's Elements |volume=1 |publisher=Cambridge University press|url=http://books.google.com/books?id=UhgPAAAAIAAJ}}<br />
<br />
;Attribution<br />
*{{EB1911|wstitle=Angle}}<br />
<br />
==External links==<br />
{{Commons category|Angles}}<br />
*[http://www.cut-the-knot.org/Curriculum/Geometry/CyQuadri.shtml Angle Bisectors in a Quadrilateral] at [[cut-the-knot]]<br />
*[http://www.cut-the-knot.org/triangle/TriangleFromBisectors.shtml Constructing a triangle from its angle bisectors] at [[cut-the-knot]]<br />
*[http://www.austinastro.org/angles.html Angle Estimation]&nbsp;– for basic [[astronomy]]<br />
*[http://www.mathopenref.com/tocs/constructionstoc.html Various angle constructions with compass and straightedge]<br />
* [http://www.mathopenref.com/anglecomplementary.html Complementary Angles animated demonstration. ] With interactive applet<br />
* [http://www.mathopenref.com/anglesupplementary.html Supplementary Angles animated demonstration. ] With interactive applet<br />
* [http://www.mathopenref.com/tocs/anglestoc.html Angle definition pages] with interactive applets that are also useful in a classroom setting. Math Open Reference<br />
<br />
[[Category:Angle| ]]<br />
<br />
{{Link FA|nl}}</div>Adminhttps://en.formulasearchengine.com/index.php?title=Amplitude_modulation&diff=218515Amplitude modulation2014-07-31T13:50:17Z<p>Admin: Reverted edits by MediaWiki spam cleanup (talk) to last revision by Interferometrist</p>
<hr />
<div>{{Modulation techniques}}<br />
'''Amplitude modulation''' ('''AM''') is a [[modulation]] technique used in electronic communication, most commonly for transmitting information via a [[radio]] [[carrier wave]]. AM works by varying the strength ([[amplitude]]) of the carrier in proportion to the waveform being sent. That waveform may, for instance, correspond to the sounds to be reproduced by a [[loudspeaker]], or the light intensity of television pixels. This contrasts with [[frequency modulation]], in which the [[frequency]] of the [[carrier signal]] is varied, and [[phase modulation]], in which its [[Phase (waves)|phase]] is varied, by the modulating signal.<br />
<br />
AM was the earliest modulation method used to transmit voice by radio. It was developed during the first two decades of the 20th century beginning with [[Reginald Fessenden]]'s [[radiotelephone]] experiments in 1900. It remains in use today in many forms of communication; for example it is used in portable [[two way radio]]s, [[Airband|VHF aircraft radio]] and in computer [[modem]]s.{{citation needed|date=January 2014}} "AM" is often used to refer to [[mediumwave]] [[AM broadcasting|AM radio broadcasting]].<br />
<br />
[[File:Amfm3-en-de.gif|thumb|right|250px|Fig 1: An audio signal (top) may be carried by an AM or [[Frequency modulation|FM]] radio wave.|alt=Animation of audio, AM and FM sine waves]]<br />
<br />
==Forms of amplitude modulation==<br />
<br />
In [[electronics]] and [[telecommunications]], [[modulation]] means varying some aspect of a higher frequency [[continuous wave]] [[carrier signal]] with an information-bearing modulation waveform, such as an [[audio signal]] which represents sound, or a [[video signal]] which represents images, so the carrier will "carry" the information. When it reaches its destination, the information signal is extracted from the modulated carrier by [[demodulation]].<br />
<br />
In amplitude modulation, the [[amplitude]] or "strength" of the carrier oscillations is what is varied. For example, in AM radio communication, a [[continuous wave]] radio-frequency signal (a [[Sine wave|sinusoid]]al [[carrier wave]]) has its [[amplitude]] [[modulation|modulated]] by an audio waveform before transmission. The audio waveform modifies the amplitude of the carrier wave and determines the ''[[Envelope (waves)|envelope]]'' of the waveform. In the [[frequency domain]], amplitude modulation produces a signal with power concentrated at the carrier frequency and two adjacent [[sideband]]s. Each sideband is equal in [[bandwidth (signal processing)|bandwidth]] to that of the modulating signal, and is a mirror image of the other. Standard AM is thus sometimes called "double-sideband amplitude modulation" (DSB-AM) to distinguish it from more sophisticated modulation methods also based on AM.<br />
<br />
One disadvantage of all amplitude modulation techniques (not only standard AM) is that the receiver amplifies and detects [[Noise (radio)|noise]] and [[electromagnetic interference]] in equal proportion to the signal. Increasing the received [[signal to noise ratio]], say, by a factor of 10 (a 10 [[decibel]] improvement), thus would require increasing the transmitter power by a factor of 10. This is in contrast to [[frequency modulation]] (FM) and [[digital radio]] where the effect of such noise following [[demodulation]] is strongly reduced so long as the received signal is well above the threshold for reception. For this reason AM broadcast is not favored for music and [[high fidelity]] broadcasting, but rather for voice communications and broadcasts (sports, news, [[talk radio]] etc.).<br />
<br />
Another disadvantage of AM is that it is inefficient in power usage; at least two-thirds of the power is concentrated in the carrier signal. The carrier signal contains none of the original information being transmitted (voice, video, data, etc.). However its presence provides a simple means of demodulation using [[envelope detector|envelope detection]], providing a frequency and phase reference to extract the modulation from the sidebands. In some modulation systems based on AM, a lower transmitter power is required through partial or total elimination of the carrier component, however receivers for these signals are more complex and costly. The receiver may regenerate a copy of the carrier frequency (usually as shifted to the [[intermediate frequency]]) from a greatly reduced "pilot" carrier (in [[reduced-carrier transmission]] or DSB-RC) to use in the demodulation process. Even with the carrier totally eliminated in [[double-sideband suppressed-carrier transmission]], carrier regeneration is possible using a [[Costas loop|Costas phase-locked loop]]. This doesn't work however for [[single-sideband suppressed-carrier transmission]] (SSB-SC), leading to the characteristic "Donald Duck" sound from such receivers when slightly detuned. Single sideband is nevertheless used widely in [[amateur radio]] and other voice communications both due to its power efficiency and bandwidth efficiency (cutting the RF bandwidth in half compared to standard AM). On the other hand, in [[medium wave]] and [[short wave]] broadcasting, standard AM with the full carrier allows for reception using inexpensive receivers. The broadcaster absorbs the extra power cost to greatly increase potential audience.<br />
<br />
An additional function provided by the carrier in standard AM, but which is lost in either single or double-sideband suppressed-carrier transmission, is that it provides an amplitude reference. In the receiver, the [[automatic gain control]] (AGC) responds to the carrier so that the reproduced audio level stays in a fixed proportion to the original modulation. On the other hand, with suppressed-carrier transmissions there is ''no'' transmitted power during pauses in the modulation, so the AGC must respond to peaks of the transmitted power during peaks in the modulation. This typically involves a so-called ''fast attack, slow decay'' circuit which holds the AGC level for a second or more following such peaks, in between syllables or short pauses in the program. This is very acceptable for communications radios, where [[Dynamic range compression|compression]] of the audio aids intelligibility. However it is absolutely undesired for music or normal broadcast programming, where a faithful reproduction of the original program, including its varying modulation levels, is expected.<br />
<br />
A trivial form of AM which can be used for transmitting [[Digital data|binary data]] is [[on-off keying]], the simplest form of ''[[amplitude-shift keying]]'', in which [[Binary numeral system|ones and zeros]] are represented by the presence or absence of a carrier. On-off keying is likewise used by radio amateurs to transmit [[Morse code]] where it is known as [[continuous wave]] (CW) operation, even though the transmission is not strictly "continuous."<br />
<br />
===ITU designations===<br />
<br />
In 1982, the [[International Telecommunication Union]] (ITU) designated the types of amplitude modulation:<br />
{|class="wikitable"<br />
|-<br />
!Designation!!Description<br />
|-<br />
|A3E||[[double sideband|double-sideband]] a full-carrier - the basic Amplitude modulation scheme<br />
|-<br />
|R3E||[[Single-sideband modulation|single-sideband]] [[Reduced-carrier transmission|reduced-carrier]]<br />
|-<br />
|H3E||[[Single-sideband modulation|single-sideband]] full-carrier<br />
|-<br />
|J3E||[[Single-sideband suppressed-carrier transmission|single-sideband suppressed-carrier]]<br />
|-<br />
|B8E||[[independent sideband|independent-sideband]] emission<br />
|-<br />
|C3F||[[vestigal sideband|vestigial-sideband]]<br />
|-<br />
|Lincompex||linked [[compander|compressor and expander]]<br />
|}<!--single mid band linked modulation<br />
[B4E side bander inter linked] [expand carrier] full reduced.aLTIDOE--><br />
<br />
==History==<br />
[[Image:Telefunken arc radiotelephone.jpg|thumb|One of the crude pre-vacuum tube AM transmitters, a Telefunken [[arc converter|arc transmitter]] from 1906. The carrier wave is generated by 6 electric arcs in the vertical tubes, connected to a [[tuned circuit]]. Modulation is done by the large carbon microphone ''(cone shape)'' in the antenna lead. ]]<br />
[[Image:Meissner radiotelephone transmitter.jpg|thumb|One of the first [[vacuum tube]] AM radio transmitters, built by Meissner in 1913 with an early triode tube by Robert von Lieben. He used it in a historic 36 km (24 mi) voice transmission from Berlin to Nauen, Germany. Compare its small size with above transmitter. ]]<br />
<br />
Although AM was used in a few crude experiments in multiplex telegraph and telephone transmission in the late 1800s,<ref name="Bray">{{cite book <br />
| last = Bray<br />
| first = John <br />
| title = Innovation and the Communications Revolution: From the Victorian Pioneers to Broadband Internet<br />
| publisher = Inst. of Electrical Engineers<br />
| year = 2002<br />
| location = <br />
| pages = 59, 61–62<br />
| url = http://books.google.com/books?id=3h7R36Y0yFUC&pg=PA61<br />
| doi = <br />
| id = <br />
| isbn = 0852962185}}</ref> the practical development of amplitude modulation is synonymous with the development between 1900 and 1920 of "[[radiotelephone]]" transmission, that is, the effort to send sound (audio) by radio waves. The first radio transmitters, called [[spark gap transmitter]]s, transmitted information by [[wireless telegraphy]], using different length pulses of carrier wave to spell out text messages in [[Morse code]]. They couldn't transmit audio because the carrier consisted of strings of [[damped wave]]s, pulses of radio waves that declined to zero, that sounded like a buzz in receivers. In effect they were already amplitude modulated.<br />
<br />
===Continuous waves===<br />
The first AM transmission was made by Canadian researcher [[Reginald Fessenden]] on 23 December 1900 using a [[spark gap transmitter]] with a specially designed high frequency 10&nbsp;kHz [[induction coil|interrupter]], over a distance of 1 mile (1.6&nbsp;km) at Cobb Island, Maryland, USA. His first transmitted words were, "Hello. One, two, three, four. Is it snowing where you are, Mr. Thiessen?". The words were barely intelligible above the background buzz of the spark.<br />
<br />
Fessenden was a significant figure in the development of AM radio. He was one of the first researchers to realize, from experiments like the above, that the existing technology for producing radio waves, the spark transmitter, was not usable for amplitude modulation, and that a new kind of transmitter, one that produced [[sinusoidal]] ''[[continuous wave]]s'', was needed. This was a radical idea at the time, because experts believed the impulsive spark was necessary to produce radio frequency waves, and Fessenden was ridiculed. He invented and helped develop one of the first continuous wave transmitters - the [[Alexanderson alternator]], with which he made what is considered the first AM public entertainment broadcast on Christmas Eve, 1906. He also discovered the principle on which AM modulation is based, [[heterodyne|heterodyning]], and invented one of the first [[detector (radio)|detector]]s able to [[rectifier|rectify]] and receive AM, the electrolytic detector or "liquid baretter", in 1902. Other radio detectors invented for wireless telegraphy, such as the [[Fleming valve]] (1904) and the [[crystal detector]] (1906) also proved able to rectify AM signals, so the technological hurdle was generating AM waves; receiving them was not a problem.<br />
<br />
===Early technologies===<br />
Early experiments in AM radio transmission, conducted by Fessenden, Valdamar Poulsen, Ernst Ruhmer, [[Quirino Majorana]], Charles Harrold, and [[Lee De Forest]], were hampered by the lack of a technology for [[amplifier|amplification]]. The first practical continuous wave AM [[transmitter]]s were based on either the huge, expensive [[Alexanderson alternator]], or versions of the [[Poulsen arc]] transmitter (arc converter), invented in 1903. The modifications necessary to transmit AM were clumsy and resulted in very low quality audio. Modulation was usually accomplished by a carbon [[microphone]] inserted directly in the antenna or ground wire; its varying resistance varied the current to the antenna. The limited power handling ability of the microphone severely limited the power of the first radiotelephones.<br />
<br />
===Vacuum tubes===<br />
The discovery in 1912 of the amplifying ability of the [[Audion]] [[vacuum tube]], invented in 1906 by [[Lee De Forest]], solved these problems. The vacuum tube [[electronic oscillator|feedback oscillator]], invented in 1912 by [[Edwin Armstrong]] and [[Alexander Meissner]], was a cheap source of [[continuous wave]]s and could be easily [[modulation|modulated]] to make an AM transmitter. Modulation did not have to be done at the output but could be applied to the signal before the final amplifier tube, so the microphone or other audio source didn't have to handle high power. Wartime research greatly advanced the art of AM modulation, and after the war the availability of cheap tubes sparked a great increase in the number of radio stations experimenting with AM transmission of news or music. The vacuum tube was responsible for the rise of [[AM broadcasting|AM radio broadcasting]] around 1920, the first electronic [[mass communication|mass entertainment]] medium. Amplitude modulation was virtually the only type used for [[radio broadcasting]] until [[FM broadcasting]] began after World War 2.<br />
<br />
At the same time as AM radio began, [[telephone company|telephone companies]] such as [[AT&T]] were developing the other large application for AM: sending multiple telephone calls through a single wire by modulating them on separate [[carrier signal|carrier]] frequencies, called ''[[frequency division multiplexing]]''.<ref name="Bray" /><br />
<br />
===Single-sideband===<br />
John Renshaw Carson in 1915 did the first mathematical analysis of amplitude modulation, showing that a signal and carrier frequency combined in a nonlinear device would create two sidebands on either side of the carrier frequency, and passing the modulated signal through another nonlinear device would extract the original baseband signal.<ref name="Bray" /> His analysis also showed only one sideband was necessary to transmit the audio signal, and Carson patented [[single-sideband modulation]] (SSB) on 1 December 1915.<ref name="Bray" /> This more advanced variant of amplitude modulation was adopted by AT&T for [[longwave]] transatlantic telephone service beginning 7 January 1927. After WW2 it was developed by the military for aircraft communication.<br />
<br />
==Simplified analysis of standard AM==<br />
[[File:Am2 spec.gif|thumb|Left part: Modulating signal. Right part: Frequency spectrum of the resulting amplitude modulated carrier]]<br />
<br />
Consider a carrier wave (sine wave) of frequency ''f<sub>c</sub>'' and amplitude ''A'' given by:<br />
<br />
:<math>c(t) = A\cdot \sin(2 \pi f_c t)\,</math>.<br />
<br />
Let ''m''(''t'') represent the modulation waveform. For this example we shall take the modulation to be simply a sine wave of a frequency ''f<sub>m</sub>'', a much lower frequency (such as an audio frequency) than ''f<sub>c</sub>'':<br />
<br />
:<math>m(t) = M\cdot \cos(2 \pi f_m t + \phi)\,</math>,<br />
<br />
where ''M'' is the amplitude of the modulation. We shall insist that ''M''<1 so that ''(1+m(t))'' is always positive. Amplitude modulation results when the carrier ''c(t)'' is multiplied by the positive quantity ''(1+m(t))'':<br />
<br />
:{|<br />
|<math>y(t)\,</math><br />
|<math>= [1 + m(t)]\cdot c(t) \,</math><br />
|-<br />
|<br />
|<math>= [1 + M\cdot \cos(2 \pi f_m t + \phi)] \cdot A \cdot \sin(2 \pi f_c t)</math><br />
|}<br />
<br />
In this simple case ''M'' is identical to the [[#Modulation Index|modulation index]], discussed below. With ''M''=0.5 the amplitude modulated signal ''y''(''t'') thus corresponds to the top graph (labelled "50% Modulation") in Figure 4.<br />
<br />
Using [[Prosthaphaeresis#The identities|prosthaphaeresis identities]], ''y''(''t'') can be shown to be the sum of three sine waves:<br />
<br />
:<math>y(t) = A\cdot \sin(2 \pi f_c t) + \begin{matrix}\frac{AM}{2} \end{matrix} \left[\sin(2 \pi (f_c + f_m) t + \phi) + \sin(2 \pi (f_c - f_m) t - \phi)\right].\,</math><br />
<br />
Therefore, the modulated signal has three components: the carrier wave ''c(t)'' which is unchanged, and two pure sine waves (known as [[sideband]]s) with frequencies slightly above and below the carrier frequency ''f<sub>c</sub>''.<br />
<br />
==Spectrum==<br />
[[File:AM spectrum.svg|thumb|400px|Fig 2: Double-sided spectra of baseband and AM signals.|alt=Diagrams of an AM signal, with formulas]]<br />
Of course a useful modulation signal ''m(t)'' will generally not consist of a single sine wave, as treated above. However by the principle of [[fourier decomposition]], ''m(t)'' can be expressed as the sum of a number of sine waves of various frequencies, amplitudes, and phases. Carrying out the multiplication of ''1+m(t)'' with ''c(t)'' as above then yields a result consisting of a sum of sine waves. Again the carrier ''c(t)'' is present unchanged, but for each frequency component of ''m'' at ''f<sub>i</sub>'' there are two sidebands at frequencies ''f<sub>c</sub> + f<sub>i</sub>'' and ''f<sub>c</sub> - f<sub>i</sub>''. The collection of the former frequencies above the carrier frequency is known as the upper sideband, and those below constitute the lower sideband. In a slightly different way of looking at it, we can consider the modulation ''m(t)'' to consist of an equal mix of positive and negative frequency components (as results from a formal [[fourier transform]] of a real valued quantity) as shown in the top of Fig. 2. Then one can view the sidebands as that modulation ''m(t)'' having simply been shifted in frequency by ''f<sub>c</sub>'' as depicted at the bottom right of Fig. 2 (formally, the modulated signal also contains identical components at negative frequencies, shown at the bottom left of Fig. 2 for completeness).<br />
<br />
[[File:AM signal.jpg|thumb|200px|right|Fig 3: The [[spectrogram]] of an AM voice broadcast shows the two sidebands (green) on either side of the carrier (red) with time proceeding in the vertical direction.|alt=Sonogram of an AM signal, showing the carrier and both sidebands vertically]]<br />
If we just look at the short-term spectrum of modulation, changing as it would for a human voice for instance, then we can plot the frequency content (horizontal axis) as a function of time (vertical axis) as in Fig. 3. It can again be seen that as the modulation frequency content varies, at any point in time there is an upper sideband generated according to those frequencies shifted ''above'' the carrier frequency, and the same content mirror-imaged in the lower sideband below the carrier frequency. At all times, the carrier itself remains constant, and of greater power than the total sideband power.<br />
<br />
==Power and spectrum efficiency==<br />
The RF bandwidth of an AM transmission (refer to Figure 2, but only considering positive frequencies) is twice the bandwidth of the modulating (or "[[baseband]]") signal, since the positive and negative sidebands around the carrier frequency each have a bandwidth as wide as the highest modulating frequency. Although the bandwidth of an AM signal is narrower than one using [[frequency modulation]] (FM), it is twice as wide as [[single-sideband]] techniques; it thus may be viewed as spectrally inefficient. Within a frequency band, only half as many transmissions (or "channels") can thus be accommodated. For this reason television employs a variant of single-sideband (known as [[vestigial sideband]], somewhat of a compromise in terms of bandwidth) in order to reduce the required channel spacing.<br />
<br />
Another improvement over standard AM is obtained through reduction or suppression of the carrier component of the modulated spectrum. In Figure 2 this is the spike in between the sidebands; even with full (100%) sine wave modulation, the power in the carrier component is twice that in the sidebands, yet it carries no unique information. Thus there is a great advantage in efficiency in reducing or totally suppressing the carrier, either in conjunction with elimination of one sideband ([[single-sideband suppressed-carrier transmission]]) or with both sidebands remaining ([[double sideband suppressed carrier]]). While these suppressed carrier transmissions are efficient in terms of transmitter power, they require more sophisticated receivers employing [[Product detector|synchronous detection]] and regeneration of the carrier frequency. For that reason, standard AM continues to be widely used, especially in broadcast transmission, to allow for the use of inexpensive receivers using [[envelope detector|envelope detection]]. Even (analog) television, with a (largely) suppressed lower sideband, includes sufficient carrier power for use of envelope detection. But for communications systems where both transmitters and receivers can be optimized, suppression of both one sideband and the carrier represent a net advantage and are frequently employed.<br />
<br />
==Modulation Index==<br />
The AM modulation index is a measure based on the ratio of the modulation excursions of the RF signal to the level of the unmodulated carrier. It is thus defined as:<br />
:<math>h = \frac{\mathrm{peak\ value\ of\ } m(t)}{A} = \frac{M}{A} </math> <br />
where <math>M\,</math> and <math>A\,</math> are the modulation amplitude and carrier amplitude, respectively; the modulation amplitude is the peak (positive or negative) change in the RF amplitude from its unmodulated value. Modulation index is normally expressed as a percentage, and may be displayed on a meter connected to an AM transmitter.<br />
<br />
So if <math>h=0.5</math>, carrier amplitude varies by 50% above (and below) its unmodulated level, as is shown in the first waveform, below. For <math>h=1.0</math>, it varies by 100% as shown in the illustration below it. With 100% modulation the wave amplitude sometimes reaches zero, and this represents full modulation using standard AM and is often a target (in order to obtain the highest possible [[signal to noise ratio]]) but mustn't be exceeded. Increasing the modulating signal beyond that point, known as [[overmodulation]], causes a standard AM modulator (see below) to fail, as the negative excursions of the wave envelope cannot become less than zero, resulting in [[distortion]] ("clipping") of the received modulation. Transmitters typically incorporate a [[limiter]] circuit to avoid overmodulation, and/or a [[Dynamic range compression|compressor]] circuit (especially for voice communications) in order to still approach 100% modulation for maximum intelligibility above the noise. Such circuits are sometimes referred to as a [[vogad]].<br />
<br />
However it is possible to talk about a modulation index exceeding 100%, without introducing distortion, in the case of [[double-sideband reduced-carrier transmission]]. In that case, negative excursions beyond zero entail a reversal of the carrier phase, as shown in the third waveform below. This cannot be produced using the efficient high-level (output stage) modulation techniques (see below) which are widely used especially in high power [[broadcast]] transmitters. Rather, a special modulator produces such a waveform at a low level followed by a [[linear amplifier]]. What's more, a standard AM receiver using an [[envelope detector]] is incapable of properly demodulating such a signal. Rather, synchronous detection is required. Thus double-sideband transmission is generally ''not'' referred to as "AM" even though it generates an identical RF waveform as standard AM as long as the modulation index is below 100%. Such systems more often attempt a radical reduction of the carrier level compared to the sidebands (where the useful information is present) to the point of [[double-sideband suppressed-carrier transmission]] where the carrier is (ideally) reduced to zero. In all such cases the term "modulation index" loses its value as it refers to the ratio of the modulation amplitude to a rather small (or zero) remaining carrier amplitude.<br />
<br />
[[File:Amplitude Modulated Wave-hm-64.svg|frame|center|Fig 4: Modulation depth|alt=Graphs illustrating how signal intelligibility increases with modulation index, but only up to 100% using standard AM.]]<br />
<br />
=={{anchor|AM modulation methods}}Modulation methods==<br />
<br />
[[File:ammodstage.png|300px|right|thumb|Anode (plate) modulation. A tetrode's plate and screen grid voltage is modulated via an audio transformer. The resistor R1 sets the grid bias; both the input and output are tuned circuits with inductive coupling.]]<br />
<br />
Modulation circuit designs may be classified as low- or high-level (depending on whether they modulate in a low-power domain—followed by amplification for transmission—or in the high-power domain of the transmitted signal).<ref><br />
{{cite book<br />
| title = Communication Engineering<br />
| author = A.P.Godse and U.A.Bakshi<br />
| publisher = Technical Publications<br />
| year = 2009<br />
| isbn = 978-81-8431-089-4<br />
| page = 36<br />
| url = http://books.google.com/books?id=coQ6ac-fh6QC&pg=PA36<br />
}}</ref><br />
<br />
===Low-level generation===<br />
In modern radio systems, modulated signals are generated via [[digital signal processing]] (DSP). With DSP many types of AM are possible with software control (including DSB with carrier, SSB suppressed-carrier and independent sideband, or ISB). Calculated digital samples are converted to voltages with a [[digital to analog converter]], typically at a frequency less than the desired RF-output frequency. The analog signal must then be shifted in frequency and [[linear amplifier|linearly amplified]] to the desired frequency and power level (linear amplification must be used to prevent modulation distortion).<ref><br />
{{cite book<br />
|publisher= American Radio Relay League<br />
|title= The ARRL Handbook for Radio Communications<br />
|editor1-last= Silver |editor1-first= Ward<br />
|edition= Eighty-eighth<br />
|year= 2011<br />
|chapter= Ch. 15 DSP and Software Radio Design<br />
|isbn= 978-0-87259-096-0}}</ref><br />
This low-level method for AM is used in many Amateur Radio transceivers.<ref>{{cite book<br />
|publisher= American Radio Relay League<br />
|title= The ARRL Handbook for Radio Communications<br />
|editor1-last= Silver |editor1-first= Ward<br />
|edition= Eighty-eighth<br />
|year= 2011<br />
|chapter= Ch. 14 Transceivers<br />
|isbn= 978-0-87259-096-0}}</ref><br />
<br />
AM may also be generated at a low level, using analog methods described in the next section.<br />
<br />
===High-level generation===<br />
High-power AM [[transmitter]]s (such as those used for [[AM broadcasting]]) are based on high-efficiency class-D and class-E [[Electronic amplifier|power amplifier]] stages, modulated by varying the supply voltage.<ref><br />
{{cite journal<br />
|author= Frederick H. Raab, et al.<br />
|title= RF and Microwave Power Amplifier and Transmitter Technologies - Part 2<br />
|journal= High Frequency Design<br />
|date=May 2003<br />
|page= p. 22ff<br />
|url= http://www.scribd.com/doc/8616046/RF-Power-Amplifier-and-Transmitter-Technologies-Part2<br />
}}</ref><br />
<br />
Older designs (for broadcast and amateur radio) also generate AM by controlling the gain of the transmitter’s final amplifier (generally class-C, for efficiency). The following types are for vacuum tube transmitters (but similar options are available with transistors):<ref><br />
{{cite book<br />
|author= Laurence Gray and Richard Graham<br />
|title= Radio Transmitters<br />
|publisher= McGraw-Hill<br />
|year= 1961<br />
|page= 141ff<br />
}}</ref><br />
<br />
* '''Plate modulation:''' In plate modulation, the plate voltage of the RF amplifier is modulated with the audio signal. The audio power requirement is 50 percent of the RF-carrier power.<br />
* '''Heising (constant-current) modulation:''' RF amplifier plate voltage is fed through a [[Choke (electronics)|“choke”]] (high-value inductor). The AM modulation tube plate is fed through the same inductor, so the modulator tube diverts current from the RF amplifier. The choke acts as a constant current source in the audio range. This system has a low power efficiency.<br />
* '''Control grid modulation:''' The operating bias and gain of the final RF amplifier can be controlled by varying the voltage of the control grid. This method requires little audio power, but care must be taken to reduce distortion.<br />
* '''Clamp tube (screen grid) modulation:''' The screen-grid bias may be controlled through a “clamp tube”, which reduces voltage according to the modulation signal. It is difficult to approach 100-percent modulation while maintaining low distortion with this system.<br />
* '''[[Doherty amplifier|Doherty modulation:]]''' One tube provides the power under carrier conditions and another operates only for positive modulation peaks. Overall efficiency is good, and distortion is low.<br />
* '''[[Ampliphase|Outphasing modulation:]]''' Two tubes are operated in parallel, but partially out of phase with each other. As they are differentially phase modulated their combined amplitude is greater or smaller. Efficiency is good and distortion low when properly adjusted.<br />
* '''[[Pulse-width modulation|Pulse width modulation (PWM) or Pulse duration modulation (PDM):]]''' A highly efficient high voltage power supply is applied to the tube plate. The output voltage of this supply is varied at an audio rate to follow the program. This system was pioneered by [[Hilmer Swanson]] and has a number of variations, all of which achieve high efficiency and sound quality.<br />
<br />
=={{anchor|AM demodulation methods}}Demodulation methods==<br />
The simplest form of AM demodulator consists of a diode which is configured to act as [[envelope detector]]. Another type of demodulator, the [[product detector]], can provide better-quality demodulation with additional circuit complexity.<br />
<br />
==See also==<br />
* [[AM broadcasting|AM radio]]<br />
* [[AM stereo]]<br />
* [[Mediumwave]] band used for AM broadcast radio<br />
* [[Longwave]] band used for AM broadcast radio<br />
* [[Frequency modulation]]<br />
* [[Shortwave radio]] almost universally uses AM, narrow FM occurring above 25&nbsp;MHz.<br />
* [[Modulation]], for a list of other modulation techniques<br />
* [[Amplitude modulation signalling system]] (AMSS), a digital system for adding low bitrate information to an AM signal.<br />
* [[Sideband]], for some explanation of what this is.<br />
* [[Types of radio emissions]], for the emission types designated by the [[ITU]]<br />
* [[Airband]]<br />
* [[Quadrature amplitude modulation]]<br />
<br />
==References==<br />
;Notes<br />
{{Reflist}}<br />
;Sources<br />
* Newkirk, David and Karlquist, Rick (2004). Mixers, modulators and demodulators. In D. G. Reed (ed.), ''The ARRL Handbook for Radio Communications'' (81st ed.), pp.&nbsp;15.1&ndash;15.36. Newington: ARRL. ISBN 0-87259-196-4.<br />
<br />
==External links==<br />
* ''[http://demonstrations.wolfram.com/AmplitudeModulation/ Amplitude Modulation]'' by Jakub Serych, [[Wolfram Demonstrations Project]].<br />
* [http://robotics.eecs.berkeley.edu/~sastry/ee20/modulation/node3.html Amplitude Modulation], by S Sastry.<br />
* [http://www.fas.org/man/dod-101/navy/docs/es310/AM.htm Amplitude Modulation], an introduction by [[Federation of American Scientists]].<br />
* [http://www.afrotechmods.com/videos/amplitude_modulation_tutorial.htm Amplitude Modulation tutorial video with example transmitter circuit.]<br />
<br />
{{Telecommunications}}<br />
{{Audio broadcasting}}<br />
<br />
{{DEFAULTSORT:Amplitude Modulation}}<br />
[[Category:Radio modulation modes]]</div>Adminhttps://en.formulasearchengine.com/index.php?title=Area&diff=218538Area2014-07-31T13:49:29Z<p>Admin: Reverted edits by MediaWiki spam cleanup (talk) to last revision by ClueBot NG</p>
<hr />
<div>{{About|the geometric quantity}}<br />
<br />
[[File:Area.svg|right|thumb|alt=Three shapes on a square grid|The combined area of these three [[shapes]] is [[approximation|approximately]] 15.57 [[square]]s.]]<br />
'''Area''' is a [[quantity]] that expresses the extent of a [[two-dimensional]] [[surface]] or [[shape]], or [[planar lamina]], in the [[Plane (geometry)|plane]]. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of [[paint]] necessary to cover the surface with a single coat.<ref name=MathWorld>{{cite web|url=http://mathworld.wolfram.com/Area.html|title=Area|publisher=[[Wolfram MathWorld]]|author=[[Eric W. Weisstein]]|accessdate=3 July 2012}}</ref> It is the two-dimensional [[analogy|analog]] of the [[length]] of a [[plane curve|curve]] (a one-dimensional concept) or the [[volume]] of a [[solid geometry|solid]] (a three-dimensional concept).<br />
<br />
The area of a shape can be measured by comparing the shape to [[square]]s of a fixed size.<ref name=AF/> In the [[International System of Units]] (SI), the standard unit of area is the [[square metre]] (written as m<sup>2</sup>), which is the area of a square whose sides are one [[metre]] long.<ref name=B>[[Bureau International des Poids et Mesures]] [http://www.bipm.org/en/CGPM/db/11/12/ Resolution 12 of the 11th meeting of the CGPM (1960)], retrieved 15 July 2012</ref> A shape with an area of three square metres would have the same area as three such squares. In [[mathematics]], the [[unit square]] is defined to have area one, and the area of any other shape or surface is a [[Dimensionless quantity|dimensionless]] [[real number]].<br />
<br />
There are several well-known [[formula]]s for the areas of simple shapes such as [[triangle]]s, [[rectangle]]s, and [[circle]]s. Using these formulas, the area of any [[polygon]] can be found by [[Polygon triangulation|dividing the polygon into triangles]].<ref name=bkos>{{Cite book |author1=Mark de Berg |author2=Marc van Kreveld |author3=Mark Overmars |author3-link=Mark Overmars |author4=Otfried Schwarzkopf |year=2000 |title=Computational Geometry |publisher=[[Springer-Verlag]] |edition=2nd revised |isbn=3-540-65620-0 |chapter=Chapter 3: Polygon Triangulation |pages=45–61 |postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</ref> For shapes with curved boundary, [[calculus]] is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the [[History of calculus|historical development of calculus]].<ref>{{cite book|first=Carl B. |last=Boyer |authorlink=Carl Benjamin Boyer |title=A History of the Calculus and Its Conceptual Development |publisher=Dover |year=1959 |isbn=0-486-60509-4}}</ref><br />
<br />
For a solid shape such as a [[sphere]], [[Cone (geometry)|cone]], or [[cylinder (geometry)|cylinder]], the area of its boundary surface is called the [[surface area]].<ref name=MathWorld/><ref name=MathWorldSurfaceArea>{{cite web|url=http://mathworld.wolfram.com/SurfaceArea.html|title=Surface Area|publisher=[[Wolfram MathWorld]]|author=[[Eric W. Weisstein]]|accessdate=3 July 2012}}</ref> Formulas for the surface areas of simple shapes were computed by the [[Greek mathematics|ancient Greeks]], but computing the surface area of a more complicated shape usually requires [[multivariable calculus]].<br />
<br />
Area plays an important role in modern mathematics. In addition to its obvious importance in [[geometry]] and calculus, area is related to the definition of [[determinant]]s in [[linear algebra]], and is a basic property of surfaces in [[differential geometry]].<ref name="doCarmo">do Carmo, Manfredo. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976. Page 98, ISBN 978-0-13-212589-5</ref> In [[analysis]], the area of a subset of the plane is defined using [[Lebesgue measure]],<ref name="Rudin">Walter Rudin, ''Real and Complex Analysis'', McGraw-Hill, 1966, ISBN 0-07-100276-6.</ref> though not every subset is measurable.<ref>Gerald Folland, Real Analysis: modern techniques and their applications, John Wiley & Sons, Inc., 1999,Page 20,ISBN 0-471-31716-0</ref> In general, area in higher mathematics is seen as a special case of [[volume]] for two-dimensional regions.<ref name=MathWorld/><br />
<br />
Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists.<br />
<br />
==Formal definition==<br />
{{see also|Jordan measure}}<br />
An approach to defining what is meant by "area" is through [[axioms]]. "Area" can be defined as a function from a collection M of special kind of plane figures (termed measurable sets) to the set of real numbers which satisfies the following properties:<br />
* For all ''S'' in ''M'', ''a''(''S'') ≥ 0.<br />
* If ''S'' and ''T'' are in ''M'' then so are ''S'' ∪ ''T'' and ''S'' ∩ ''T'', and also ''a''(''S''∪''T'') = ''a''(''S'') + ''a''(''T'') − ''a''(''S''∩''T'').<br />
* If ''S'' and ''T'' are in ''M'' with ''S'' ⊆ ''T'' then ''T'' − ''S'' is in ''M'' and ''a''(''T''−''S'') = ''a''(''T'') − ''a''(''S'').<br />
* If a set ''S'' is in ''M'' and ''S'' is congruent to ''T'' then ''T'' is also in ''M'' and ''a''(''S'') = ''a''(''T'').<br />
* Every rectangle ''R'' is in ''M''. If the rectangle has length ''h'' and breadth ''k'' then ''a''(''R'') = ''hk''.<br />
* Let ''Q'' be a set enclosed between two step regions ''S'' and ''T''. A step region is formed from a finite union of adjacent rectangles resting on a common base, i.e. ''S'' ⊆ ''Q'' ⊆ ''T''. If there is a unique number ''c'' such that ''a''(''S'') ≤ c ≤ ''a''(''T'') for all such step regions ''S'' and ''T'', then ''a''(''Q'') = ''c''.<br />
<br />
It can be proved that such an area function actually exists.<ref name=Moise>{{cite book|last=Moise|first=Edwin|title=Elementary Geometry from an Advanced Standpoint|url=http://books.google.com/?id=7nUNAQAAIAAJ|accessdate=15 July 2012|year=1963|publisher= Addison-Wesley Pub. Co.|isbn=|page=}}</ref><br />
<br />
==Units==<br />
[[Image:SquareMeterQuadrat.JPG|thumb|right|alt=A square made of PVC pipe on grass|A square metre [[quadrat]] made of PVC pipe.]]<br />
Every [[unit of length]] has a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured in [[square metre]]s (m<sup>2</sup>), square centimetres (cm<sup>2</sup>), square millimetres (mm<sup>2</sup>), [[square kilometre]]s (km<sup>2</sup>), [[square foot|square feet]] (ft<sup>2</sup>), [[square yard]]s (yd<sup>2</sup>), [[square mile]]s (mi<sup>2</sup>), and so forth.<ref name=BIPM2006Ch5/> Algebraically, these units can be thought of as the [[square (algebra)|squares]] of the corresponding length units.<br />
<br />
The SI unit of area is the square metre, which is considered an [[SI derived unit]].<ref name=B/><br />
<br />
===Conversions===<br />
[[Image:Area conversion - square mm in a square cm.png|thumb|right|320px|alt=A diagram showing the conversion factor between different areas|Although there are 10 mm in 1 cm, there are 100 mm<sup>2</sup> in 1 cm<sup>2</sup>.]]<br />
The conversion between two square units is the [[square (algebra)|square]] of the conversion between the corresponding length units. For example, since<br />
:1 [[foot (unit)|foot]] = 12 [[inch]]es,<br />
the relationship between square feet and square inches is<br />
:1 square foot = 144 square inches,<br />
where 144 = 12<sup>2</sup> = 12 × 12. Similarly:<br />
* 1 square kilometer = [[million|1,000,000]] square meters<br />
* 1 square meter = [[10000 (number)|10,000]] square centimetres = 1,000,000 square millimetres<br />
* 1 square centimetre = [[100 (number)|100]] square millimetres<br />
* 1 square yard = [[9 (number)|9]] square feet<br />
* 1 square mile = 3,097,600 square yards = 27,878,400 square feet<br />
In addition,<br />
* 1 square inch = 6.4516 square centimetres<br />
* 1 square foot = {{gaps|0.092|903|04}} square metres<br />
* 1 square yard = {{gaps|0.836|127|36}} square metres<br />
* 1 square mile = {{gaps|2.589|988|110|336}} square kilometres<br />
<br />
===Other units===<br />
{{See also|Category:Units of area}}<br />
There are several other common units for area. The "[[Are (unit)|Are]]" was the original unit of area in the [[metric system]], with; <br />
*1 are = 100 square metres<br />
Though the are has fallen out of use, the [[hectare]] is still commonly used to measure land:<ref name=BIPM2006Ch5>{{Cite journal|author= Bureau international des poids et mesures|year=2006 |url=http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf |title=The International System of Units (SI)|version= 8th ed.|accessdate=2008-02-13}} Chapter 5.</ref><br />
*1 hectare = 100 ares = 10,000 square metres = 0.01 square kilometres<br />
Other uncommon metric units of area include the [[tetrad (unit of area)|tetrad]], the [[hectad]], and the [[myriad]].<br />
<br />
The [[acre]] is also commonly used to measure land areas, where<br />
*1 acre = 4,840 square yards = 43,560 square feet.<br />
An acre is approximately 40% of a hectare.<br />
<br />
On the atomic scale, area is measured in units of [[Barn (unit)|barns]], such that:<ref name=BIPM2006Ch5/><br />
*1 barn = 10<sup>−28</sup> square meters.<br />
The barn is commonly used in describing the cross sectional area of interaction in [[nuclear physics]].<ref name=BIPM2006Ch5/><br />
<br />
In India,<br />
<br />
*20 Dhurki = 1 Dhur<br />
<br />
*20 Dhur = 1 Khatha<br />
<br />
*20 Khata = 1 Bigha<br />
<br />
*32 Khata = 1 Acre<br />
<br />
==Area formulae==<br />
<br />
===Polygon formulae===<br />
<br />
====Rectangles====<br />
[[Image:RectangleLengthWidth.svg|thumb|right|180px|alt=A rectangle with length and width labelled|The area of this rectangle is&nbsp;{{mvar|lw}}.]]<br />
The most basic area formula is the formula for the area of a [[rectangle]]. Given a rectangle with length {{mvar|l}} and width {{mvar|w}}, the formula for the area is:</big><ref name=AF>{{cite web|url=http://www.math.com/tables/geometry/areas.htm|title=Area Formulas|publisher=Math.com|accessdate=2 July 2012}}</ref><br />
<br />
:{{bigmath|''A'' {{=}} ''lw''}}&nbsp;(rectangle)<br />
That is, the area of the rectangle is the length multiplied by the width. As a special case, as {{math|''l'' {{=}} ''w''}} in the case of a square, the area of a square with side length {{mvar|s}} is given by the formula:<ref name=MathWorld/><ref name=AF/><br />
:{{bigmath|''A'' {{=}} ''s''<sup>2</sup>}}&nbsp;(square)<br />
<br />
The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a [[definition]] or [[axiom]]. On the other hand, if [[geometry]] is developed before [[arithmetic]], this formula can be used to define [[multiplication]] of [[real number]]s.<br />
<br />
[[Image:ParallelogramArea.svg|thumb|left|180px|alt=A diagram showing how a parallelogram can be re-arranged into the shape of a rectangle|Equal area figures.]]<br />
<br />
====Dissection formulae====<br />
Most other simple formulae for area follow from the method of [[dissection (geometry)|dissection]].<br />
This involves cutting a shape into pieces, whose areas must [[addition|sum]] to the area of the original shape.<br />
<br />
For an example, any [[parallelogram]] can be subdivided into a [[trapezoid]] and a [[right triangle]], as shown in figure to the left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle:<ref name=AF/><br />
:{{bigmath|''A'' {{=}} ''bh''}} <big>&nbsp;(parallelogram).</big><br />
[[Image:TriangleArea.svg|thumb|right|180px|alt=A parallelogram split into two equal triangles|Two equal triangles.]]However, the same parallelogram can also be cut along a [[diagonal]] into two [[congruence (geometry)|congruent]] triangles, as shown in the figure to the right. It follows that the area of each triangle is half the area of the parallelogram:<ref name=AF/> <br />
:<math>A = \frac{1}{2}bh</math> <big>&nbsp;(triangle).</big><br />
Similar arguments can be used to find area formulae for the [[trapezoid]] and the [[rhombus]], as well as more complicated [[polygon]]s.{{citation needed|date=July 2012}}<br />
<br />
===Area of curved shapes===<br />
<br />
====Circles====<br />
[[Image:CircleArea.svg|thumb|right|alt=A circle divided into many sectors can be re-arranged roughly to form a parallelogram|A circle can be divided into [[Circular sector|sector]]s which rearrange to form an approximate [[parallelogram]].]]<br />
{{main|Area of a circle}}<br />
The formula for the area of a [[circle]] (more properly called [[area of a disk]]) is based on a similar method. Given a circle of radius {{math|''r''}}, it is possible to partition the circle into [[Circular sector|sector]]s, as shown in the figure to the right. Each sector is approximately triangular in shape, and the sectors can be rearranged to form and approximate parallelogram. The height of this parallelogram is {{math|''r''}}, and the width is half the [[circumference]] of the circle, or {{math|π''r''}}. Thus, the total area of the circle is {{math|''r'' × π''r''}}, or {{math|π''r''<sup>2</sup>}}:<ref name=AF/><br />
:{{bigmath|''A'' {{=}} π''r''<sup>2</sup>}} <big>&nbsp;(circle).</big><br />
Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. The [[limit (mathematics)|limit]] of the areas of the approximate parallelograms is exactly {{math|π''r''<sup>2</sup>}}, which is the area of the circle.<ref name=Surveyor/><br />
<br />
This argument is actually a simple application of the ideas of [[calculus]]. In ancient times, the [[method of exhaustion]] was used in a similar way to find the area of the circle, and this method is now recognized as a precursor to [[integral calculus]]. Using modern methods, the area of a circle can be computed using a [[definite integral]]:<br />
:<math>A \;=\; \int_{-r}^r 2\sqrt{r^2 - x^2}\,dx \;=\; \pi r^2</math><br />
<br />
====Ellipses====<br />
{{main|Ellipse#Area}}<br />
The formula for the area of an [[ellipse]] is related to the formula of a circle; for an ellipse with [[semi-major axis|semi-major]] and [[semi-minor axis|semi-minor]] axes {{math|''x''}} and {{math|''y''}} the formula is:<ref name=AF/><br />
:<math>A = \pi xy \,\!</math><br />
<br />
====Surface area====<br />
{{main|Surface area}}<br />
[[Image:Archimedes sphere and cylinder.svg|right|thumb|180px|alt=A blue sphere inside a cylinder of the same height and radius|[[Archimedes]] showed that the surface area and volume of a [[sphere]] is exactly 2/3 of the area and volume of the surrounding [[cylinder (geometry)|cylindrical]] surface.]]<br />
Most basic formulae for [[surface area]] can be obtained by cutting surfaces and flattening them out. For example, if the side surface of a [[cylinder (geometry)|cylinder]] (or any [[prism (geometry)|prism]]) is cut lengthwise, the surface can be flattened out into a rectangle. Similarly, if a cut is made along the side of a [[cone (geometry)|cone]], the side surface can be flattened out into a sector of a circle, and the resulting area computed.<br />
<br />
The formula for the surface area of a [[sphere]] is more difficult to derive: because a sphere has nonzero [[Gaussian curvature]], it cannot be flattened out. The formula for the surface area of a sphere was first obtained by [[Archimedes]] in his work ''[[On the Sphere and Cylinder]]''. The formula is:<ref name=MathWorldSurfaceArea/><br />
:{{bigmath|''A'' {{=}} 4''πr''<sup>2</sup>}} <big>&nbsp;(sphere).</big><br />
where {{math|''r''}} is the radius of the sphere. As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar to [[calculus]].<br />
<br />
===General formulae===<br />
<br />
====Areas of 2-dimensional figures====<br />
*A [[triangle]]: <math>\tfrac12Bh</math> (where ''B'' is any side, and ''h'' is the distance from the line on which ''B'' lies to the other vertex of the triangle). This formula can be used if the height ''h'' is known. If the lengths of the three sides are known then ''[[Heron's formula]]'' can be used: <math>\sqrt{s(s-a)(s-b)(s-c)}</math> where ''a'', ''b'', ''c'' are the sides of the triangle, and <math>s = \tfrac12(a + b + c)</math> is half of its perimeter.<ref name=AF/> If an angle and its two included sides are given, the area is <math>\tfrac12 a b \sin(C)</math> where {{math|''C''}} is the given angle and {{math|''a''}} and {{math|''b''}} are its included sides.<ref name=AF/> If the triangle is graphed on a coordinate plane, a matrix can be used and is simplified to the absolute value of <math>\tfrac12(x_1 y_2 + x_2 y_3 + x_3 y_1 - x_2 y_1 - x_3 y_2 - x_1 y_3)</math>. This formula is also known as the [[shoelace formula]] and is an easy way to solve for the area of a coordinate triangle by substituting the 3 points ''(x<sub>1</sub>,y<sub>1</sub>)'', ''(x<sub>2</sub>,y<sub>2</sub>)'', and ''(x<sub>3</sub>,y<sub>3</sub>)''. The shoelace formula can also be used to find the areas of other polygons when their vertices are known. Another approach for a coordinate triangle is to use [[Infinitesimal calculus]] to find the area.<br />
*A [[simple polygon]] constructed on a grid of equal-distanced points (i.e., points with [[integer]] coordinates) such that all the polygon's vertices are grid points: <math>i + \frac{b}{2} - 1</math>, where ''i'' is the number of grid points inside the polygon and ''b'' is the number of boundary points.<ref name=Pick>{{cite journal |last=Trainin |first=J. |title=An elementary proof of Pick's theorem |journal=[[Mathematical Gazette]] |volume=91 |issue=522 |date=November 2007 |pages=536–540}}</ref> This result is known as [[Pick's theorem]].<ref name=Pick/><br />
<br />
====Area in calculus====<br />
[[File:Integral as region under curve.svg|left|thumb|280px|alt=A diagram showing the area between a given curve and the x-axis|Integration can be thought of as measuring the area under a curve, defined by ''f''(''x''), between two points (here ''a'' and ''b'').]]<br />
[[File:Areabetweentwographs.svg|thumb|287px|alt=A diagram showing the area between two functions|The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions]]<br />
<br />
*The area between a positive-valued curve and the horizontal axis, measured between two values ''a'' and ''b'' (b is defined as the larger of the two values) on the horizontal axis, is given by the integral from ''a'' to ''b'' of the function that represents the curve:<ref name=MathWorld/><br />
:<math> A = \int_a^{b} f(x) \, dx</math><br />
*The area between the [[graph of a function|graph]]s of two functions is [[equality (mathematics)|equal]] to the [[integral]] of one [[function (mathematics)|function]], ''f''(''x''), [[subtraction|minus]] the integral of the other function, ''g''(''x''):<br />
:<math> A = \int_a^{b} ( f(x) - g(x) ) \, dx </math> where <math> f(x) </math> is the curve with the greater y-value.<br />
*An area bounded by a function ''r'' = ''r''(θ) expressed in [[polar coordinates]] is:<ref name=MathWorld/><br />
:<math>A = {1 \over 2} \int r^2 \, d\theta </math><br />
*The area enclosed by a [[parametric curve]] <math>\vec u(t) = (x(t), y(t)) </math> with endpoints <math> \vec u(t_0) = \vec u(t_1) </math> is given by the [[line integral]]s:<br />
::<math> \oint_{t_0}^{t_1} x \dot y \, dt = - \oint_{t_0}^{t_1} y \dot x \, dt = {1 \over 2} \oint_{t_0}^{t_1} (x \dot y - y \dot x) \, dt </math><br />
(see [[Green's theorem]]) or the ''z''-component of<br />
<br />
:<math>{1 \over 2} \oint_{t_0}^{t_1} \vec u \times \dot{\vec u} \, dt.</math><br />
<br />
====Surface area of 3-dimensional figures====<br />
*[[Cone (geometry)|cone]]:<ref name=MathWorldCone>{{cite web|url=http://mathworld.wolfram.com/Cone.html|title=Cone|publisher=[[Wolfram MathWorld]]|author=[[Eric W. Weisstein]]|accessdate=6 July 2012}}</ref> <math>\pi r\left(r + \sqrt{r^2 + h^2}\right)</math>, where ''r'' is the radius of the circular base, and ''h'' is the height. That can also be rewritten as <math>\pi r^2 + \pi r l </math><ref name=MathWorldCone/> or <math>\pi r (r + l) \,\!</math> where ''r'' is the radius and ''l'' is the slant height of the cone. <math>\pi r^2 </math> is the base area while <math>\pi r l </math> is the lateral surface area of the cone.<ref name=MathWorldCone/><br />
*[[cube (geometry)|cube]]: <math>6s^2</math>, where ''s'' is the length of an edge.<ref name=MathWorldSurfaceArea/><br />
*[[cylinder (geometry)|cylinder]]: <math>2\pi r(r + h)</math>, where ''r'' is the radius of a base and ''h'' is the height. The ''2<math>\pi</math>r'' can also be rewritten as ''<math>\pi</math> d'', where ''d'' is the diameter.<br />
*[[Prism (geometry)|prism]]: 2B + Ph, where ''B'' is the area of a base, ''P'' is the perimeter of a base, and ''h'' is the height of the prism.<br />
*[[pyramid (geometry)|pyramid]]: <math>B + \frac{PL}{2}</math>, where ''B'' is the area of the base, ''P'' is the perimeter of the base, and ''L'' is the length of the slant.<br />
*[[rectangular prism]]: <math>2 (\ell w + \ell h + w h)</math>, where <math>\ell</math> is the length, ''w'' is the width, and ''h'' is the height.<br />
<br />
====General formula====<br />
The general formula for the surface area of the graph of a continuously differentiable function <math>z=f(x,y),</math> where <math>(x,y)\in D\subset\mathbb{R}^2</math> and <math>D</math> is a region in the xy-plane with the smooth boundary:<br />
: <math> A=\iint_D\sqrt{\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2+1}\,dx\,dy. </math><br />
Even more general formula for the area of the graph of a [[parametric surface]] in the vector form <math>\mathbf{r}=\mathbf{r}(u,v),</math> where <math>\mathbf{r}</math> is a continuously differentiable vector function of <math>(u,v)\in D\subset\mathbb{R}^2</math>:<ref name="doCarmo"/><br />
: <math> A=\iint_D \left|\frac{\partial\mathbf{r}}{\partial u}\times\frac{\partial\mathbf{r}}{\partial v}\right|\,du\,dv. </math><br />
<br />
===List of formulas===<br />
<!-- NOTICE TO CONTRIBUTORS<br />
This section is regularly edited to change "formulae" to "formula". "Formula" is the singular, whilst "formulae" a plural form. Since there are multiple formulae listed, "formula" is incorrect in this case. Thank you for your understanding.<br />
Please raise any further issues you may have on the talk page.--><br />
<br />
There are [[formula]]e for many different regular and irregular polygons, and those additional to the ones above are listed here.<br />
<br />
{| class="wikitable"<br />
|+ Additional common formulae for area:<br />
! Shape<br />
! Formula<br />
! Variables<br />
|-<br />
|Regular [[triangle]] ([[equilateral triangle]])<br />
||<math>\frac\sqrt{3}{4}s^2\,\!</math><br />
||<math>s</math> is the length of one side of the triangle.<br />
|-<br />
|[[Triangle]]<ref name=MathWorld/><br />
|<math>\sqrt{s(s-a)(s-b)(s-c)}\,\!</math><br />
|<math> s </math> is half the perimeter, <math>a</math>, <math>b</math> and <math>c</math> are the length of each side.<br />
|-<br />
|[[Triangle]]<ref name=AF/><br />
|<math>\tfrac12 a b \sin(C)\,\!</math><br />
|<math>a</math> and <math>b</math> are any two sides, and <math>C</math> is the angle between them.<br />
|-<br />
|[[Triangle]]<ref name=MathWorld/><br />
|<math>\tfrac12bh \,\!</math><br />
|<math>b</math> and <math>h</math> are the [[Base (geometry)|base]] and [[Altitude (triangle)|altitude]] (measured perpendicular to the base), respectively.<br />
|-<br />
|[[Isosceles triangle]]<br />
|<math>\frac{1}{2}b\sqrt{a^2-\frac{b^2}{4}}</math><br />
|<math>a</math> is the length of an equal side and <math>b</math> is the length of a different side.<br />
|-<br />
|[[Rhombus]]<br />
|<math>\tfrac12ab</math><br />
|<math>a</math> and <math>b</math> are the lengths of the two [[diagonals]] of the rhombus.<br />
|-<br />
|[[Parallelogram]]<br />
|<math>bh\,\!</math><br />
|<math>b</math> is the length of the base and <math>h</math> is the perpendicular height.<br />
|-<br />
|[[Trapezoid]]<br />
|<math>\frac{(a+b)h}{2} \,\!</math><br />
|<math>a</math> and <math>b</math> are the parallel sides and <math>h</math> the distance (height) between the parallels.<br />
|-<br />
|Regular [[hexagon]]<br />
|<math>\frac{3}{2} \sqrt{3}s^2\,\!</math><br />
|<math>s</math> is the length of one side of the hexagon.<br />
|-<br />
|Regular [[octagon]]<br />
|<math>2(1+\sqrt{2})s^2\,\!</math><br />
|<math>s</math> is the length of one side of the octagon.<br />
|-<br />
| [[Regular polygon]]<br />
|<math>\frac{1}{4}nl^2\cdot \cot(\pi/n)\,\!</math><br />
|<math> l </math> is the side length and <math>n</math> is the number of sides.<br />
|-<br />
| Regular polygon<br />
|<math>\frac{1}{4n}p^2\cdot \cot(\pi/n)\,\!</math><br />
|<math> p </math> is the perimeter and <math>n</math> is the number of sides.<br />
|-<br />
| Regular polygon<br />
|<math>\frac{1}{2}nR^2\cdot \sin(2\pi/n) = nr^2 \tan(\pi/n)\,\!</math><br />
|<math> R </math> is the radius of a circumscribed circle, <math>r</math> is the radius of an inscribed circle, and <math>n</math> is the number of sides.<br />
|-<br />
| Regular polygon<br />
|<math>\tfrac12 ap = \tfrac12 nsa \,\!</math><br />
|<math>n</math> is the number of sides, <math>s</math> is the side length, <math>a</math> is the [[apothem]], or the radius of an inscribed circle in the polygon, and <math>p</math> is the perimeter of the polygon.<br />
|-<br />
|[[Circle]]<br />
|<math>\pi r^2\ \text{or}\ \frac{\pi d^2}{4} \,\!</math><br />
|<math>r</math> is the radius and <math>d</math> the [[diameter]].<br />
|-<br />
|[[Circular sector]]<br />
|<math>\frac{\theta}{2}r^2\ \text{or}\ \frac{L \cdot r}{2}\,\!</math><br />
|<math>r</math> and <math>\theta</math> are the radius and angle (in [[radian]]s), respectively and <math>L</math> is the length of the perimeter.<br />
|-<br />
|[[Ellipse]]<ref name=AF/><br />
|<math>\pi ab \,\!</math><br />
|<math>a</math> and <math>b</math> are the [[semi-major axis|semi-major]] and [[semi-minor axis|semi-minor]] axes, respectively.<br />
|-<br />
|Total surface area of a [[Cylinder (geometry)|cylinder]]<br />
|<math>2\pi r (r + h)\,\!</math><br />
|<math>r</math> and <math>h</math> are the radius and height, respectively.<br />
|-<br />
|Lateral surface area of a cylinder<br />
|<math>2 \pi r h \,\!</math><br />
|<math>r</math> and <math>h</math> are the radius and height, respectively.<br />
|-<br />
|Total surface area of a [[sphere (geometry)|sphere]]<ref name=MathWorldSurfaceArea/><br />
|<math>4\pi r^2\ \text{or}\ \pi d^2\,\!</math><br />
|<math>r</math> and <math>d</math> are the radius and diameter, respectively.<br />
|-<br />
|Total surface area of a [[pyramid (geometry)|pyramid]]<ref name=MathWorldSurfaceArea/><br />
|<math>B+\frac{P L}{2}\,\!</math><br />
|<math>B</math> is the base area, <math>P</math> is the base perimeter and <math>L</math> is the slant height.<br />
|-<br />
|Total surface area of a [[pyramid (geometry)|pyramid]] [[frustum]]<ref name=MathWorldSurfaceArea/><br />
|<math>B+\frac{P L}{2}\,\!</math><br />
|<math>B</math> is the base area, <math>P</math> is the base perimeter and <math>L</math> is the slant height. <br />
|-<br />
|[[Square]] to circular area conversion<br />
|<math>\frac{4}{\pi} A\,\!</math><br />
|<math>A</math> is the area of the [[square]] in square units.<br />
|-<br />
|[[Circle|Circular]] to square area conversion<br />
|<math>\frac{1}{4} C\pi\,\!</math><br />
|<math>C</math> is the area of the [[circle]] in circular units.<br />
|-<br />
|[[Reuleaux Triangle]]<br />
|<math>\frac{\pi x^2}{6}-\frac{3 \sqrt{x^2-(\frac{x}{2})^2}}{2}+\frac{\sqrt{x^2-(\frac{x}{2})^2}}{2}</math><br />
|<math>x</math> is the side of the triangle inside the reuleaux triangle.<br />
<!--<br />
|-<br />
|A revolution of f(x) about the x-axis<br />
|<math>2 \pi \int_{a}^{b} |f(x)| \sqrt{1+(f'(x))^2}dx</math><br />
|-<br />
|Area of surface of revolution of f(x) about the y-axis<br />
|<math>2 \pi \int_{a}^{b} |x| \sqrt{1+(f'(x))^2}dx</math><br />
--><br />
|}<br />
<br />
The above calculations show how to find the area of many common [[shapes]].<br />
<br />
The areas of irregular polygons can be calculated using the "[[Surveyor's formula]]".<ref name=Surveyor>{{cite journal|last1=Braden|first1=Bart|date=September 1986|title= The Surveyor's Area Formula|journal=The College Mathematics Journal|volume=17|issue=4|pages=326–337|publisher=|doi=10.2307/2686282|url=http://www.maa.org/pubs/Calc_articles/ma063.pdf|accessdate=15 July 2012}}</ref><br />
<br />
==Optimization==<br />
Given a wire contour, the surface of least area spanning ("filling") it is a [[minimal surface]]. Familiar examples include [[soap bubble]]s.<br />
<br />
The question of the [[filling area conjecture|filling area]] of the [[Riemannian circle]] remains open.{{citation needed|date=October 2012}}<br />
<br />
==See also==<br />
*[[2 × 2 real matrices#Equi-areal mapping|Equi-areal mapping]]<br />
*[[Integral]]<br />
*[[Orders of magnitude (area)]]&mdash;A list of areas by size.<br />
*[[Perimeter]]<br />
*[[Planimeter]], an instrument for measuring small areas, e.g. on maps.<br />
*[[Volume]]<br />
<br />
==References==<br />
{{reflist|30em}}<br />
<br />
==External links==<br />
{{Commons category|Area}}<br />
{{Wiktionary}}<br />
* [http://www.area-of-a-circle.com Area Calculator]<br />
* [http://www.sengpielaudio.com/calculator-cross-section.htm Conversion cable diameter to circle cross-sectional area and vice versa]<br />
* [http://milloz.com/site/index.php?q=Free-Tools/Area-Measurement-Tool Geographical Area Calculator using Satellite Maps View]<br />
<br />
[[Category:Area|*]]</div>Adminhttps://en.formulasearchengine.com/index.php?title=Atomic_orbital&diff=218533Atomic orbital2014-07-31T13:48:06Z<p>Admin: Reverted edits by MediaWiki spam cleanup (talk) to last revision by AnomieBOT</p>
<hr />
<div>[[File:neon orbitals.JPG|right|thumb|442px|upright|The shapes of the first five atomic orbitals: 1s, 2s, 2p<sub>''x''</sub>, 2p<sub>''y''</sub>, and 2p<sub>''z''</sub>. The colors show the wave function phase. These are graphs of {{math|ψ(''x'', ''y'', ''z'')}}<!--Please don't italicize a bracket. --> functions which depend on the coordinates of one electron. To see the elongated shape of {{math|ψ(''x'', ''y'', ''z'')<sup>2</sup>}} functions that show probability density more directly, see the graphs of d-orbitals below.]]<br />
<br />
An '''atomic orbital''' is a [[Function (mathematics)|mathematical function]] that describes the wave-like behavior of either one [[electron]] or a pair of electrons in an [[atom]].<ref>{{cite book|first1= Milton|last1= Orchin|first2=Roger S.|last2=Macomber|first3=Allan|last3= Pinhas|first4= R. Marshall|last4= Wilson| year=2005| url=http://media.wiley.com/product_data/excerpt/81/04716802/0471680281.pdf|title= Atomic Orbital Theory}}</ref> This function can be used to calculate the probability of finding any electron of an atom in any specific region around the [[Atomic nucleus|atom's nucleus]]. The term may also refer to the physical region or space where the electron can be calculated to be present, as defined by the particular mathematical form of the orbital.<ref>{{Cite book|last=Daintith|first= J. |title=Oxford Dictionary of Chemistry|location=New York | publisher=Oxford University Press|year=2004|isbn=0-19-860918-3}}</ref> <br />
<br />
Each orbital in an atom is characterized by a unique set of values of the three [[quantum numbers]] {{mvar|n|size=120%}}, {{mvar|ℓ|size=120%}}, and {{mvar|m|size=120%}}, which correspond to the electron's [[energy]], [[angular momentum]], and an angular momentum [[vector component]], respectively. Any orbital can be occupied by a maximum of two electrons, each with its own [[spin quantum number]]. The simple names '''s orbital''', '''p orbital''', '''d orbital''' and '''f orbital''' refer to orbitals with angular momentum quantum number {{math|1=''ℓ'' = 0, 1, 2}} and {{math|3}} respectively. These names, together with the value of&nbsp;{{mvar|n}}, are used to describe the [[electron configuration]]s. They are derived from the description by early spectroscopists of certain series of alkali metal [[spectroscopic lines]] as '''s'''harp, '''p'''rincipal, '''d'''iffuse, and '''f'''undamental. Orbitals for {{mvar|ℓ|size=120%}} > 3 are named in alphabetical order (omitting&nbsp;j).<ref>{{Cite book<br />
| first=David | last=Griffiths | year=1995<br />
| title=Introduction to Quantum Mechanics | pages=190–191<br />
| publisher=Prentice Hall<br />
| isbn=0-13-124405-1}}</ref><ref>{{Cite book<br />
| first=Ira| last=Levine | year=2000<br />
| title=Quantum Chemistry | edition=5 | pages=144–145<br />
| publisher=Prentice Hall<br />
| isbn=0-13-685512-1}}</ref><ref>{{Cite book<br />
| first1=Keith J. | last1=Laidler | first2=John H. | last2=Meiser| year=1982<br />
| title=Physical Chemistry | page=488<br />
| publisher=Benjamin/Cummings<br />
| isbn=0-8053-5682-7}}</ref><br />
<br />
Atomic orbitals are the basic building blocks of the '''atomic orbital model''' (alternatively known as the electron cloud or wave mechanics model), a modern framework for visualizing the submicroscopic behavior of electrons in matter. In this model the electron cloud of a multi-electron atom may be seen as being built up (in approximation) in an [[electron configuration]] that is a product of simpler [[hydrogen atom|hydrogen-like]] atomic orbitals. The repeating ''periodicity'' of the blocks of 2, 6, 10, and 14 [[Chemical element|elements]] within sections of the [[periodic table]] arises naturally from the total number of electrons that occupy a complete set of '''s''', '''p''', '''d''' and '''f''' atomic orbitals, respectively.<br />
<br />
== Electron properties ==<br />
With the development of [[quantum mechanics]], it was found that the orbiting electrons around a nucleus could not be fully described as particles, but needed to be explained by the [[Wave–particle duality|wave-particle duality]]. In this sense, the electrons have the following properties:<br />
<br />
Wave-like properties:<br />
# The electrons do not orbit the nucleus in the sense of a planet orbiting the sun, but instead exist as [[standing wave]]s. The lowest possible energy an electron can take is therefore analogous to the fundamental frequency of a wave on a string. Higher energy states are then similar to [[harmonics]] of the [[fundamental frequency]].<br />
# The electrons are never in a single point location, although the probability of interacting with the electron at a single point can be found from the wave function of the electron.<br />
<br />
Particle-like properties:<br />
# There is always an integer number of electrons orbiting the nucleus.<br />
# Electrons jump between orbitals in a particle-like fashion. For example, if a single [[photon]] strikes the electrons, only a single electron changes states in response to the photon.<br />
# The electrons retain particle like-properties such as: each wave state has the same electrical charge as the electron particle. Each wave state has a single discrete spin (spin up or spin down).<br />
<br />
Thus, despite the obvious analogy to planets revolving around the Sun, electrons cannot be described simply as solid particles. In addition, atomic orbitals do not closely resemble a planet's elliptical path in ordinary atoms. A more accurate analogy might be that of a large and often oddly shaped "atmosphere" (the electron), distributed around a relatively tiny planet (the atomic nucleus). Atomic orbitals exactly describe the shape of this "atmosphere" only when a single electron is present in an atom. When more electrons are added to a single atom, the additional electrons tend to more evenly fill in a volume of space around the nucleus so that the resulting collection (sometimes termed the atom’s “electron cloud”<ref>{{cite book| title=The Feynman Lectures on Physics -The Definitive Edition, Vol 1 lect 6| page=11| year=2006| publisher= Pearson PLC, Addison Wesley|isbn =0-8053-9046-4|last1= Feynman|first1= Richard|last2= Leighton |first2=Robert B. |first3= Matthew |last3=Sands<br />
}}</ref>) tends toward a generally spherical zone of probability describing where the atom’s electrons will be found.<br />
<br />
===Formal quantum mechanical definition===<br />
Atomic orbitals may be defined more precisely in formal [[quantum mechanics|quantum mechanical]] language. Specifically, in quantum mechanics, the state of an atom, i.e. an [[eigenstate]] of the atomic [[Hamiltonian (quantum mechanics)|Hamiltonian]], is approximated by an expansion (see [[configuration interaction]] expansion and [[basis set (chemistry)|basis set]]) into [[linear combination]]s of anti-symmetrized products ([[Slater determinant]]s) of one-electron functions. The spatial components of these one-electron functions are called atomic orbitals. (When one considers also their [[spin (physics)|spin]] component, one speaks of '''atomic spin orbitals'''.) A state is actually a function of the coordinates of all the electrons, so that their motion is correlated, but this is often approximated by this [[Nuclear structure#The independent-particle model|independent-particle model]] of products of single electron wave functions.<ref>[[Roger Penrose]], ''[[The Road to Reality]]''</ref> (The [[London dispersion force]], for example, depends on the correlations of the motion of the electrons.)<br />
<br />
In [[atomic physics]], the [[atomic spectral line]]s correspond to transitions ([[Atomic electron transition|quantum leap]]s) between [[quantum state]]s of an atom. These states are labeled by a set of [[quantum number]]s summarized in the [[term symbol]] and usually associated with particular electron configurations, i.e., by occupation schemes of atomic orbitals (for example, 1s<sup>2</sup>&nbsp;2s<sup>2</sup>&nbsp;2p<sup>6</sup> for the ground state of [[neon]]-term symbol: <sup>1</sup>S<sub>0</sub>).<br />
<br />
This notation means that the corresponding Slater determinants have a clear higher weight in the [[configuration interaction]] expansion. The atomic orbital concept is therefore a key concept for visualizing the excitation process associated with a given [[Atomic electron transition|transition]]. For example, one can say for a given transition that it corresponds to the excitation of an electron from an occupied orbital to a given unoccupied orbital. Nevertheless, one has to keep in mind that electrons are [[fermion]]s ruled by the [[Pauli exclusion principle]] and cannot be distinguished from the other electrons in the atom. Moreover, it sometimes happens that the configuration interaction expansion converges very slowly and that one cannot speak about simple one-determinant wave function at all. This is the case when [[electron correlation]] is large.<br />
<br />
Fundamentally, an atomic orbital is a one-electron wave function, even though most electrons do not exist in one-electron atoms, and so the one-electron view is an approximation. When thinking about orbitals, we are often given an orbital vision which (even if it is not spelled out) is heavily influenced by this [[Hartree–Fock]] approximation, which is one way to reduce the complexities of [[molecular orbital theory]].<br />
<br />
===Types of orbitals===<br />
Atomic orbitals can be the hydrogen-like "orbitals" which are exact solutions to the [[Schrödinger equation]] for a [[Hydrogen-like atom|hydrogen-like "atom"]] (i.e., an atom with one electron). Alternatively, atomic orbitals refer to functions that depend on the coordinates of one electron (i.e. orbitals) but are used as starting points for approximating wave functions that depend on the simultaneous coordinates of all the electrons in an atom or molecule. The [[coordinate system]]s chosen for atomic orbitals are usually [[spherical coordinates]] {{math|(''r'', θ, φ)}} in atoms and [[Cartesian coordinate system|cartesians]] {{math|(x, y, z)}} in polyatomic molecules. The advantage of spherical coordinates (for atoms) is that an orbital wave function is a product of three factors each dependent on a single coordinate: {{math|1=ψ(''r'', θ, φ) = ''R''(''r'') Θ(θ) Φ(φ)}}.<br />
<br />
The angular factors of atomic orbitals Θ(θ) Φ(φ) generate s, p, d, etc. functions as [[Spherical harmonics#Real form|real combinations]] of [[spherical harmonics]] {{math|''Y''<sub>''ℓm''</sub>(θ, φ)}} (where {{mvar|ℓ}} and {{mvar|m}} are quantum numbers). There are typically three mathematical forms for the radial functions&nbsp;{{math|''R''(''r'')}} which can be chosen as a starting point for the calculation of the properties of atoms and molecules with many electrons. <br />
<br />
# the ''hydrogen-like atomic orbitals'' are derived from the exact solution of the Schrödinger Equation for one electron and a nucleus, for a [[hydrogen-like atom]]. The part of the function that depends on the distance from the nucleus has nodes (radial nodes) and decays as [[exponential function|{{math|''e''<sup>−(constant × distance)</sup>}}]].<br />
# The [[Slater-type orbital]] (STO) is a form without radial nodes but decays from the nucleus as does the hydrogen-like orbital.<br />
# The form of the [[Gaussian orbital|Gaussian type orbital]] (Gaussians) has no radial nodes and decays as {{math|''e''<sup>(−distance squared)</sup>}}.<br />
<br />
Although hydrogen-like orbitals are still used as pedagogical tools, the advent of computers has made STOs preferable for atoms and diatomic molecules since combinations of STOs can replace the nodes in hydrogen-like atomic orbital. Gaussians are typically used in molecules with three or more atoms. Although not as accurate by themselves as STOs, combinations of many Gaussians can attain the accuracy of hydrogen-like orbitals.<br />
<br />
==History==<br />
{{main|Atomic theory}}<br />
<br />
The term "orbital" was coined by [[Robert Mulliken]] in 1932 as an abbreviation for ''one-electron orbital wave function''.<ref><br />
{{cite journal<br />
| last=Mulliken | first=Robert S.<br />
| title=Electronic Structures of Polyatomic Molecules and Valence. II. General Considerations<br />
|date=July 1932<br />
| journal=[[Physical Review]]<br />
| volume=41 | issue=1 | pages=49–71<br />
| bibcode = 1932PhRv...41...49M<br />
| doi = 10.1103/PhysRev.41.49<br />
}}</ref> However, the idea that electrons might revolve around a compact nucleus with definite angular momentum was convincingly argued at least 19 years earlier by [[Niels Bohr]],<ref name="Bohr 1913 476">{{cite journal<br />
| last=Bohr | first=Niels<br />
| title=On the Constitution of Atoms and Molecules<br />
| journal=Philosophical Magazine | year=1913<br />
| volume=26 | issue=1 | page=476<br />
|url=http://www.chemteam.info/Chem-History/Bohr/Bohr-1913a.html<br />
| bibcode=1914Natur..93..268N<br />
| doi=10.1038/093268a0}}</ref> and the Japanese physicist [[Hantaro Nagaoka]] published an orbit-based hypothesis for electronic behavior as early as 1904.<ref name="Nagaoka 1904 445–455">{{cite journal <br />
| first=Hantaro | last=Nagaoka<br />
| title=Kinetics of a System of Particles illustrating the Line and the Band Spectrum and the Phenomena of Radioactivity<br />
| journal=Philosophical Magazine |date=May 1904<br />
| volume=7 | pages=445–455<br />
| url=http://www.chemteam.info/Chem-History/Nagaoka-1904.html <br />
| doi=10.1080/14786440409463141 <br />
| issue=41}}</ref> <br />
Explaining the behavior of these electron "orbits" was one of the driving forces behind the development of [[quantum mechanics]].<ref>{{cite book <br />
| first=Bill | last=Bryson | year=2003<br />
| title=A Short History of Nearly Everything | pages=141–143<br />
| publisher=Broadway Books<br />
| isbn=0-7679-0818-X }}</ref><br />
<br />
===Early models===<br />
With [[J.J. Thomson]]'s discovery of the electron in 1897,<ref name="referenceC">{{Cite journal|first= J. J. |last=Thomson |year=1897|title=Cathode rays|journal=Philosophical Magazine|volume= 44|page=293|doi= 10.1080/14786449708621070|issue= 269}}</ref> it became clear that atoms were not the smallest building blocks of nature, but were rather composite particles. The newly discovered structure within atoms tempted many to imagine how the atom's constituent parts might interact with each other. Thomson theorized that multiple electrons revolved in orbit-like rings within a positively charged jelly-like substance,<ref><br />
{{cite journal<br />
|first=J. J.|last= Thomson<br />
|title=On the Structure of the Atom: an Investigation of the Stability and Periods of Oscillation of a number of Corpuscles arranged at equal intervals around the Circumference of a Circle; with Application of the Results to the Theory of Atomic Structure<br />
|url=http://www.chemteam.info/Chem-History/Thomson-Structure-Atom.html | doi = 10.1080/14786440409463107<br />
|journal=[[Philosophical Magazine]] Series 6<br />
|volume=7 |issue=39 |pages=237<br />
|format=extract of paper<br />
|year=1904<br />
}}</ref> and between the electron's discovery and 1909, this "[[plum pudding model]]" was the most widely accepted explanation of atomic structure.<br />
<br />
Shortly after Thomson's discovery, [[Hantaro Nagaoka]], a Japanese physicist, predicted a different model for electronic structure.<ref name="Nagaoka 1904 445–455"/> Unlike the plum pudding model, the positive charge in Nagaoka's "Saturnian Model" was concentrated into a central core, pulling the electrons into circular orbits reminiscent of Saturn's rings. Few people took notice of Nagaoka's work at the time,<ref><br />
{{Cite book<br />
| last = Rhodes<br />
| first = Richard<br />
| title = The Making of the Atomic Bomb<br />
| publisher = Simon & Schuster<br />
| year = 1995<br />
| pages = 50–51<br />
| url = http://books.google.com/?id=aSgFMMNQ6G4C&printsec=frontcover&dq=making+of+the+atomic+bomb#v=onepage&q&f=false<br />
| isbn = 978-0-684-81378-3 }}</ref><br />
and Nagaoka himself recognized a fundamental defect in the theory even at its conception, namely that a classical charged object cannot sustain orbital motion because it is accelerating and therefore loses energy due to electromagnetic radiation.<ref>{{cite journal <br />
| first=Hantaro | last=Nagaoka<br />
| title=Kinetics of a System of Particles illustrating the Line and the Band Spectrum and the Phenomena of Radioactivity<br />
| journal=Philosophical Magazine |date=May 1904<br />
| volume=7 | page=446<br />
| url=http://www.chemteam.info/Chem-History/Nagaoka-1904.html}}</ref> Nevertheless, the [[Saturnian model]] turned out to have more in common with modern theory than any of its contemporaries.<br />
<br />
===Bohr atom===<br />
In 1909, [[Ernest Rutherford]] discovered that bulk of the atomic mass was tightly condensed into a nucleus, which was also found to be positively charged. It became clear from his analysis in 1911 that the plum pudding model could not explain atomic structure. Shortly after, in 1913, Rutherford's post-doctoral student [[Niels Bohr]] proposed a new model of the atom, wherein electrons orbited the nucleus with classical periods, but were only permitted to have discrete values of angular momentum, quantized in units [[reduced Planck constant|''h''/2π]].<ref name="Bohr 1913 476"/> This constraint automatically permitted only certain values of electron energies. The [[Bohr model]] of the atom fixed the problem of energy loss from radiation from a ground state (by declaring that there was no state below this), and more importantly explained the origin of spectral lines.<br />
<br />
[[File:Bohr-atom-PAR.svg|thumb|The [[Bohr model|Rutherford–Bohr model]] of the hydrogen atom.]]<br />
After Bohr's use of [[Albert Einstein|Einstein]]'s explanation of the [[photoelectric effect]] to relate energy levels in atoms with the wavelength of emitted light, the connection between the structure of electrons in atoms and the [[Emission spectra|emission]] and [[absorption spectra]] of atoms became an increasingly useful tool in the understanding of electrons in atoms. The most prominent feature of emission and absorption spectra (known experimentally since the middle of the 19th century), was that these atomic spectra contained discrete lines. The significance of the Bohr model was that it related the lines in emission and absorption spectra to the energy differences between the orbits that electrons could take around an atom. This was, however, ''not'' achieved by Bohr through giving the electrons some kind of wave-like properties, since the idea that electrons could behave as [[matter waves]] was not suggested until eleven years later. Still, the Bohr model's use of quantized angular momenta and therefore quantized energy levels was a significant step towards the understanding of electrons in atoms, and also a significant step towards the development of [[quantum mechanics]] in suggesting that quantized restraints must account for all discontinuous energy levels and spectra in atoms.<br />
<br />
With [[Louis de Broglie|de Broglie]]'s suggestion of the existence of electron matter waves in 1924, and for a short time before the full 1926 [[Schrödinger equation]] treatment of [[hydrogen-like atom]], a Bohr electron "wavelength" could be seen to be a function of its momentum, and thus a Bohr orbiting electron was seen to orbit in a circle at a multiple of its half-wavelength (this physically incorrect Bohr model is still often taught to beginning students). The Bohr model for a short time could be seen as a classical model with an additional constraint provided by the 'wavelength' argument. However, this period was immediately superseded by the full three-dimensional wave mechanics of 1926. In our current understanding of physics, the Bohr model is called a semi-classical model because of its quantization of angular momentum, not primarily because of its relationship with electron wavelength, which appeared in hindsight a dozen years after the Bohr model was proposed.<br />
<br />
The Bohr model was able to explain the emission and absorption spectra of [[hydrogen]]. The energies of electrons in the ''n'' = 1, 2, 3, etc. states in the Bohr model match those of current physics. However, this did not explain similarities between different atoms, as expressed by the periodic table, such as the fact that [[helium]] (two electrons), neon (10 electrons), and [[argon]] (18 electrons) exhibit similar chemical inertness. Modern [[quantum mechanics]] explains this in terms of [[electron shell]]s and subshells which can each hold a number of electrons determined by the [[Pauli exclusion principle]]. Thus the ''n'' = 1 state can hold one or two electrons, while the ''n'' = 2 state can hold up to eight electrons in 2s and 2p subshells. In helium, all ''n'' = 1 states are fully occupied; the same for ''n'' = 1 and ''n'' = 2 in neon. In argon the 3s and 3p subshells are similarly fully occupied by eight electrons; quantum mechanics also allows a 3d subshell but this is at higher energy than the 3s and 3p in argon (contrary to the situation in the hydrogen atom) and remains empty.<br />
<br />
===Modern conceptions and connections to the Heisenberg Uncertainty Principle===<br />
Immediately after [[Werner Heisenberg|Heisenberg]] discovered his [[Uncertainty principle|uncertainty relation]],<ref>{{cite journal<br />
| last=Heisenberg | first=W.<br />
| title=Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik<br />
| journal=[[Zeitschrift für Physik A]] |date=March 1927<br />
| volume=43 | pages=172–198<br />
|doi=10.1007/BF01397280 | url=http://www.springerlink.com/content/t8173612621026q5/|bibcode = 1927ZPhy...43..172H<br />
| issue=3–4 }}</ref><br />
it was noted by [[Niels Bohr|Bohr]] that the existence of any sort of [[wave packet]] implies uncertainty in the wave frequency and wavelength, since a spread of frequencies is needed to create the packet itself.<ref>{{cite journal<br />
| last=Bohr | first=Niels<br />
| title=The Quantum Postulate and the Recent Development of Atomic Theory<br />
| journal=Nature |date=April 1928<br />
| volume=121 | pages=580–590<br />
|doi=10.1038/121580a0 | url=http://www.nature.com/doifinder/10.1038/121580a0|bibcode = 1928Natur.121..580B<br />
| issue=3050 }}</ref><br />
In quantum mechanics, where all particle momenta are associated with waves, it is the formation of such a wave packet which localizes the wave, and thus the particle, in space. In states where a quantum mechanical particle is bound, it must be localized as a wave packet, and the existence of the packet and its minimum size implies a spread and minimal value in particle wavelength, and thus also momentum and energy. In quantum mechanics, as a particle is localized to a smaller region in space, the associated compressed wave packet requires a larger and larger range of momenta, and thus larger kinetic energy. Thus, the binding energy to contain or trap a particle in a smaller region of space, increases without bound, as the region of space grows smaller. Particles cannot be restricted to a geometric point in space, since this would require an infinite particle momentum. <br />
<br />
In chemistry, [[Erwin Schrödinger|Schrödinger]], [[Linus Pauling|Pauling]], [[Robert S. Mulliken|Mulliken]] and others noted that the consequence of Heisenberg's relation was that the electron, as a wave packet, could not be considered to have an exact location in its orbital. [[Max Born]] suggested that the electron's position needed to be described by a [[probability distribution]] which was connected with finding the electron at some point in the wave-function which described its associated wave packet. The new quantum mechanics did not give exact results, but only the probabilities for the occurrence of a variety of possible such results. Heisenberg held that the path of a moving particle has no meaning if we cannot observe it, as we cannot with electrons in an atom. <br />
<br />
In the quantum picture of Heisenberg, Schrödinger and others, the Bohr atom number&nbsp;''n'' for each orbital became known as an ''n-sphere''{{citation needed|date=January 2013}} in a three dimensional atom and was pictured as the mean energy of the probability cloud of the electron's wave packet which surrounded the atom.<br />
<br />
==Orbital names==<br />
Orbitals are given names in the form:<br />
:<math>X \, \mathrm{type}^y \ </math><br />
where ''X'' is the energy level corresponding to the [[principal quantum number]] {{mvar|n}}, '''type''' is a lower-case letter denoting the shape or [[Electron shell#Subshells|subshell]] of the orbital and it corresponds to the [[angular quantum number]]&nbsp;{{mvar|ℓ}}, and {{mvar|y}} is the number of electrons in that orbital.<br />
<br />
For example, the orbital 1s<sup>2</sup> (pronounced "one ess two") has two electrons and is the lowest energy level ({{math|1=''n'' = 1}}) and has an angular quantum number of {{math|1=''ℓ'' = 0}}. In [[X-ray notation]], the principal quantum number is given a letter associated with it. For {{math|1=''n'' = 1, 2, 3, 4, 5, …}}, the letters associated with those numbers are K, L, M, N, O, ... respectively.<br />
<br />
==Hydrogen-like orbitals==<br />
{{Main|Hydrogen-like atom}}<br />
<br />
The simplest atomic orbitals are those that are calculated for systems with a single electron, such as the [[hydrogen atom]]. An atom of any other element [[ion]]ized down to a single electron is very similar to hydrogen, and the orbitals take the same form. In the Schrödinger equation for this system of one negative and one positive particle, the atomic orbitals are the [[eigenstates]] of the [[Hamiltonian operator]] for the energy. They can be obtained analytically, meaning that the resulting orbitals are products of a polynomial series, and exponential and trigonometric functions. (see [[hydrogen atom]]). <br />
<br />
For atoms with two or more electrons, the governing equations can only be solved with the use of methods of iterative approximation. Orbitals of multi-electron atoms are ''qualitatively'' similar to those of hydrogen, and in the simplest models, they are taken to have the same form. For more rigorous and precise analysis, the numerical approximations must be used.<br />
<br />
A given (hydrogen-like) atomic orbital is identified by unique values of three quantum numbers: [[Principal quantum number|{{mvar|n|size=120%}}]], [[Azimuthal quantum number|{{mvar|ℓ|size=120%}}]], and [[magnetic quantum number|{{mvar|m<sub>ℓ</sub>|size=120%}}]]. The rules restricting the values of the quantum numbers, and their energies (see below), explain the electron configuration of the atoms and the [[periodic table]].<br />
<br />
The stationary states ([[quantum state]]s) of the hydrogen-like atoms are its atomic orbitals.{{Clarify|date=November 2011|hydrogen-like atoms and orbitals}} However, in general, an electron's behavior is not fully described by a single orbital. Electron states are best represented by time-depending "mixtures" ([[linear combination]]s) of multiple orbitals. See [[Linear combination of atomic orbitals molecular orbital method]].<br />
<br />
The quantum number {{mvar|n}} first appeared in the [[Bohr model]] where it determines the radius of each circular electron orbit. In modern quantum mechanics however, {{mvar|n}} determines the mean distance of the electron from the nucleus; all electrons with the same value of ''n'' lie at the same average distance. For this reason, orbitals with the same value of ''n'' are said to comprise a "[[electron shell|shell]]". Orbitals with the same value of ''n'' and also the same value of&nbsp;{{mvar|ℓ}} are even more closely related, and are said to comprise a "[[electron subshell|subshell]]".<br />
<br />
==Quantum numbers==<br />
{{main|Quantum number}}<br />
<br />
Because of the quantum mechanical nature of the electrons around a nucleus, atomic orbitals can be uniquely defined by a set of integers known as quantum numbers. These quantum numbers only occur in certain combinations of values, and their physical interpretation changes depending on whether real or complex versions of the atomic orbitals are employed.<br />
<br />
===Complex orbitals===<br />
In physics, the most common orbital descriptions are based on the solutions to the hydrogen atom, where orbitals are given by the product between a radial function and a pure spherical harmonic. The quantum numbers, together with the rules governing their possible values, are as follows:<br />
<br />
The [[principal quantum number]] {{mvar|n}} describes the energy of the electron and is always a [[positive integer]]. In fact, it can be any positive integer, but for reasons discussed below, large numbers are seldom encountered. Each atom has, in general, many orbitals associated with each value of ''n''; these orbitals together are sometimes called ''[[electron shells]]''.<br />
<br />
The [[azimuthal quantum number]] {{mvar|ℓ}} describes the orbital angular momentum of each electron and is a non-negative integer. Within a shell where {{mvar|n}} is some integer {{math|''n''<sub>0</sub>}}, {{mvar|ℓ}} ranges across all (integer) values satisfying the relation <math>0 \le \ell \le n_0-1</math>. For instance, the {{math|1=''n'' = 1}}&nbsp;shell has only orbitals with <math>\ell=0</math>, and the {{math|1=''n'' = 2}}&nbsp;shell has only orbitals with <math>\ell=0</math>, and <math>\ell=1</math>. The set of orbitals associated with a particular value of&nbsp;{{mvar|ℓ}} are sometimes collectively called a ''subshell''.<br />
<br />
The [[magnetic quantum number]], <math>m_\ell</math>, describes the magnetic moment of an electron in an arbitrary direction, and is also always an integer. Within a subshell where <math>\ell</math> is some integer <math>\ell_0</math>, <math>m_\ell</math> ranges thus: <math>-\ell_0 \le m_\ell \le \ell_0</math>.<br />
<br />
The above results may be summarized in the following table. Each cell represents a subshell, and lists the values of <math>m_\ell</math> available in that subshell. Empty cells represent subshells that do not exist.<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! {{math|1=''ℓ'' = 0}}<br />
! {{math|1=''ℓ'' = 1}}<br />
! {{math|1=''ℓ'' = 2}}<br />
! {{math|1=''ℓ'' = 3}}<br />
! {{math|1=''ℓ'' = 4}}<br />
! ...<br />
|-<br />
! {{math|1=''n'' = 1}}<br />
| <math>m_\ell=0</math><br />
| || || || ||<br />
|-<br />
! {{math|1=''n'' = 2}}<br />
| 0 || −1, 0, 1<br />
| || || ||<br />
|-<br />
! {{math|1=''n'' = 3}}<br />
| 0 || −1, 0, 1 || −2, −1, 0, 1, 2<br />
| || ||<br />
|-<br />
! {{math|1=''n'' = 4}}<br />
| 0 || −1, 0, 1 || −2, −1, 0, 1, 2 || −3, −2, −1, 0, 1, 2, 3<br />
| ||<br />
|-<br />
! {{math|1=''n'' = 5}}<br />
| 0 || −1, 0, 1 || −2, −1, 0, 1, 2 || −3, −2, −1, 0, 1, 2, 3 || −4, −3, −2, −1, 0, 1, 2, 3, 4<br />
|<br />
|-<br />
! ...<br />
| ... || ... || ... || ... || ... || ...<br />
|}<br />
<br />
Subshells are usually identified by their <math>n</math>- and <math>\ell</math>-values. <math>n</math> is represented by its numerical value, but <math>\ell</math> is represented by a letter as follows: 0 is represented by 's', 1 by 'p', 2 by 'd', 3 by 'f', and 4 by 'g'. For instance, one may speak of the subshell with <math>n=2</math> and <math>\ell=0</math> as a '2s subshell'.<br />
<br />
Each electron also has a [[spin quantum number]], '''s''', which describes the spin of each electron (spin up or spin down). The number '''s''' can be +{{frac|1|2}} or −{{frac|1|2}}.<br />
<br />
The [[Pauli exclusion principle]] states that no two electrons can occupy the same quantum state: every electron in an atom must have a unique combination of quantum numbers.<br />
<br />
The above conventions imply a preferred axis (for example, the ''z'' direction in Cartesian coordinates), and they also imply a preferred direction along this preferred axis. Otherwise there would be no sense in distinguishing {{math|1=''m'' = +1}} from {{math|1=''m'' = −1}}. As such, the model is most useful when applied to physical systems that share these symmetries. The [[Stern–Gerlach experiment]] — where an atom is exposed to a magnetic field — provides one such example.<ref><br />
{{cite journal<br />
|last=Gerlach |first=W.<br />
|last2=Stern |first2=O.<br />
|title=Das magnetische Moment des Silberatoms<br />
|journal=[[Zeitschrift für Physik]]<br />
|volume=9 |pages=353–355<br />
|year=1922<br />
|doi=10.1007/BF01326984<br />
|bibcode = 1922ZPhy....9..353G }}</ref><br />
<br />
===Real orbitals===<br />
An atom that is embedded in a crystalline solid feels multiple preferred axes, but no preferred direction. Instead of building atomic orbitals out of the product of radial functions and a single spherical harmonic, linear combinations of spherical harmonics are typically used, designed so that the imaginary part of the spherical harmonics cancel out. These '''real orbitals''' are the building blocks most commonly shown in orbital visualizations.<br />
<br />
In the real hydrogen-like orbitals, for example, {{mvar|n}} and {{mvar|ℓ}} have the same interpretation and significance as their complex counterparts, but {{mvar|m}} is no longer a good quantum number (though its absolute value is). The orbitals are given new names based on their shape with respect to a standardized Cartesian basis. The real hydrogen-like p&nbsp;orbitals are given by the following<ref name=Levine00>{{cite book|last=Levine|first=Ira|title=Quantum Chemistry|year=2000|publisher=Prentice-Hall|location=Upper Saddle River, NJ|isbn=0-13-685512-1|pages=148}}</ref><ref>C.D.H. Chisholm (1976). Group theoretical techniques in quantum chemistry. New York: Academic Press. ISBN 0-12-172950-8.</ref><ref>{{Cite journal|author=Blanco, Miguel A.; Flórez, M.; Bermejo, M.|date= December 1997|title=Evaluation of the rotation matrices in the basis of real spherical harmonics|journal=Journal of Molecular Structure: THEOCHEM |volume=419 |issue=1-3|pages=19–27|doi=10.1016/S0166-1280(97)00185-1}}</ref><br />
:<br />
<br />
:<math>p_z = p_0</math><br />
<br />
:<math>p_x = \frac{1}{\sqrt{2}} \left(p_1 + p_{-1} \right) </math><br />
<br />
:<math>p_y = \frac{1}{i\sqrt{2}} \left( p_1 - p_{-1} \right) </math><br />
<br />
where {{math|1=''p''<sub>0</sub> = ''R''<sub>''n'' 1</sub> ''Y''<sub>1 0</sub>}}, {{math|1=''p''<sub>1</sub> = ''R''<sub>''n'' 1</sub> ''Y''<sub>1 1</sub>}}, and {{math|1=''p''<sub>−1</sub> = ''R''<sub>''n'' 1</sub> ''Y''<sub>1 −1</sub>}}, are the complex orbitals corresponding to {{math|1=''ℓ'' = 1}}.<br />
<br />
==Shapes of orbitals==<br />
[[File:HydrogenOrbitalsN6L0M0.png|thumb|Cross-section of computed hydrogen atom orbital ({{math|1=ψ(''r'', θ, φ)<sup>2</sup>}}) for the 6s {{math|1=(''n'' = 6, ''ℓ'' = 0, ''m'' = 0)}} orbital. Note that s orbitals, though spherically symmetrical, have radially placed wave-nodes for {{math|''n'' > 1}}. However, only s orbitals invariably have a center anti-node; the other types never do.]]<br />
<br />
Simple pictures showing orbital shapes are intended to describe the angular forms of regions in space where the electrons occupying the orbital are likely to be found. The diagrams cannot, however, show the entire region where an electron can be found, since according to quantum mechanics there is a non-zero probability of finding the electron (almost) anywhere in space. Instead the diagrams are approximate representations of boundary or [[contour line|contour surfaces]] where the probability density {{math|{{!}} ψ(''r'', θ, φ) {{!}}<sup>2</sup>}} has a constant value, chosen so that there is a certain probability (for example 90%) of finding the electron within the contour. Although {{math|{{!}} ψ {{!}}<sup>2</sup>}} as the square of an [[absolute value]] is everywhere non-negative, the sign of the [[wave function]] {{math|ψ(''r'', θ, φ)}} is often indicated in each subregion of the orbital picture.<br />
<br />
Sometimes the {{math|ψ}} function will be graphed to show its phases, rather than the {{math|{{!}} ψ(''r'', θ, φ) {{!}}<sup>2</sup>}} which shows probability density but has no phases (which have been lost in the process of taking the absolute value, since {{math|ψ(''r'', θ, φ)}} is a complex number). {{math|{{!}} ψ(''r'', θ, φ) {{!}}<sup>2</sup>}} orbital graphs tend to have less spherical, thinner lobes than {{math|ψ(''r'', θ, φ)}} graphs, but have the same number of lobes in the same places, and otherwise are recognizable. This article, in order to show wave function phases, shows mostly {{math|ψ(''r'', θ, φ)}} graphs.<br />
<br />
The lobes can be viewed as [[interference (wave propagation)|interference]] patterns between the two counter rotating "{{mvar|m}}" and "{{math|−''m''}}" modes, with the projection of the orbital onto the xy plane having a resonant "{{mvar|m}}" wavelengths around the circumference. For each {{mvar|m}} there are two of these {{math|⟨''m''⟩+⟨−''m''⟩}} and {{math|⟨''m''⟩−⟨−''m''⟩}}. For the case where {{math|1=''m'' = 0}} the orbital is vertical, counter rotating information is unknown, and the orbital is z-axis symmetric. For the case where {{math|1=''ℓ'' = 0}} there are no counter rotating modes. There are only radial modes and the shape is spherically symmetric. For any given {{mvar|n}}, the smaller {{mvar|ℓ}} is, the more radial nodes there are. Loosely speaking ''n'' is energy, {{mvar|ℓ}} is analogous to [[orbital eccentricity|eccentricity]], and {{mvar|m}} is orientation.<br />
<br />
Generally speaking, the number {{mvar|n}} determines the size and energy of the orbital for a given nucleus: as {{mvar|n}} increases, the size of the orbital increases. However, in comparing different elements, the higher nuclear charge {{mvar|Z}} of heavier elements causes their orbitals to contract by comparison to lighter ones, so that the overall size of the whole atom remains very roughly constant, even as the number of electrons in heavier elements (higher {{mvar|Z}}) increases.<br />
<br />
Also in general terms, {{mvar|ℓ}} determines an orbital's shape, and {{mvar|m<sub>ℓ</sub>}} its orientation. However, since some orbitals are described by equations in [[complex number]]s, the shape sometimes depends on {{mvar|m<sub>ℓ</sub>}} also. Together, the the whole set of orbitals for a given {{mvar|ℓ}} and {{mvar|n}} fill space as symmetrically as possible, though with increasingly complex sets of lobes and nodes.<br />
<br />
The single s-orbitals (<math>\ell=0</math>) are shaped like spheres. For {{math|1=''n'' = 1}} it is roughly a [[ball (mathematics)|solid ball]] (it is most dense at the center and fades exponentially outwardly), but for {{math|1=''n'' = 2}} or more, each single s-orbital is composed of spherically symmetric surfaces which are nested shells (i.e., the "wave-structure" is radial, following a sinusoidal radial component as well). See illustration of a cross-section of these nested shells, at right. The s-orbitals for all {{mvar|n}} numbers are the only orbitals with an anti-node (a region of high wave function density) at the center of the nucleus. All other orbitals (p, d, f, etc.) have angular momentum, and thus avoid the nucleus (having a wave node ''at'' the nucleus).<br />
<br />
The three p-orbitals for {{math|1=''n'' = 2}} have the form of two [[ellipsoid]]s with a [[point of tangency]] at the [[atomic nucleus|nucleus]] (the two-lobed shape is sometimes referred to as a "[[dumbbell]]"—there are two lobes pointing in opposite directions from each other). The three p-orbitals in each [[Electron shell|shell]] are oriented at right angles to each other, as determined by their respective linear combination of values of&nbsp;{{mvar|m<sub>ℓ</sub>}}. The overall result is a lobe pointing along each direction of the primary axes.<br />
<br />
[[File:D orbitals.svg|thumb|250px|The five d orbitals in {{math|ψ(''x'', ''y'', ''z'')<!--Please don't italicize a bracket --><sup>2</sup>}} form, with a combination diagram showing how they fit together to fill space around an atomic nucleus.]]<br />
Four of the five d-orbitals for {{math|1=''n'' = 3}} look similar, each with four pear-shaped lobes, each lobe tangent at right angles to two others, and the centers of all four lying in one plane. Three of these planes are the xy-, xz-, and yz-planes—the lobes are between the pairs of primary axes—and the fourth has the centres along the on the x and y axes themselves. The fifth and final d-orbital consists of three regions of high probability density: a [[torus]] with two pear-shaped regions placed symmetrically on its z axis. The overall total of 18 directional lobes point in every primaxy axis direction and between every pair.<br />
<br />
There are seven f-orbitals, each with shapes more complex than those of the d-orbitals.<br />
<br />
Additionally, as is the case with the s orbitals, individual p, d, f and g orbitals with {{mvar|n}} values higher than the lowest possible value, exhibit an additional radial node structure which is reminiscent of harmonic waves of the same type, as compared with the lowest (or fundamental) mode of the wave. As with s orbitals, this phenomenon provides p, d, f, and g orbitals at the next higher possible value of {{mvar|n}} (for example, 3p orbitals vs. the fundamental 2p), an additional node in each lobe. Still higher values of {{mvar|n}} further increase the number of radial nodes, for each type of orbital.<br />
<br />
The shapes of atomic orbitals in one-electron atom are related to 3-dimensional [[spherical harmonics]]. These shapes are not unique, and any linear combination is valid, like a transformation to [[cubic harmonic]]s, in fact it is possible to generate sets where all the d's are the same shape, just like the {{math|''p''<sub>''x''</sub>, ''p''<sub>''y''</sub>,}} and {{math|''p''<sub>''z''</sub>}} are the same shape.<ref>{{Cite journal| doi = 10.1021/ed045p45 | title = The five equivalent d orbitals | year = 1968 | last1 = Powell | first1 = Richard E. | journal = Journal of Chemical Education | volume = 45| issue = 1 | page = 45|bibcode = 1968JChEd..45...45P }}</ref><ref>{{Cite journal| doi =10.1063/1.1750628 | title =Directed Valence | year =1940 | last1 =Kimball | first1 =George E. | journal =The Journal of Chemical Physics | volume =8| issue =2 | page =188|bibcode = 1940JChPh...8..188K }}</ref><br />
<br />
===Orbitals table===<br />
This table shows all orbital configurations for the real hydrogen-like wave functions up to 7s, and therefore covers the simple electronic configuration for all elements in the periodic table up to [[radium]]. "ψ" graphs are shown with '''−''' and '''+''' [[wave function]] phases shown in two different colors (arbitrarily red and blue). The {{math|''p''<sub>''z''</sub>}} orbital is the same as the {{math|''p''<sub>''0''</sub>}} orbital, but the {{math|''p''<sub>''x''</sub>}} and {{math|''p''<sub>''y''</sub>}} are formed by taking linear<br />
combinations of the {{math|''p''<sub>''+1''</sub>}} and {{math|''p''<sub>''−1''</sub>}} orbitals (which is why they are listed under the {{math|1=''m'' = ±1}} label). Also, the {{math|''p''<sub>''+1''</sub>}} and {{math|''p''<sub>''−1''</sub>}} are not<br />
the same shape as the {{math|''p''<sub>''0''</sub>}}, since they are pure [[spherical harmonics]].<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! s ({{math|1=''ℓ'' = 0}})<br />
! colspan="3" |p ({{math|1=''ℓ'' = 1}})<br />
! colspan="5" |d ({{math|1=''ℓ'' = 2}})<br />
! colspan="7" |f ({{math|1=''ℓ'' = 3}})<br />
|-<br />
!<br />
! {{math|1=''m'' = 0}}<br />
! {{math|1=''m'' = 0}}<br />
! colspan="2" |{{math|1=''m'' = ±1}}<br />
! {{math|1=''m'' = 0}}<br />
! colspan="2" |{{math|1=''m'' = ±1}}<br />
! colspan="2" |{{math|1=''m'' = ±2}}<br />
! {{math|1=''m'' = 0}}<br />
! colspan="2" |{{math|1=''m'' = ±1}}<br />
! colspan="2" |{{math|1=''m'' = ±2}}<br />
! colspan="2" |{{math|1=''m'' = ±3}}<br />
|-<br />
!<br />
! ''s''<br />
! ''p''<sub>''z''</sub><br />
! ''p''<sub>''x''</sub><br />
! ''p''<sub>''y''</sub><br />
! ''d''<sub>''z<sup>2</sup>''</sub><br />
! ''d''<sub>''xz''</sub><br />
! ''d''<sub>''yz''</sub><br />
! ''d''<sub>''xy''</sub><br />
! ''d''<sub>''x<sup>2</sup>−y<sup>2</sup>''</sub><br />
! ''f''<sub>''z<sup>3</sup>''</sub><br />
! ''f''<sub>''xz<sup>2</sup>''</sub><br />
! ''f''<sub>''yz<sup>2</sup>''</sub><br />
! ''f''<sub>''xyz''</sub><br />
! ''f''<sub>''z(x<sup>2</sup>−y<sup>2</sup>)''</sub><br />
! ''f''<sub>''x(x<sup>2</sup>−3y<sup>2</sup>)''</sub><br />
! ''f''<sub>''y(3x<sup>2</sup>−y<sup>2</sup>)''</sub><br />
|-<br />
!{{math|1=''n'' = 1}}<br />
| [[File:S1M0.png|50px]]<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
!{{math|1=''n'' = 2}}<br />
| [[File:S2M0.png|50px]]<br />
| [[File:P2M0.png|50px]]<br />
| [[File:P2M1.png|50px]]<br />
| [[File:P2M-1.png|50px]]<br />
|<br />
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|-<br />
!{{math|1=''n'' = 3}}<br />
| [[File:S3M0.png|50px]]<br />
| [[File:P3M0.png|50px]]<br />
| [[File:P3M1.png|50px]]<br />
| [[File:P3M-1.png|50px]]<br />
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!{{math|1=''n'' = 4}}<br />
| [[File:S4M0.png|50px]]<br />
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| [[File:P4M1.png|50px]]<br />
| [[File:P4M-1.png|50px]]<br />
| [[File:D4M0.png|50px]]<br />
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| [[File:D4M2.png|50px]]<br />
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| [[File:F4M0.png|50px]]<br />
| [[File:F4M1.png|50px]]<br />
| [[File:F4M-1.png|50px]]<br />
| [[File:F4M2.png|50px]]<br />
| [[File:F4M-2.png|50px]]<br />
| [[File:F4M3.png|50px]]<br />
| [[File:F4M-3.png|50px]]<br />
|-<br />
!{{math|1=''n'' = 5}}<br />
| [[File:S5M0.png|50px]]<br />
| [[File:P5M0.png|50px]]<br />
| [[File:P5M1.png|50px]]<br />
| [[File:P5M-1.png|50px]]<br />
| [[File:D5M0.png|50px]]<br />
| [[File:D5M1.png|50px]]<br />
| [[File:D5M-1.png|50px]]<br />
| [[File:D5M2.png|50px]]<br />
| [[File:D5M-2.png|50px]]<br />
| '''. . .'''<br />
| '''. . .'''<br />
| '''. . .'''<br />
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| '''. . .'''<br />
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| '''. . .'''<br />
|-<br />
!{{math|1=''n'' = 6}}<br />
| [[File:S6M0.png|50px]]<br />
| [[File:P6M0.png|50px]]<br />
| [[File:P6M1.png|50px]]<br />
| [[File:P6M-1.png|50px]]<br />
| '''. . .'''<br />
| '''. . .'''<br />
| '''. . .'''<br />
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| '''. . .'''<br />
| '''. . .'''<br />
| '''. . .'''<br />
| '''. . .'''<br />
| '''. . .'''<br />
| '''. . .'''<br />
| '''. . .'''<br />
| '''. . .'''<br />
|-<br />
!{{math|1=''n'' = 7}}<br />
| [[File:S7M0.png|50px]]<br />
| '''. . .'''<br />
| '''. . .'''<br />
| '''. . .'''<br />
| '''. . .'''<br />
| '''. . .'''<br />
| '''. . .'''<br />
| '''. . .'''<br />
| '''. . .'''<br />
| '''. . .'''<br />
| '''. . .'''<br />
| '''. . .'''<br />
| '''. . .'''<br />
| '''. . .'''<br />
| '''. . .'''<br />
| '''. . .'''<br />
|}<br />
<br />
===Qualitative understanding of shapes===<br />
The shapes of atomic orbitals can be understood qualitatively by considering the analogous case of [[Vibrations of a circular drum|standing waves on a circular drum]].<ref>{{cite journal<br />
| last=Cazenave, Lions | first=T., P. <br />
| title=Orbital stability of standing waves for some nonlinear Schrödinger equations<br />
| journal=Communications in Mathematical Physics | year=1982<br />
| volume=85 | issue=4 | pages= 549–561<br />
|doi = 10.1007/BF01403504 |bibcode = 1982CMaPh..85..549C<br />
| last2=Lions<br />
| first2=P. L. }}</ref> To see the analogy, the mean vibrational displacement of each bit of drum membrane from the equilibrium point over many cycles (a measure of average drum membrane velocity and momentum at that point) must be considered relative to that point's distance from the center of the drum head. If this displacement is taken as being analogous to the probability of finding an electron at a given distance from the nucleus, then it will be seen that the many modes of the vibrating disk form patterns that trace the various shapes of atomic orbitals. The basic reason for this correspondence lies in the fact that the distribution of kinetic energy and momentum in a matter-wave is predictive of where the particle associated with the wave will be. That is, the probability of finding an electron at a given place is also a function of the electron's average momentum at that point, since high electron momentum at a given position tends to "localize" the electron in that position, via the properties of electron wave-packets (see the [[Heisenberg uncertainty principle]] for details of the mechanism). <br />
<br />
This relationship means that certain key features can be observed in both drum membrane modes and atomic orbitals. For example, in all of the modes analogous to '''s'''&nbsp;orbitals (the top row in the animated illustration below), it can be seen that the very center of the drum membrane vibrates most strongly, corresponding to the [[antinode]] in all '''s'''&nbsp;orbitals in an atom. This antinode means the electron is most likely to be at the physical position of the nucleus (which it passes straight through without scattering or striking it), since it is moving (on average) most rapidly at that point, giving it maximal momentum. <br />
<br />
A mental "planetary orbit" picture closest to the behavior of electrons in '''s'''&nbsp;orbitals, all of which have no angular momentum, might perhaps be that of a [[Keplerian orbit]] with the [[orbital eccentricity]] of 1 but a finite major axis, not physically possible (because [[particle]]s were to collide), but can be imagined as a [[limit (mathematics)|limit]] of orbits with equal major axes but increasing eccentricity.<!-- could somebody make an illustration? --><br />
<br />
Below, a number of drum membrane vibration modes are shown. The analogous wave functions of the hydrogen atom are indicated. A correspondence can be considered where the wave functions of a vibrating drum head are for a two-coordinate system {{math|ψ(''r'', θ)}} and the wave functions for a vibrating sphere are three-coordinate {{math|ψ(''r'', θ, φ)}}.<br />
<br />
<center | align = center><br />
'''s-type modes'''<br />
<gallery widths="200px"><br />
Image:Drum vibration mode01.gif|Mode <math>u_{01}</math> (1s orbital)<br />
Image:Drum vibration mode02.gif|Mode <math>u_{02}</math> (2s orbital)<br />
Image:Drum vibration mode03.gif|Mode <math>u_{03}</math> (3s orbital)<br />
</gallery><br />
</center><br />
<br />
None of the other sets of modes in a drum membrane have a central antinode, and in all of them the center of the drum does not move. These correspond to a node at the nucleus for all non-'''s''' orbitals in an atom. These orbitals all have some angular momentum, and in the planetary model, they correspond to particles in orbit with eccentricity less than 1.0, so that they do not pass straight through the center of the primary body, but keep somewhat away from it. <br />
<br />
In addition, the drum modes analogous to '''p''' and '''d''' modes in an atom show spatial irregularity along the different radial directions from the center of the drum, whereas all of the modes analogous to '''s'''&nbsp;modes are perfectly symmetrical in radial direction. The non radial-symmetry properties of non-'''s''' orbitals are necessary to localize a particle with angular momentum and a wave nature in an orbital where it must tend to stay away from the central attraction force, since any particle localized at the point of central attraction could have no angular momentum. For these modes, waves in the drum head tend to avoid the central point. Such features again emphasize that the shapes of atomic orbitals are a direct consequence of the wave nature of electrons. <br />
<br />
<center | align = center><br />
'''p-type modes'''<br />
<gallery widths="200px"><br />
Image:Drum vibration mode11.gif|Mode <math>u_{11}</math> (2p orbital)<br />
Image:Drum vibration mode12.gif|Mode <math>u_{12}</math> (3p orbital)<br />
Image:Drum vibration mode13.gif|Mode <math>u_{13}</math> (4p orbital)<br />
</gallery><br />
'''d-type modes'''<br />
<gallery widths="200px"><br />
Image:Drum vibration mode21.gif|Mode <math>u_{21}</math> (3d orbital)<br />
Image:Drum vibration mode22.gif|Mode <math>u_{22}</math> (4d orbital)<br />
Image:Drum vibration mode23.gif|Mode <math>u_{23}</math> (5d orbital)<br />
</gallery><br />
</center><br />
<br />
==Orbital energy==<br />
{{main|Electron shell}}<br />
In atoms with a single electron ([[hydrogen-like atom]]s), the energy of an orbital (and, consequently, of any electrons in the orbital) is determined exclusively by <math>n</math>. The <math>n=1</math> orbital has the lowest possible energy in the atom. Each successively higher value of <math>n</math> has a higher level of energy, but the difference decreases as <math>n</math> increases. For high <math>n</math>, the level of energy becomes so high that the electron can easily escape from the atom. In single electron atoms, all levels with different <math>\ell</math> within a given <math>n</math> are (to a good approximation) degenerate, and have the same energy. This approximation is broken to a slight extent by the effect of the magnetic field of the nucleus, and by [[quantum electrodynamics]] effects. The latter induce tiny binding energy differences especially for '''s'''&nbsp;electrons that go nearer the nucleus, since these feel a very slightly different nuclear charge, even in one-electron atoms; see [[Lamb shift]].<br />
<br />
In atoms with multiple electrons, the energy of an electron depends not only on the intrinsic properties of its orbital, but also on its interactions with the other electrons. These interactions depend on the detail of its spatial probability distribution, and so the [[energy level]]s of orbitals depend not only on <math>n</math> but also on <math>\ell</math>. Higher values of <math>\ell</math> are associated with higher values of energy; for instance, the 2p state is higher than the 2s state. When <math>\ell = 2</math>, the increase in energy of the orbital becomes so large as to push the energy of orbital above the energy of the s-orbital in the next higher shell; when <math>\ell = 3</math> the energy is pushed into the shell two steps higher. The filling of the 3d orbitals does not occur until the 4s orbitals have been filled. <br />
<br />
The increase in energy for subshells of increasing angular momentum in larger atoms is due to electron–electron interaction effects, and it is specifically related to the ability of low angular momentum electrons to penetrate more effectively toward the nucleus, where they are subject to less screening from the charge of intervening electrons. Thus, in atoms of higher atomic number, the <math>\ell</math> of electrons becomes more and more of a determining factor in their energy, and the principal quantum numbers <math>n</math> of electrons becomes less and less important in their energy placement.<br />
<br />
The energy sequence of the first 24&nbsp;subshells (e.g., 1s, 2p, 3d, etc.) is given in the following table. Each cell represents a subshell with <math>n</math> and <math>\ell</math> given by its row and column indices, respectively. The number in the cell is the subshell's position in the sequence. For a linear listing of the subshells in terms of increasing energies in multielectron atoms, see the section below.<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! s<br />
! p<br />
! d<br />
! f<br />
! g<br />
|-<br />
! 1<br />
| 1 || || || ||<br />
|-<br />
! 2<br />
| 2 || 3 || || ||<br />
|-<br />
! 3<br />
| 4 || 5 || 7 || ||<br />
|-<br />
! 4<br />
| 6 || 8 || 10 || 13 ||<br />
|-<br />
! 5<br />
| 9 || 11 || 14 || 17 || ''21''<br />
|-<br />
! 6<br />
|12 || 15 || 18 || ''22'' || <br />
|-<br />
! 7<br />
|16 || 19 || ''23'' |||| <br />
|-<br />
! 8<br />
|''20'' || ''24'' |||||| <br />
|}<br />
<br />
''Note: empty cells indicate non-existent sublevels, while numbers in italics indicate sublevels that could exist, but which do not hold electrons in any element currently known.''<br />
<br />
==Electron placement and the periodic table==<br />
[[File:Electron orbitals.svg|right|thumb|350px| [[Electron]] atomic and [[molecular orbital|molecular]] orbitals. The chart of orbitals ('''left''') is arranged by increasing energy (see [[Madelung rule]]). ''Note that atomic orbits are functions of three variables (two angles, and the distance&nbsp;{{mvar|r}} from the nucleus). These images are faithful to the angular component of the orbital, but not entirely representative of the orbital as a whole.'']]<br />
<br />
[[File:Atomic orbitals and periodic table construction.ogv|thumb|Atomic orbitals and periodic table construction]]<br />
<br />
{{main|electron configuration|electron shell}}<br />
<br />
Several rules govern the placement of electrons in orbitals (''[[electron configuration]]''). The first dictates that no two electrons in an atom may have the same set of values of quantum numbers (this is the [[Pauli exclusion principle]]). These quantum numbers include the three that define orbitals, as well as [[Spin quantum number|{{mvar|s}}]], or [[spin quantum number]]. Thus, two electrons may occupy a single orbital, so long as they have different values of&nbsp;{{mvar|s}}. However, ''only'' two electrons, because of their spin, can be associated with each orbital.<br />
<br />
Additionally, an electron always tends to fall to the lowest possible energy state. It is possible for it to occupy any orbital so long as it does not violate the Pauli exclusion principle, but if lower-energy orbitals are available, this condition is unstable. The electron will eventually lose energy (by releasing a [[photon]]) and drop into the lower orbital. Thus, electrons fill orbitals in the order specified by the energy sequence given above.<br />
<br />
This behavior is responsible for the structure of the [[periodic table]]. The table may be divided into several rows (called 'periods'), numbered starting with 1 at the top. The presently known elements occupy seven periods. If a certain period has number ''i'', it consists of elements whose outermost electrons fall in the ''i''th shell. [[Niels Bohr]] was the first to propose (1923) that the [[Periodic table|periodicity]] in the properties of the elements might be explained by the periodic filling of the electron energy levels, resulting in the electronic structure of the atom.<ref name="Bohr">{{cite journal | last = Bohr | first = Niels | authorlink = Niels Bohr | title = Über die Anwendung der Quantumtheorie auf den Atombau. I | journal = [[Zeitschrift für Physik]] | year = 1923 | volume = 13 | page = 117|bibcode = 1923ZPhy...13..117B |doi = 10.1007/BF01328209 }}</ref><br />
<br />
The periodic table may also be divided into several numbered rectangular '[[Periodic table block|blocks]]'. The elements belonging to a given block have this common feature: their highest-energy electrons all belong to the same {{mvar|ℓ}}-state (but the {{mvar|n}} associated with that {{mvar|ℓ}}-state depends upon the period). For instance, the leftmost two columns constitute the 's-block'. The outermost electrons of [[Lithium|Li]] and [[Beryllium|Be]] respectively belong to the 2s&nbsp;subshell, and those of [[sodium|Na]] and [[magnesium|Mg]] to the 3s&nbsp;subshell.<br />
<br />
The following is the order for filling the "subshell" orbitals, which also gives the order of the "blocks" in the periodic table:<br />
<br />
:'''1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p'''<br />
<br />
The "periodic" nature of the filling of orbitals, as well as emergence of the '''s''', '''p''', '''d''' and '''f''' "blocks", is more obvious if this order of filling is given in matrix form, with increasing principal quantum numbers starting the new rows ("periods") in the matrix. Then, each subshell (composed of the first two quantum numbers) is repeated as many times as required for each pair of electrons it may contain. The result is a compressed periodic table, with each entry representing two successive elements:<br />
<center><br />
{|<br />
|-<br />
|<pre><br />
1s <br />
2s 2p 2p 2p<br />
3s 3p 3p 3p<br />
4s 3d 3d 3d 3d 3d 4p 4p 4p<br />
5s 4d 4d 4d 4d 4d 5p 5p 5p<br />
6s (4f) 5d 5d 5d 5d 5d 6p 6p 6p<br />
7s (5f) 6d 6d 6d 6d 6d 7p 7p 7p<br />
</pre><br />
|}<br />
</center><br />
<br />
The number of electrons in an electrically neutral atom increases with the [[atomic number]]. The electrons in the outermost shell, or ''[[valence electron]]s'', tend to be responsible for an element's chemical behavior. Elements that contain the same number of valence electrons can be grouped together and display similar chemical properties.<br />
<br />
===Relativistic effects===<br />
{{Main|Relativistic quantum chemistry}}<br />
<br />
For elements with high atomic number {{mvar|Z}}, the effects of relativity become more pronounced, and especially so for s&nbsp;electrons, which move at relativistic velocities as they penetrate the screening electrons near the core of high-{{mvar|Z}} atoms. This relativistic increase in momentum for high speed electrons causes a corresponding decrease in wavelength and contraction of 6s orbitals relative to 5d orbitals (by comparison to corresponding s and d electrons in lighter elements in the same column of the periodic table); this results in 6s valence electrons becoming lowered in energy.<br />
<br />
Examples of significant physical outcomes of this effect include the lowered melting temperature of [[mercury (element)|mercury]] (which results from 6s electrons not being available for metal bonding) and the golden color of [[gold]] and [[caesium]] (which results from narrowing of 6s to 5d transition energy to the point that visible light begins to be absorbed).<ref>{{Cite web|url=http://www.chem1.com/acad/webtut/atomic/qprimer/#Q26|title= Primer on Quantum Theory of the Atom|first=Stephen |last=Lower}}</ref> <br />
<br />
In the [[Bohr Model]], an {{math|''n'' = 1}}&nbsp;electron has a velocity given by <math>v = Z \alpha c</math>, where {{mvar|Z}} is the atomic number, <math>\alpha</math> is the [[fine-structure constant]], and {{math|''c''}} is the speed of light. In non-relativistic quantum mechanics, therefore, any atom with an atomic number greater than 137 would require its 1s electrons to be traveling faster than the speed of light. Even in the [[Dirac equation]], which accounts for relativistic effects, the wavefunction of the electron for atoms with {{math|'''''Z'' > 137'''}} is oscillatory and [[unbounded]]. The significance of element 137, also known as [[untriseptium]], was first pointed out by the physicist [[Richard Feynman]]. Element 137 is sometimes informally called [[feynmanium]] (symbol Fy) {{citation needed|date=February 2014}}. However, Feynman's approximation fails to predict the exact critical value of&nbsp;{{mvar|Z}} due to the non-point-charge nature of the nucleus and very small orbital radius of inner electrons, resulting in a potential seen by inner electrons which is effectively less than {{mvar|Z}}. The critical {{mvar|Z}}&nbsp;value which makes the atom unstable with regard to high-field breakdown of the vacuum and production of electron-positron pairs, does not occur until {{mvar|Z}} is about 173. These conditions are not seen except transiently in collisions of very heavy nuclei such as lead or uranium in accelerators, where such electron-positron production from these effects has been claimed to be observed. See [[Extension of the periodic table beyond the seventh period]].<br />
<br />
There are no nodes in relativistic orbital densities, although individual components of the wavefunction will have nodes.<ref>{{cite journal|doi=10.1021/ed046p678|title=Contour diagrams for relativistic orbitals|year=1969|last1=Szabo|first1=Attila|journal=Journal of Chemical Education|volume=46|issue=10|pages=678|bibcode = 1969JChEd..46..678S }}</ref><br />
<br />
== Transitions between orbitals ==<br />
{{main|Atomic electron transition}}<br />
Under quantum mechanics, each quantum state has a well-defined energy. When applied to atomic orbitals, this means that each state has a specific energy, and that if an electron is to move between states, the energy difference is also very fixed.<br />
<br />
Consider two states of the Hydrogen atom:<br />
<br />
State 1) {{math|1=''n'' = 1}}, {{math|1=''ℓ'' = 0}}, {{math|1=''m''<sub>''ℓ''</sub> = 0}} and {{math|1=''s'' = +}}{{frac|1|2}}<br />
<br />
State 2) {{math|1=''n'' = 2}}, {{math|1=''ℓ'' = 0}}, {{math|1=''m''<sub>''ℓ''</sub> = 0}} and {{math|1=''s'' = +}}{{frac|1|2}}<br />
<br />
By quantum theory, state&nbsp;1 has a fixed energy of {{math|''E''<sub>1</sub>}}, and state&nbsp;2 has a fixed energy of {{math|''E''<sub>2</sub>}}. Now, what would happen if an electron in state&nbsp;1 were to move to state&nbsp;2? For this to happen, the electron would need to gain an energy of exactly {{math|''E''<sub>2</sub> − ''E''<sub>1</sub>}}. If the electron receives energy that is less than or greater than this value, it cannot jump from state&nbsp;1 to state&nbsp;2. Now, suppose we irradiate the atom with a broad-spectrum of light. Photons that reach the atom that have an energy of exactly {{math|''E''<sub>2</sub> − ''E''<sub>1</sub>}} will be absorbed by the electron in state&nbsp;1, and that electron will jump to state&nbsp;2. However, photons that are greater or lower in energy cannot be absorbed by the electron, because the electron can only jump to one of the orbitals, it cannot jump to a state between orbitals. The result is that only photons of a specific frequency will be absorbed by the atom. This creates a line in the spectrum, known as an absorption line, which corresponds to the energy difference between states 1 and 2.<br />
<br />
The atomic orbital model thus predicts line spectra, which are observed experimentally. This is one of the main validations of the atomic orbital model.<br />
<br />
The atomic orbital model is nevertheless an approximation to the full quantum theory, which only recognizes many electron states. The predictions of line spectra are qualitatively useful but are not quantitatively accurate for atoms and ions other than those containing only one electron.<br />
<br />
==See also==<br />
* [[Atomic electron configuration table]]<br />
* [[Condensed matter physics]]<br />
* [[Electron configuration]]<br />
* [[Energy level]]<br />
* [[Hund's rules]]<br />
* [[Molecular orbital]]<br />
* [[Quantum chemistry]]<br />
* [[Quantum chemistry computer programs]]<br />
* [[Solid state physics]]<br />
* [[Orbital resonance]]<br />
* [[Wave function collapse]]<br />
<br />
==References==<br />
{{reflist|30em}}<br />
<br />
==Further reading==<br />
* {{Cite book| last1=Tipler | first1=Paul | first2=Ralph|last2= Llewellyn | year=2003 | title=Modern Physics | edition=4 | location=New York |publisher=W. H. Freeman and Company | isbn=0-7167-4345-0}}<br />
* {{Cite book| last=Scerri | first=Eric | year=2007 | title=The Periodic Table, Its Story and Its Significance |location=New York |publisher=Oxford University Press | isbn=978-0-19-530573-9}}<br />
* {{Cite book| last=Levine | first=Ira | year=2000 | title=Quantum Chemistry |location=Upper Saddle River, New Jersey |publisher=Prentice Hall | isbn=0-13-685512-1}}<br />
* {{Cite book| last=Griffiths | first=David | year=2000 | title=Introduction to Quantum Mechanics |publisher=Benjamin Cummings |edition=2 | isbn=978-0-13-111892-8}}<br />
* {{Cite journal | last1 = Cohen | first1 = Irwin | first2 = Thomas |last2=Bustard | title = Atomic Orbitals: Limitations and Variations | journal = J. Chem. Educ. | volume = 43 | page = 187 | year = 1966 | url = http://pubs.acs.org/doi/pdfplus/10.1021/ed043p187|bibcode = 1966JChEd..43..187C |doi = 10.1021/ed043p187 | issue = 4 }}<br />
<br />
==External links==<br />
* [http://www.chemguide.co.uk/atoms/properties/atomorbs.html Guide to atomic orbitals]<br />
* [http://wps.prenhall.com/wps/media/objects/602/616516/Chapter_07.html Covalent Bonds and Molecular Structure]<br />
* [http://strangepaths.com/atomic-orbital/2008/04/20/en/ Animation of the time evolution of an hydrogenic orbital]<br />
* [http://www.shef.ac.uk/chemistry/orbitron/ The Orbitron], a visualization of all common and uncommon atomic orbitals, from 1s to 7g<br />
* [http://www.orbitals.com/orb/orbtable.htm Grand table] Still images of many orbitals<br />
* David Manthey's [http://www.orbitals.com/orb/index.html Orbital Viewer] renders orbitals with ''n''&nbsp;≤&nbsp;30<br />
* [http://www.falstad.com/qmatom/ Java orbital viewer applet]<br />
* [http://www.hydrogenlab.de/elektronium/HTML/einleitung_hauptseite_uk.html What does an atom look like? Orbitals in 3D]<br />
* [http://taras-zavedy.narod.ru/PROGRAMMS/ATOM_ORBITALS_v_1_5_ENG/Atom_Orbitals_v_1_5_ENG.html Atom Orbitals v.1.5 visualization software]<br />
<br />
{{Atomic models}}<br />
<br />
{{DEFAULTSORT:Atomic Orbital}}<br />
[[Category:Atomic physics]]<br />
[[Category:Chemical bonding]]<br />
[[Category:Electron states]]<br />
[[Category:Quantum chemistry]]</div>Adminhttps://en.formulasearchengine.com/index.php?title=Adiabatic_process&diff=218562Adiabatic process2014-07-31T13:46:28Z<p>Admin: Reverted edits by MediaWiki spam cleanup (talk) to last revision by 5.249.53.64</p>
<hr />
<div>{{Thermodynamics|cTopic=[[Thermodynamic system|Systems]]}}<br />
An '''adiabatic process''' ({{IPAc-en|ˌ|æ|d|i|ə|ˈ|b|æ|t|ɪ|k}}; from the [[Greek language|Greek]] [[privative a|privative]] "a" + "diavaton") is a process that occurs without the transfer of [[heat]] or matter between a system and its surroundings.<ref>[[Constantin Carathéodory|Carathéodory, C.]] (1909). Untersuchungen über die Grundlagen der Thermodynamik, ''Mathematische Annalen'', '''67''': 355–386, {{doi|10.1007/BF01450409}}. A translation may be found [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf here]. Also a mostly reliable [http://books.google.com.au/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations translation is to be found] at Kestin, J. (1976). ''The Second Law of Thermodynamics'', Dowden, Hutchinson & Ross, Stroudsburg PA.</ref><ref>Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics Press, New York, ISBN 0-88318-797-3, p. 21.</ref> A key concept in [[thermodynamics]], adiabatic transfer provides a rigorous conceptual basis for the theory used to expound the [[first law of thermodynamics]]. It is also key in a practical sense, that many rapid chemical and physical processes are described using the adiabatic approximation; such processes are usually followed or preceded by events that do involve heat transfer.<br />
<br />
Adiabatic processes are primarily and exactly defined for a system contained by walls that are completely thermally insulating and impermeable to matter; such walls are said to be [[Adiabatic enclosure|adiabatic]]. An '''adiabatic transfer''' is a transfer of energy as work across an adiabatic wall or sector of a boundary.<br />
<br />
Approximately, a transfer may be regarded as adiabatic if it happens in an extremely short time, so that there is no opportunity for significant heat exchange.<ref>http://buphy.bu.edu/~duffy/semester1/c27_process_adiabatic_sim.html</ref><br />
<br />
The [[adiabatic flame temperature]] is a virtual quantity. It is the temperature that would be achieved by a [[fire|flame]] in the absence of heat loss to the surroundings.<br />
<br />
==Etymology==<br />
<br />
The term ''adiabatic'' literally means 'not to be passed'. It is formed from the [[privative a|privative]] "α" ("not") + ''διαβατός'', "able to be passed through", in turn deriving from ''διὰ-'' ("through"), and ''βαῖνειν'' ("to pass"), thus ''ἀδιάβατος ''.<ref>Liddell, H.G., Scott, R. (1940). ''A Greek-English Lexicon'', Clarendon Press, Oxford UK.</ref> According to [[James Clerk Maxwell|Maxwell]], the term was introduced by [[William John Macquorn Rankine|Rankine]].<ref><br />
{{Citation<br />
| last = Maxwell<br />
| first = J.C.<br />
| author-link = James Clerk Maxwell<br />
| last2 =<br />
| first2 =<br />
| author2-link =<br />
| other =<br />
| title = Theory of Heat<br />
| place = London<br />
| publisher = [[Longman|Longmans, Green and Co.]]<br />
| series =<br />
| volume =<br />
| origyear =<br />
| year = 1871<br />
| month=<br />
| edition = first<br />
| page = 129<br />
| language =<br />
| url = http://archive.org/details/theoryheat04maxwgoog <br />
| archiveurl =<br />
| archivedate =<br />
| doi =<br />
| id =<br />
| isbn =<br />
}}</ref><ref>Rankine, W.J.McQ. (1866). On the theory of explosive gas engines, ''The Engineeer'', July 27, 1866; at page 467 of the reprint in ''[https://archive.org/details/miscellaneoussci00rank Miscellaneous Scientific Papers]'', edited by W.J. Millar, 1881, Charles Griffin, London.</ref><br />
<br />
The etymological origin corresponds here to an impossibility of [[Heat|transfer of energy as heat]] and of transfer of matter across the wall.<br />
<br />
==Description==<br />
<br />
An adiabatic transfer of energy as work may be described by the notation {{math|''Q'' {{=}} 0}} where {{math|''Q''}} is the quantity of energy transferred as heat across the adiabatic boundary or wall.<br />
<br />
An ideal or fictive adiabatic transfer of energy as work that occurs without friction or viscous dissipation within the system is said to be [[Isentropic process|isentropic]], with {{math|Δ''S'' {{=}} 0}}.<br />
<br />
For a natural process of transfer of energy as heat, driven by a finite temperature difference, entropy is both transferred with the heat and generated within the system. Such a process is in general neither adiabatic nor isentropic, having {{math|''Q'' ≠ 0}} and {{math|Δ''S'' ≠ 0}}.<br />
<br />
For a general fictive quasi-static transfer of energy as heat, driven by an ideally infinitesimal temperature difference, the [[second law of thermodynamics]] provides that {{math|δ''Q'' {{=}} ''T'' d<sub>e</sub>''S''}}, where {{math|δ''Q''}} denotes an infinitesimal element of transfer of energy as heat into the system from its surroundings, {{math|''T''}} denotes the practically common temperature of system and surroundings at which the transfer takes place, and {{math|d<sub>e</sub>''S''}} denotes the infinitesimal element of entropy transferred into the system from the surroundings with the heat transfer. For an adiabatic fictive quasi-static process, {{math|δ''Q'' {{=}} 0}} and {{math|d<sub>e</sub>''S'' {{=}} 0}}.<br />
<br />
For a natural process of transfer of energy as heat, driven by a finite temperature difference, there is generation of entropy within the system, in addition to entropy that is transferred into the system from the surroundings. If the process is fairly slow, so that it can be described near enough by differentials, the second law of thermodynamics observes that {{math|δ''Q'' < ''T'' d''S''}}. Here {{math|''T''}} denotes the temperature of the system to which heat is transferred. Entropy {{math|d<sub>i</sub>''S''}} is thereby generated internally within the system, in addition to the entropy {{math|d<sub>e</sub>''S''}} transferred with the heat. Thus the total entropy increment within the system is given by {{math|d''S'' {{=}} d<sub>i</sub>''S'' + d<sub>e</sub>''S''}}.<ref>Kondepudi, D., [[Ilya Prigogine|Prigogine, I.]] (1998). ''Modern Thermodynamics: From Heat Engines to Dissipative Structures'', John Wiley & Sons, Chichester, ISBN 0–471–97393–9, p. 88.</ref><br />
<br />
A natural adiabatic process is irreversible and is not isentropic. Adiabatic transfer of energy as work can be analyzed into two extreme component kinds. One extreme kind is without friction or viscous dissipation within the system, and this is usually pressure-volume work, denoted customarily by {{math|''P'' d''V''}}. This is an ideal case that does not exactly occur in nature. It may be regarded as "reversible". The other extreme kind is isochoric work, for which {{math|d''V'' {{=}} 0}}, solely through friction or viscous dissipation within the system. Isochoric work is irreversible.<ref>Münster, A. (1970), ''Classical Thermodynamics'', translated by E.S. Halberstadt, Wiley–Interscience, London, ISBN 0-471-62430-6, p. 45.</ref> The second law of thermodynamics observes that a natural process of transfer of energy as work, exactly considered, always consists at least of isochoric work and often of both of these extreme kinds of work. Every natural process, exactly considered, is irreversible, however slight may be the friction or viscosity.<br />
<br />
==Adiabatic heating and cooling==<!-- This section is linked from [[Water]] --><br />
Adiabatic changes in temperature occur due to changes in [[pressure]] of a [[gas]] while not adding or subtracting any [[heat]]. In contrast, [[free expansion]] is an [[isothermal]] process for an ideal gas.<br />
<br />
'''Adiabatic heat''' occurs when the pressure of a gas is increased from work done on it by its surroundings, e.g., a [[piston]] compressing a gas contained within an adiabatic cylinder. This finds practical application in [[Diesel engines]] which rely on the lack of quick heat dissipation during their compression stroke to elevate the fuel vapor temperature sufficiently to ignite it.<br />
<br />
Adiabatic heating also occurs in the [[Earth's atmosphere]] when an [[air mass]] descends, for example, in a [[katabatic wind]] or [[Foehn wind|Foehn]] or [[chinook wind|chinook]] wind flowing downhill over a mountain range. When a parcel of air descends, the pressure on the parcel increases. Due to this increase in pressure, the parcel's volume decreases and its temperature increases, thus increasing the internal energy.<br />
<br />
'''Adiabatic cooling''' occurs when the pressure of a substance is decreased as it does work on its surroundings. Adiabatic cooling occurs in the [[Earth's atmosphere]] with [[orographic lifting]] and [[lee waves]], and this can form [[Pileus (meteorology)|pileus]] or [[lenticular cloud]]s if the air is cooled below the [[dew point]]. When the pressure applied on a parcel of air decreases, the air in the parcel is allowed to expand; as the volume increases, the temperature falls and internal energy decreases.<br />
<br />
Adiabatic cooling does not have to involve a fluid. One technique used to reach very low temperatures (thousandths and even millionths of a degree above absolute zero) is [[adiabatic demagnetization|adiabatic demagnetisation]], where the change in [[magnetic field]] on a magnetic material is used to provide adiabatic cooling. Also, the contents of an [[expanding universe]] (to first order) can be described as an adiabatically cooling fluid. ''(See - [[Heat death of the universe]])''<br />
<br />
Rising magma also undergoes adiabatic cooling before eruption, particularly significant in the case of magmas that rise quickly from great depths such as [[kimberlite]]s.<ref name="Kavanagh">{{cite journal|last=Kavanagh|first=J.L.|author2=Sparks R.S.J.|year=2009|title=Temperature changes in ascending kimberlite magmas|journal=Earth and Planetary Science Letters|publisher=[[Elsevier]]|volume=286|issue=3&ndash;4|pages=404&ndash;413|doi=10.1016/j.epsl.2009.07.011|url=http://monash.academia.edu/JanineKavanagh/Papers/114092/Temperature_changes_in_ascending_kimberlite_magma|accessdate=18 February 2012|bibcode = 2009E&PSL.286..404K }}</ref><br />
<br />
Such temperature changes can be quantified using the [[ideal gas law]], or the [[hydrostatic equation]] for atmospheric processes.<br />
<br />
In practice, no process is truly adiabatic. Many processes rely on a large difference in time scales of the process of interest and the rate of heat dissipation across a system boundary, and thus are approximated by using an adiabatic assumption. There is always some heat loss, as no perfect insulators exist.<br />
<br />
==Ideal gas (reversible process)==<br />
{{main|Reversible adiabatic process}}<br />
<br />
[[Image:Adiabatic.svg|thumb|341px|For a simple substance, during an adiabatic process in which the volume increases, the [[internal energy]] of the working substance must decrease]]<br />
The mathematical equation for an [[ideal gas]] undergoing a reversible (i.e., no entropy generation) adiabatic process is<br />
: <math> P V^{\gamma} = \operatorname{constant} \qquad </math><br />
where ''P'' is pressure, ''V'' is volume, and<br />
: <math> \gamma = {C_{P} \over C_{V}} = \frac{f + 2}{f}, </math><br />
<math> C_{P} </math> being the [[specific heat]] for constant pressure,<br />
<math> C_{V} </math> being the specific heat for constant volume, <math> \gamma </math> is the [[adiabatic index]], and <math> f </math> is the number of [[Degrees of freedom (physics and chemistry)|degrees of freedom]] (3 for monatomic gas, 5 for diatomic gas and collinear molecules e.g. carbon dioxide).<br />
<br />
For a monatomic ideal gas, <math> \gamma = 5/3 \,</math>, and for a diatomic gas (such as [[nitrogen]] and [[oxygen]], the main components of [[Earth's atmosphere|air]]) <math> \gamma = 7/5 \,</math>.<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/adiab.html Adiabatic Processes]</ref> Note that the above formula is only applicable to classical ideal gases and not [[Bose–Einstein condensate|Bose–Einstein]] or [[Fermionic condensate|Fermi gases]].<br />
<br />
For reversible adiabatic processes, it is also true that<br />
<br />
: <math> P^{1-\gamma}T^{\gamma}= \operatorname{constant}</math><br />
<br />
: <math> VT^{f/2} = \operatorname{constant} </math><br />
<br />
where ''T'' is an absolute temperature.<br />
<br />
This can also be written as<br />
<br />
: <math> TV^{\gamma - 1} = \operatorname{constant} </math><br />
<br />
===Example of adiabatic compression===<br />
Let's now look at a common example of adiabatic compression- the compression stroke in a [[gasoline engine]]. We will make a few simplifying assumptions: that the uncompressed volume of the cylinder is 1000 cm3 (one liter), that the gas within is nearly pure nitrogen (thus a diatomic gas with five degrees of freedom and so <math>\gamma </math> = 7/5), and that the compression ratio of the engine is 10:1 (that is, the 1000 cm3 volume of uncompressed gas will compress down to 100 cm3 when the piston goes from bottom to top). The uncompressed gas is at approximately room temperature and pressure (a warm room temperature of ~27 ºC or 300 K, and a pressure of 1 bar ~ 100 kPa, or about 14.7 PSI, or typical sea-level atmospheric pressure).<br />
<br />
<math> P V^{\gamma} = \operatorname{constant} = 100,000 \operatorname{pa} * 1000^{7/5} = 100 \times 10^3 * 15.8 \times 10^3 = 1.58 \times 10^9 </math><br />
<br />
so our adiabatic constant for this experiment is about 1.58 billion.<br />
<br />
The gas is now compressed to a 100cc volume (we will assume this happens quickly enough that no heat can enter or leave the gas). The new volume is 100 ccs, but the constant for this experiment is still 1.58 billion:<br />
<br />
<math> P * V^{\gamma} = \operatorname{constant} = 1.58 \times 10^9 = P * 100^{7/5} </math><br />
<br />
so solving for P:<br />
<br />
<math> P = 1.58 \times 10^9 / {100^{7/5}} = 1.58 \times 10^9 / 630.9 = 2.50 \times 10^6 \operatorname{ Pa} </math><br />
<br />
or about 362 PSI or 24.5 atm. Note that this pressure increase is more than a simple 10:1 compression ratio would indicate; this is because the gas is not only compressed, but the work done to compress the gas has also heated the gas and the hotter gas will have a greater pressure even if the volume had not changed.<br />
<br />
We can solve for the temperature of the compressed gas in the engine cylinder as well, using the ideal gas law.<br />
Our initial conditions are 100,000 pa of pressure, 1000 cc volume, and 300 K of temperature, so our experimental constant is:<br />
<br />
<math> {P V \over T} = \operatorname {constant} = {{10^5 * 10^3 } \over {300} } = 3.33 \times 10^5 </math><br />
<br />
We know the compressed gas has V = 100 cc and P = 2.50E6 pascals, so we can solve for temperature by simple algebra:<br />
<br />
<math> {P V \over {\operatorname{constant}}} = T = {{2.50 \times 10^6 * 100} \over {3.33 \times 10^5}} = 751 </math><br />
<br />
That's a final temperature of 751 K, or 477 °C, or 892 °F, well above the ignition point of many fuels. This is why a high compression engine requires fuels specially formulated to not self-ignite (which would cause [[engine knocking]] when operated under these conditions of temperature and pressure), or that a [[supercharger]] and [[inter cooler]] to provide a lower temperature at the same pressure would be advantageous. A [[diesel engine]] operates under even more extreme conditions, with compression ratios of 20:1 or more being typical, in order to provide a very high gas temperature which ensures immediate ignition of injected fuel.<br />
<br />
===Adiabatic free expansion of a gas===<br />
{{See also|Free expansion}}<br />
For an adiabatic free expansion of an ideal gas, the gas is contained in an insulated container and then allowed to expand in a vacuum. Because there is no external pressure for the gas to expand against, the work done by or on the system is zero. Since this process does not involve any heat transfer or work, the First Law of Thermodynamics then implies that the net internal energy change of the system is zero. For an ideal gas, the temperature remains constant because the internal energy only depends on temperature in that case. Since at constant temperature, the entropy is proportional to the volume, the entropy increases in this case, therefore this process is irreversible.<br />
<br />
===Derivation of continuous formula for adiabatic heating and cooling===<br />
The definition of an adiabatic process is that heat transfer to the system is zero, <math>\delta Q=0 </math>. Then, according to the [[first law of thermodynamics]],<br />
<br />
:<math> \text{(1)} \qquad d U + \delta W = \delta Q = 0, </math><br />
<br />
where ''dU'' is the change in the internal energy of the system and ''δW'' is work done<br />
''by'' the system. Any work (''δW'') done must be done at the expense of internal energy ''U'', since no heat ''δQ'' is being supplied from the surroundings. Pressure-volume work ''δW'' done ''by'' the system is defined as<br />
<br />
:<math> \text{(2)} \qquad \delta W = P \, dV. </math><br />
<br />
However, ''P'' does not remain constant during an adiabatic process but<br />
instead changes along with ''V''.<br />
<br />
It is desired to know how the values of ''dP'' and<br />
''dV'' relate to each other as the adiabatic process proceeds.<br />
For an ideal gas the internal energy is given by<br />
<br />
:<math> \text{(3)} \qquad U = \alpha n R T, </math><br />
<br />
where <big><math>{\alpha}</math></big> is the number of [[Degrees of freedom (physics and chemistry)|degrees of freedom]] divided by two, ''R'' is the [[universal gas constant]] and ''n'' is the number of moles in the system (a constant).<br />
<br />
Differentiating Equation (3) and use of the [[ideal gas law]], <math>P V = n R T</math>, yields<br />
<br />
:<math> \text{(4)} \qquad d U = \alpha n R \, dT<br />
= \alpha \, d (P V)<br />
= \alpha (P \, dV + V \, dP). </math><br />
<br />
Equation (4) is often expressed as <math> d U = n C_{V} \, d T </math><br />
because <math> C_{V} = \alpha R </math>.<br />
<br />
Now substitute equations (2) and (4) into equation (1) to obtain<br />
<br />
: <math> -P \, dV = \alpha P \, dV + \alpha V \, dP,</math><br />
<br />
factorize :<math> -P \, dV,</math>:<br />
<br />
: <math> - (\alpha + 1) P \, dV = \alpha V \, dP,</math><br />
<br />
and divide both sides by ''PV'':<br />
<br />
: <math> -(\alpha + 1) {d V \over V} = \alpha {d P \over P}. </math><br />
<br />
After integrating the left and right sides from <math>V_0</math> to V and from <math>P_0</math> to P and changing the sides respectively,<br />
<br />
: <math> \ln \left( {P \over P_0} \right) = {-{\alpha + 1 \over \alpha}} \ln \left( {V \over V_0} \right). </math><br />
<br />
Exponentiate both sides, and substitute <math>{\alpha + 1 \over \alpha}</math> with <math>\gamma</math>, the [[heat capacity ratio]]<br />
<br />
: <math> \left( {P \over P_0} \right) = \left( {V \over V_0} \right)^{-{\gamma}}, </math><br />
<br />
and eliminate the negative sign to obtain<br />
<br />
: <math> \left( {P \over P_0} \right) = \left( {V_0 \over V} \right)^{\gamma}. </math><br />
<br />
Therefore,<br />
<br />
: <math> \left( {P \over P_0} \right) \left( {V \over V_0} \right)^{\gamma} = 1</math><br />
<br />
and<br />
<br />
: <math> P_0 V_0^{\gamma} = P V^\gamma = \operatorname{constant}. </math><br />
<br />
===Derivation of discrete formula===<br />
The change in internal energy of a system, measured from state 1 to state 2, is equal to<br />
<br />
:<math> \text{(1)} \qquad \Delta U = \alpha R nT_2 - \alpha R nT_1 = \alpha Rn \Delta T </math><br />
<br />
At the same time, the work done by the pressure-volume changes as a result from this process, is equal to<br />
<br />
:<math> \text{(2)} \qquad W = \int_{V_1}^{V_2}P\, dV </math><br />
<br />
Since we require the process to be adiabatic, the following equation needs to be true<br />
<br />
:<math> \text{(3)} \qquad \Delta U + W = 0 </math><br />
<br />
By the previous derivation,<br />
<br />
:<math> \text{(4)} \qquad P V^\gamma = \text{constant} = P_1 V_1^\gamma </math><br />
<br />
Rearranging (4) gives<br />
<br />
:<math> P = P_1 \left(\frac{V_1}{V} \right)^\gamma </math><br />
<br />
Substituting this into (2) gives<br />
<br />
:<math> W = \int_{V_1}^{V_2}P_1 \left(\frac{V_1}{V} \right)^\gamma\, dV </math><br />
<br />
Integrating,<br />
<br />
:<math> W = P_1 V_1^\gamma \frac{V_2^{1-\gamma}-V_1^{1-\gamma}}{1-\gamma} </math><br />
<br />
Substituting <math> \gamma = \frac{\alpha+1}{\alpha} </math>,<br />
<br />
:<math> W = - \alpha P_1 V_1^{\gamma} \left( V_2^{1-\gamma} - V_1^{1-\gamma} \right) </math><br />
<br />
Rearranging,<br />
<br />
:<math> W = - \alpha P_1 V_1 \left( \left( \frac{V_2}{V_1} \right)^{1-\gamma} - 1 \right) </math><br />
<br />
Using the ideal gas law and assuming a constant molar quantity (as often happens in practical cases),<br />
<br />
:<math> W = - \alpha n R T_1 \left( \left( \frac{V_2}{V_1} \right)^{1-\gamma} - 1 \right) </math><br />
<br />
By the continuous formula,<br />
<br />
:<math> \frac{P_2}{P_1}=\left(\frac{V_2}{V_1}\right)^{-\gamma} </math><br />
<br />
Or,<br />
<br />
:<math> \left(\frac{P_2}{P_1}\right)^{-1 \over \gamma}=\frac{V_2}{V_1} </math><br />
<br />
Substituting into the previous expression for <math> W </math>,<br />
<br />
:<math> W = - \alpha n R T_1 \left( \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1 \right) </math><br />
<br />
Substituting this expression and (1) in (3) gives<br />
<br />
:<math> \alpha n R (T_2 - T_1) = \alpha n R T_1 \left( \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1 \right) </math><br />
<br />
Simplifying,<br />
<br />
:<math> T_2 - T_1 = T_1 \left( \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1 \right) </math><br />
<br />
Simplifying,<br />
<br />
:<math> \frac{T_2}{T_1}-1 = \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1 </math><br />
<br />
Simplifying,<br />
<br />
:<math> T_2 = T_1 \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} </math><br />
<br />
==Graphing adiabats==<br />
An adiabat is a curve of constant [[entropy]] on the P-V diagram. Properties of adiabats on a P-V diagram are:<br />
# Every adiabat [[Asymptote|asymptotically approaches]] both the V axis and the P axis (just like [[isotherms]]).<br />
# Each adiabat intersects each isotherm exactly once.<br />
# An adiabat looks similar to an isotherm, except that during an expansion, an adiabat loses more pressure than an isotherm, so it has a steeper inclination (more vertical).<br />
# If isotherms are concave towards the "north-east" direction (45 °), then adiabats are concave towards the "east north-east" (31 °).<br />
# If adiabats and isotherms are graphed severally at regular changes of entropy and temperature, respectively (like altitude on a contour map), then as the eye moves towards the axes (towards the south-west), it sees the density of isotherms stay constant, but it sees the density of adiabats grow. The exception is very near absolute zero, where the density of adiabats drops sharply and they become rare (see [[Nernst's theorem]]).<br />
<br />
The following diagram is a P-V diagram with a superposition of adiabats and isotherms:<br />
<br />
[[Image:Entropyandtemp.PNG]]<br />
<br />
The isotherms are the red curves and the adiabats are the black curves.<br />
<br />
The adiabats are isentropic.<br />
<br />
Volume is the horizontal axis and pressure is the vertical axis.<br />
<br />
==See also==<br />
* [[Cyclic process]]<br />
* [[First law of thermodynamics]]<br />
* [[Heat burst]]<br />
* [[Isobaric process]]<br />
* [[Isenthalpic process]]<br />
* [[Isentropic process]]<br />
* [[Isochoric process]]<br />
* [[Isothermal process]]<br />
* [[Polytropic process]]<br />
* [[Entropy (classical thermodynamics)]]<br />
* [[Quasistatic equilibrium]]<br />
* [[Total air temperature]]<br />
* [[Adiabatic engine]]<br />
* [[Magnetic refrigeration]]<br />
<br />
==References==<br />
{{reflist|2}}<br />
* {{cite book |first=Robert J. |last=Silbey |last2=''et al.'' |year=2004 |title=Physical chemistry |location=Hoboken |publisher=Wiley |page=55 |isbn=978-0-471-21504-2 }}<br />
* Broholm, Collin. "Adiabatic free expansion." Physics & Astronomy @ Johns Hopkins University. N.p., 26 Nov. 1997. Web. 14 Apr. *Nave, Carl Rod. "Adiabatic Processes." HyperPhysics. N.p., n.d. Web. 14 Apr. 2011. [http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/adiab.html].<br />
* Thorngren, Dr. Jane R.. "Adiabatic Processes." Daphne – A Palomar College Web Server. N.p., 21 July 1995. Web. 14 Apr. 2011. [http://daphne.palomar.edu/jthorngren/adiabatic_processes.htm].<br />
<br />
==External links==<br />
* [http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/adiab.html#c1: Article in HyperPhysics Encyclopaedia]<br />
<br />
{{DEFAULTSORT:Adiabatic Process}}<br />
[[Category:Thermodynamic processes]]<br />
[[Category:Atmospheric thermodynamics]]<br />
<br />
{{Link GA|ru}}</div>Adminhttps://en.formulasearchengine.com/index.php?title=Advanced_Encryption_Standard&diff=218548Advanced Encryption Standard2014-07-31T13:44:47Z<p>Admin: Reverted edits by MediaWiki spam cleanup (talk) to last revision by Trappist the monk</p>
<hr />
<div>{{Infobox block cipher<br />
| name = Advanced Encryption Standard<br/>(Rijndael)<br />
| image = [[Image:AES-SubBytes.svg|280px|center]]<br />
| caption = The <tt>SubBytes</tt> step, one of four stages in a round of AES<br />
| designers = [[Vincent Rijmen]], [[Joan Daemen]]<br />
| publish date = 1998<br />
| derived from = [[Square (cipher)|Square]]<br />
| derived to = [[Anubis (cipher)|Anubis]], [[Grand Cru (cipher)|Grand Cru]]<br />
| related to =<br />
| certification = [[Advanced Encryption Standard process|AES]] winner, [[CRYPTREC]], [[NESSIE]], [[National Security Agency|NSA]]<br />
| key size = 128, 192 or 256 bits<ref name="keysize">Key sizes of 128, 160, 192, 224, and 256 bits are supported by the Rijndael algorithm, but only the 128, 192, and 256-bit key sizes are specified in the AES standard.</ref><br />
| block size = 128 bits<ref name="blocksize">Block sizes of 128, 160, 192, 224, and 256 bits are supported by the Rijndael algorithm, but only the 128-bit block size is specified in the AES standard.</ref><br />
| structure = [[Substitution-permutation network]]<br />
| rounds = 10, 12 or 14 (depending on key size)<br />
| cryptanalysis = Attacks have been published that are computationally faster than a full [[brute force attack]], though none as of 2013 are computationally feasible:<ref name="aesbc">{{cite web|url=http://research.microsoft.com/en-us/projects/cryptanalysis/aesbc.pdf|title=Biclique Cryptanalysis of the Full AES|accessdate=July 23, 2013}}</ref><br />
<br />
For AES-128, the key can be recovered with a computational complexity of 2<sup>126.1</sup> using [[biclique]]s. For biclique attacks on AES-192 and AES-256, the computational complexities of 2<sup>189.7</sup> and 2<sup>254.4</sup> respectively apply. [[Related-key attack]]s can break AES-192 and AES-256 with complexities 2<sup>176</sup> and 2<sup>99.5</sup>, respectively.<br />
}}<br />
<br />
The '''Advanced Encryption Standard''' ('''AES''') is a specification for the [[encryption]] of electronic data established by the U.S. [[National Institute of Standards and Technology]] (NIST) in 2001.<ref name="fips-197">{{cite web|url=http://csrc.nist.gov/publications/fips/fips197/fips-197.pdf|title=Announcing the ADVANCED ENCRYPTION STANDARD (AES)|publisher=United States National Institute of Standards and Technology (NIST)|work=Federal Information Processing Standards Publication 197|date=November 26, 2001|accessdate=October 2, 2012}}</ref> It is based on the '''Rijndael''' [[cipher]]<ref>{{cite web |url=http://csrc.nist.gov/archive/aes/rijndael/Rijndael-ammended.pdf#page=1 |title=AES Proposal: Rijndael |last1=Daemen |first1=Joan |last2=Rijmen |first2=Vincent |date=9/04/2003 |publisher=National Institute of Standards and Technology |page=1 |accessdate=21 February 2013}}</ref> developed by two [[Belgium|Belgian]] cryptographers, [[Joan Daemen]] and [[Vincent Rijmen]], who submitted a proposal to NIST during the AES selection process.<ref>{{Cite news |title=U.S. Selects a New Encryption Technique |author=John Schwartz |newspaper=New York Times |date=October 3, 2000 |url=http://www.nytimes.com/2000/10/03/business/technology-us-selects-a-new-encryption-technique.html }}</ref> Rijndael is a family of ciphers with different key and block sizes. For AES, NIST selected three members of the Rijndael family, each with a block size of 128 bits, but three different key lengths: 128, 192 and 256 bits.<br />
<br />
AES has been adopted by the [[Federal government of the United States|U.S. government]] and is now used worldwide. It supersedes the [[Data Encryption Standard]] (DES),<ref>{{cite news |url=http://www.findarticles.com/p/articles/mi_m0IKZ/is_3_107?pnum=2&opg=90984479 |title=NIST reports measurable success of Advanced Encryption Standard | work=Journal of Research of the National Institute of Standards and Technology | first=Harold B. | last=Westlund | year=2002}}</ref> which was published in 1977. The algorithm described by AES is a [[symmetric-key algorithm]], meaning the same key is used for both encrypting and decrypting the data.<br />
<br />
In the [[United States]], AES was announced by the NIST as U.S. [[Federal Information Processing Standard|FIPS]] PUB 197 (FIPS 197) on November 26, 2001.<ref name="fips-197" /> This announcement followed a five-year standardization process in which fifteen competing designs were presented and evaluated, before the Rijndael cipher was selected as the most suitable (see [[Advanced Encryption Standard process]] for more details). It became effective as a federal government standard on May 26, 2002 after approval by the [[United States Secretary of Commerce|Secretary of Commerce]]. AES is included in the ISO/IEC 18033-3 standard. AES is available in many different encryption packages, and is the first publicly accessible and open [[cipher]] approved by the [[National Security Agency]] (NSA) for [[Classified information|top secret]] information when used in an NSA approved cryptographic module (see [[Advanced Encryption Standard#Security|Security of AES]], below).<br />
<br />
The name ''Rijndael'' ({{IPA-nl|ˈrɛindaːl}}) is a play on the names of the two inventors (Joan Daemen and Vincent Rijmen). <br />
<br />
== Description of the cipher ==<br />
AES is based on a design principle known as a substitution-permutation network, and is fast in both software and hardware.<ref>{{cite web |url=http://www.schneier.com/paper-twofish-final.pdf |title=The Twofish Team's Final Comments on AES Selection |author=Bruce Schneier, John Kelsey, Doug Whiting, David Wagner, Chris Hall, Niels Ferguson, Tadayoshi Kohno, Mike Stay |date=May 2000}}</ref> Unlike its predecessor DES, AES does not use a [[Feistel network]]. AES is a variant of Rijndael which has a fixed [[block size (cryptography)|block size]] of 128 [[bit]]s, and a [[key size]] of 128, 192, or 256 bits. By contrast, the Rijndael specification ''per se'' is specified with block and key sizes that may be any multiple of 32 bits, both with a minimum of 128 and a maximum of 256 bits.<br />
<br />
AES operates on a 4×4 [[column-major order]] matrix of bytes, termed the ''state'', although some versions of Rijndael have a larger block size and have additional columns in the state. Most AES calculations are done in a special [[Finite field arithmetic|finite field]].<br />
<br />
The key size used for an AES cipher specifies the number of repetitions of transformation rounds that convert the input, called the plaintext, into the final output, called the ciphertext. The number of cycles of repetition are as follows:<br />
<br />
* 10 cycles of repetition for 128-bit keys.<br />
* 12 cycles of repetition for 192-bit keys.<br />
* 14 cycles of repetition for 256-bit keys.<br />
<br />
Each round consists of several processing steps, each containing four similar but different stages, including one that depends on the encryption key itself. A set of reverse rounds are applied to transform ciphertext back into the original plaintext using the same encryption key.<br />
<br />
=== High-level description of the algorithm ===<br />
<br />
# KeyExpansion—round keys are derived from the cipher key using [[Rijndael key schedule|Rijndael's key schedule]]. AES requires a separate 128-bit round key block for each round plus one more.<br />
# InitialRound<br />
## <tt>AddRoundKey</tt>—each byte of the state is combined with a block of the round key using bitwise xor.<br />
# Rounds<br />
## <tt>SubBytes</tt>—a non-linear substitution step where each byte is replaced with another according to a [[Rijndael S-box|lookup table]].<br />
## <tt>ShiftRows</tt>—a transposition step where the last three rows of the state are shifted cyclically a certain number of steps.<br />
## <tt>MixColumns</tt>—a mixing operation which operates on the columns of the state, combining the four bytes in each column.<br />
## <tt>AddRoundKey</tt><br />
# Final Round (no <tt>MixColumns</tt>)<br />
## <tt>SubBytes</tt><br />
## <tt>ShiftRows</tt><br />
## <tt>AddRoundKey</tt>.<br />
<br />
=== The <tt>SubBytes</tt> step ===<br />
[[Image:AES-SubBytes.svg|right|320px|thumbnail|In the <tt>SubBytes</tt> step, each byte in the state is replaced with its entry in a fixed 8-bit lookup table, ''S''; ''b<sub>ij</sub>'' = ''S(a<sub>ij</sub>)''.]]<br />
In the <tt>SubBytes</tt> step, each byte <math>a_{i,j}</math> in the ''state'' matrix is replaced with a <tt>SubByte</tt> <math>S(a_{i,j})</math> using an 8-bit [[substitution box]], the [[Rijndael S-box]]. This operation provides the non-linearity in the [[cipher]]. The S-box used is derived from the [[multiplicative inverse]] over '''[[Finite field|GF]]'''(''2<sup>8</sup>''), known to have good non-linearity properties. To avoid attacks based on simple algebraic properties, the S-box is constructed by combining the inverse function with an invertible [[affine transformation]]. The S-box is also chosen to avoid any fixed points (and so is a [[derangement]]), i.e., <math> S(a_{i,j}) \neq a_{i,j} </math>, and also any opposite fixed points, i.e., <math> S(a_{i,j}) \oplus a_{i,j} \neq \text{0xFF} </math>.<br />
While performing the decryption, Inverse SubBytes step is used, which requires first taking the affine transformation and then finding the multiplicative inverse (just reversing the steps used in SubBytes step).<br />
<br />
=== The <tt>ShiftRows</tt> step ===<br />
[[Image:AES-ShiftRows.svg|right|320px|thumbnail|In the <tt>ShiftRows</tt> step, bytes in each row of the state are shifted cyclically to the left. The number of places each byte is shifted differs for each row.]]<br />
The <tt>ShiftRows</tt> step operates on the rows of the state; it cyclically shifts the bytes in each row by a certain [[Offset (computer science)|offset]]. For AES, the first row is left unchanged. Each byte of the second row is shifted one to the left. Similarly, the third and fourth rows are shifted by offsets of two and three respectively. For blocks of sizes 128 bits and 192 bits, the shifting pattern is the same. Row n is shifted left circular by n-1 bytes. In this way, each column of the output state of the <tt>ShiftRows</tt> step is composed of bytes from each column of the input state. (Rijndael variants with a larger block size have slightly different offsets). For a 256-bit block, the first row is unchanged and the shifting for the second, third and fourth row is 1 byte, 3 bytes and 4 bytes respectively—this change only applies for the Rijndael cipher when used with a 256-bit block, as AES does not use 256-bit blocks. The importance of this step is to avoid the columns being linearly independent, in which case, AES degenerates into four independent block ciphers.<br />
<br />
=== The <tt>MixColumns</tt> step ===<br />
[[Image:AES-MixColumns.svg|right|320px|thumbnail|In the <tt>MixColumns</tt> step, each column of the state is multiplied with a fixed polynomial ''c(x)''.]]<br />
In the <tt>MixColumns</tt> step, the four bytes of each column of the state are combined using an invertible [[linear transformation]]. The <tt>MixColumns</tt> function takes four bytes as input and outputs four bytes, where each input byte affects all four output bytes. Together with <tt>ShiftRows</tt>, <tt>MixColumns</tt> provides [[diffusion (cryptography)|diffusion]] in the cipher.<br />
<br />
During this operation, each column is multiplied by a fixed matrix:<br />
<br />
::<math><br />
\begin{bmatrix}<br />
2 & 3 & 1 & 1 \\<br />
1 & 2 & 3 & 1 \\<br />
1 & 1 & 2 & 3 \\<br />
3 & 1 & 1 & 2<br />
\end{bmatrix}<br />
</math><br />
<br />
Matrix multiplication is composed of multiplication and addition of the entries, and here the multiplication operation can be defined as this: multiplication by 1 means no change, multiplication by 2 means shifting to the left, and multiplication by 3 means shifting to the left and then performing [[Exclusive or|XOR]] with the initial unshifted value. After shifting, a conditional [[Exclusive or|XOR]] with 0x1B should be performed if the shifted value is larger than 0xFF. (These are special cases of the usual multiplication in '''GF'''(''2<sup>8</sup>'').) Addition is simply XOR.<br />
<br />
In more general sense, each column is treated as a polynomial over '''GF'''(''2<sup>8</sup>'') and is then multiplied modulo x<sup>4</sup>+1 with a fixed polynomial c(x) = 0x03 · x<sup>3</sup> + x<sup>2</sup> + x + 0x02. The coefficients are displayed in their [[hexadecimal]] equivalent of the binary representation of bit polynomials from '''GF'''(2)[x]. The <tt>MixColumns</tt> step can also be viewed as a multiplication by the shown particular [[MDS matrix]] in the [[finite field]] '''GF'''(''2<sup>8</sup>''). This process is described further in the article [[Rijndael mix columns]].<br />
<br />
=== The <tt>AddRoundKey</tt> step ===<br />
[[Image:AES-AddRoundKey.svg|right|320px|thumbnail|In the <tt>AddRoundKey</tt> step, each byte of the state is combined with a byte of the round subkey using the [[Exclusive or|XOR]] operation (⊕).]]<br />
In the <tt>AddRoundKey</tt> step, the subkey is combined with the state. For each round, a subkey is derived from the main [[key (cryptography)|key]] using [[Rijndael key schedule|Rijndael's key schedule]]; each subkey is the same size as the state. The subkey is added by combining each byte of the state with the corresponding byte of the subkey using bitwise [[Exclusive or|XOR]].<br />
<br />
=== Optimization of the cipher ===<br />
On systems with 32-bit or larger words, it is possible to speed up execution of this cipher by combining the <tt>SubBytes</tt> and <tt>ShiftRows</tt> steps with the <tt>MixColumns</tt> step by transforming them into a sequence of table lookups. This requires four 256-entry 32-bit tables, and utilizes a total of four kilobytes (4096 bytes) of memory — one kilobyte for each table. A round can then be done with 16 table lookups and 12 32-bit exclusive-or operations, followed by four 32-bit exclusive-or operations in the <tt>AddRoundKey</tt> step.<ref>[http://www.springerlink.com/index/UVX5NQGNN55VK199.pdf "Efficient software implementation of AES on 32-bit platforms".] Lecture Notes in Computer Science: 2523. 2003</ref><br />
<br />
If the resulting four-kilobyte table size is too large for a given target platform, the table lookup operation can be performed with a single 256-entry 32-bit (i.e. 1 kilobyte) table by the use of circular rotates.<br />
<br />
Using a byte-oriented approach, it is possible to combine the <tt>SubBytes</tt>, <tt>ShiftRows</tt>, and <tt>MixColumns</tt> steps into a single round operation.<ref>{{cite web|url=http://code.google.com/p/byte-oriented-aes |title=byte-oriented-aes - A public domain byte-oriented implementation of AES in C - Google Project Hosting |publisher=Code.google.com |date= |accessdate=2012-12-23}}</ref><br />
<br />
== Security ==<br />
Until May 2009, the only successful published attacks against the full AES were [[side-channel attack]]s on some specific implementations. The [[National Security Agency]] (NSA) reviewed all the AES finalists, including Rijndael, and stated that all of them were secure enough for U.S. Government non-classified data. In June 2003, the U.S. Government announced that AES could be used to protect [[classified information]]:<br />
<blockquote>The design and strength of all key lengths of the AES algorithm (i.e., 128, 192 and 256) are sufficient to protect classified information up to the SECRET level. TOP SECRET information will require use of either the 192 or 256 key lengths. The implementation of AES in products intended to protect national security systems and/or information must be reviewed and certified by NSA prior to their acquisition and use.<ref>{{cite web|url=http://csrc.nist.gov/groups/ST/toolkit/documents/aes/CNSS15FS.pdf |title=National Policy on the Use of the Advanced Encryption Standard (AES) to Protect National Security Systems and National Security Information |author=Lynn Hathaway |date=June 2003 |format=PDF |accessdate=2011-02-15}}</ref></blockquote><br />
<br />
AES has 10 rounds for 128-bit keys, 12 rounds for 192-bit keys, and 14 rounds for 256-bit keys. By 2006, the best known attacks were on 7 rounds for 128-bit keys, 8 rounds for 192-bit keys, and 9 rounds for 256-bit keys.<ref name=improved>[[John Kelsey (cryptanalyst)|John Kelsey]], [[Stefan Lucks]], [[Bruce Schneier]], [[Mike Stay]], [[David A. Wagner|David Wagner]], and [[Doug Whiting]], ''Improved Cryptanalysis of Rijndael'', [[Fast Software Encryption]], 2000 pp213–230 [http://www.schneier.com/paper-rijndael.html]</ref><br />
<br />
=== Known attacks ===<br />
For cryptographers, a [[cryptanalysis|cryptographic]] "break" is anything faster than a [[Brute-force attack|brute force]]—performing one trial decryption for each key (see [[Cryptanalysis#Computational resources required|Cryptanalysis]]). This includes results that are infeasible with current technology. The largest successful publicly known [[brute force attack]] against any block-cipher encryption was against a 64-bit [[RC5]] key by [[distributed.net]] in 2006.<ref name=ZD20060430>{{cite web<br />
| url = http://www.zdnet.com/blog/ou/is-encryption-really-crackable/204<br />
| title = Is encryption really crackable?<br />
| first1 = George<br />
| last1 = Ou<br />
| publisher = Ziff-Davis<br />
| date = April 30, 2006<br />
| archiveurl = http://www.webcitation.org/5rocpRxhN<br />
| archivedate = August 7, 2010<br />
| accessdate = August 7, 2010 }}</ref><br />
<br />
AES has a fairly simple algebraic description.<ref>{{cite web|url=http://www.isg.rhul.ac.uk/~sean/ |title=Sean Murphy |publisher=University of London |date= |accessdate=2008-11-02}}</ref> In 2002, a theoretical attack, termed the "[[XSL attack]]", was announced by [[Nicolas Courtois]] and [[Josef Pieprzyk]], purporting to show a weakness in the AES algorithm due to its simple description.<ref>{{cite web | url = http://www.schneier.com/crypto-gram-0209.html | title = AES News, Crypto-Gram Newsletter, September 15, 2002 | author = Bruce Schneier | accessdate = 2007-07-27| archiveurl= http://web.archive.org/web/20070707105715/http://www.schneier.com/crypto-gram-0209.html| archivedate= 7 July 2007 <!--DASHBot-->| deadurl= no}}</ref> Since then, other papers have shown that the attack as originally presented is unworkable; see [[XSL attack#Application to block ciphers|XSL attack on block ciphers]].<br />
<br />
During the AES process, developers of competing algorithms wrote of Rijndael, "...we are concerned about [its] use...in security-critical applications."<ref name="rijndael-algebraic"><br />
{{cite conference<br />
| author = [[Niels Ferguson]], [[Richard Schroeppel]], Doug Whiting<br />
| title = A simple algebraic representation of Rijndael<br />
| booktitle = Proceedings of [[Selected Areas in Cryptography]], 2001, Lecture Notes in Computer Science<br />
| pages = 103–111<br />
| publisher = [[Springer-Verlag]]<br />
| year = 2001<br />
| location =<br />
| url = http://www.macfergus.com/pub/rdalgeq.html<br />
| doi =<br />
| format = PDF/[[PostScript]]<br />
| accessdate = 2006-10-06<br />
| archiveurl= http://web.archive.org/web/20061104080748/http://www.macfergus.com/pub/rdalgeq.html<br />
| archivedate= 4 November 2006}}</ref> However, in October 2000 at the end of the AES selection process, [[Bruce Schneier]], a developer of the competing algorithm [[Twofish]], wrote that while he thought successful academic attacks on Rijndael would be developed someday, "I do not believe that anyone will ever discover an attack that will allow someone to read Rijndael traffic."<ref>Bruce Schneier, [http://www.schneier.com/crypto-gram-0010.html AES Announced], October 15, 2000</ref><br />
<br />
On July 1, 2009, Bruce Schneier blogged<ref>{{cite web<br />
|url=http://www.schneier.com/blog/archives/2009/07/new_attack_on_a.html<br />
|title=New Attack on AES<br />
|author=Bruce Schneier<br />
|date=2009-07-01<br />
|work=Schneier on Security, A blog covering security and security technology<br />
|accessdate=2010-03-11| archiveurl= http://web.archive.org/web/20100208155652/http://www.schneier.com/blog/archives/2009/07/new_attack_on_a.html| archivedate= 8 February 2010 <!--DASHBot-->| deadurl= no}}</ref><br />
about a [[related-key attack]] on the 192-bit and 256-bit versions of AES, discovered by [[Alex Biryukov]] and Dmitry Khovratovich,<ref>{{cite web<br />
|url=http://eprint.iacr.org/2009/317<br />
|title=Related-key Cryptanalysis of the Full AES-192 and AES-256<br />
|author=Biryukov, Alex<br />
|author2=Khovratovich, Dmitry<br />
|date=2009-12-04<br />
|accessdate=2010-03-11}}</ref><br />
which exploits AES's somewhat simple key schedule and has a complexity of 2<sup>119</sup>. In December 2009 it was improved to 2<sup>99.5</sup>. This is a follow-up to an attack discovered earlier in 2009 by Alex Biryukov, Dmitry Khovratovich, and Ivica Nikolić, with a complexity of 2<sup>96</sup> for one out of every 2<sup>35</sup> keys.<ref>{{cite book<br />
|title=Advances in Cryptology – CRYPTO 2009<br />
|chapter=Distinguisher and Related-Key Attack on the Full AES-256<br />
|last1=Nikolić<br />
|first1=Ivica<br />
|year=2009<br />
|publisher=Springer Berlin / Heidelberg<br />
|isbn=978-3-642-03355-1<br />
|pages=231–249<br />
|doi=10.1007/978-3-642-03356-8_14<br />
|accessdate=2010-03-11}}</ref><br />
<br />
Another attack was blogged by Bruce Schneier<ref>{{cite web<br />
|url=http://www.schneier.com/blog/archives/2009/07/another_new_aes.html<br />
|title=Another New AES Attack<br />
|author=Bruce Schneier<br />
|date=2009-07-30<br />
|work=Schneier on Security, A blog covering security and security technology<br />
|accessdate=2010-03-11}}</ref><br />
on July 30, 2009 and released as a preprint<ref>{{cite web<br />
|url=http://eprint.iacr.org/2009/374<br />
|title=Key Recovery Attacks of Practical Complexity on AES Variants With Up To 10 Rounds<br />
|author=Alex Biryukov<br />
|author2=Orr Dunkelman|author3= Nathan Keller|author4= Dmitry Khovratovich|author5= Adi Shamir<br />
|date=2009-08-19<br />
|accessdate=2010-03-11| archiveurl= http://web.archive.org/web/20100128050656/http://eprint.iacr.org/2009/374| archivedate= 28 January 2010 <!--DASHBot-->| deadurl= no}}</ref><br />
on August 3, 2009. This new attack, by Alex Biryukov, Orr Dunkelman, Nathan Keller, Dmitry Khovratovich, and [[Adi Shamir]], is against AES-256 that uses only two related keys and 2<sup>39</sup> time to recover the complete 256-bit key of a 9-round version, or 2<sup>45</sup> time for a 10-round version with a stronger type of related subkey attack, or 2<sup>70</sup> time for an 11-round version. 256-bit AES uses 14 rounds, so these attacks aren't effective against full AES.<br />
<br />
In November 2009, the first known-key [[distinguishing attack]] against a reduced 8-round version of AES-128 was released as a preprint.<ref>{{cite web<br />
|url=http://eprint.iacr.org/2009/531<br />
|title=Super-Sbox Cryptanalysis: Improved Attacks for AES-like permutations<br />
|author=Henri Gilbert<br />
|author2=Thomas Peyrin<br />
|date=2009-11-09<br />
|accessdate=2010-03-11}}</ref><br />
This known-key distinguishing attack is an improvement of the rebound or the start-from-the-middle attacks for AES-like permutations, which view two consecutive rounds of permutation as the application of a so-called Super-Sbox. It works on the 8-round version of AES-128, with a time complexity of 2<sup>48</sup>, and a memory complexity of 2<sup>32</sup>.<br />
<br />
In July 2010 Vincent Rijmen published an ironic paper on "chosen-key-relations-in-the-middle" attacks on AES-128.<ref>{{cite web |url=http://eprint.iacr.org/2010/337.pdf |title=Practical-Titled Attack on AES-128 Using Chosen-Text Relations |author=Vincent Rijmen |year=2010}}</ref><br />
<br />
The first [[key-recovery attack]]s on full AES were due to Andrey Bogdanov, Dmitry Khovratovich, and Christian Rechberger, and were published in 2011.<ref>{{cite web |url=http://research.microsoft.com/en-us/projects/cryptanalysis/aesbc.pdf |title=Biclique Cryptanalysis of the Full AES |author=Andrey Bogdanov, Dmitry Khovratovich, and Christian Rechberger |year=2011}}</ref> The attack is based on bicliques and is faster than brute force by a factor of about four. It requires 2<sup>126.1</sup> operations to recover an AES-128 key. For AES-192 and AES-256, 2<sup>189.7</sup> and 2<sup>254.4</sup> operations are needed, respectively.<br />
<br />
=== Side-channel attacks ===<!-- possibly out of date? --><br />
[[Side-channel attacks]] do not attack the underlying cipher, and thus are not related to security in that context. They rather attack implementations of the cipher on systems which inadvertently leak data. There are several such known attacks on certain implementations of AES.<br />
<br />
In April 2005, [[Daniel J. Bernstein|D.J. Bernstein]] announced a cache-timing attack that he used to break a custom server that used [[OpenSSL]]'s AES encryption.<ref name="bernstein_timing">{{cite web|url=http://cr.yp.to/papers.html#cachetiming |title=Index of formal scientific papers |publisher=Cr.yp.to |date= |accessdate=2008-11-02}}</ref> The attack required over 200 million chosen plaintexts.<ref>{{cite web | url = http://www.schneier.com/blog/archives/2005/05/aes_timing_atta_1.html | title = AES Timing Attack | author = Bruce Schneier | accessdate = 2007-03-17| archiveurl= http://web.archive.org/web/20070212015727/http://www.schneier.com/blog/archives/2005/05/aes_timing_atta_1.html| archivedate= 12 February 2007 <!--DASHBot-->| deadurl= no}}</ref> The custom server was designed to give out as much timing information as possible (the server reports back the number of machine cycles taken by the encryption operation); however, as Bernstein pointed out, "reducing the precision of the server's timestamps, or eliminating them from the server's responses, does not stop the attack: the client simply uses round-trip timings based on its local clock, and compensates for the increased noise by averaging over a larger number of samples."<ref name="bernstein_timing" /><br />
<br />
In October 2005, Dag Arne Osvik, [[Adi Shamir]] and Eran Tromer presented a paper demonstrating several cache-timing attacks against AES.<ref>{{cite journal|url=http://www.wisdom.weizmann.ac.il/~tromer/papers/cache.pdf |title=Cache Attacks and Countermeasures: the Case of AES |format=PDF |date=2005-11-20 |author=Dag Arne Osvik1|author2=Adi Shamir2|author3=Eran Tromer2 |accessdate=2008-11-02}}</ref> One attack was able to obtain an entire AES key after only 800 operations triggering encryptions, in a total of 65 milliseconds. This attack requires the attacker to be able to run programs on the same system or platform that is performing AES.<br />
<br />
In December 2009 an attack on some hardware implementations was published that used [[differential fault analysis]] and allows recovery of a key with a complexity of 2<sup>32</sup>.<ref>{{cite journal|url=http://eprint.iacr.org/2009/581.pdf |title=A Diagonal Fault Attack on the Advanced Encryption Standard |author=Dhiman Saha, Debdeep Mukhopadhyay, Dipanwita RoyChowdhury |format=PDF |accessdate=2009-12-08| archiveurl= http://web.archive.org/web/20091222070135/http://eprint.iacr.org/2009/581.pdf| archivedate= 22 December 2009 <!--DASHBot-->| deadurl= no}}</ref><br />
<br />
In November 2010 Endre Bangerter, David Gullasch and Stephan Krenn published a paper which described a practical approach to a "near real time" recovery of secret keys from AES-128 without the need for either cipher text or plaintext. The approach also works on AES-128 implementations that use compression tables, such as OpenSSL.<ref>{{cite web |url=http://eprint.iacr.org/2010/594.pdf |title=Cache Games – Bringing Access-Based Cache Attacks on AES to Practice |author=Endre Bangerter, David Gullasch and Stephan Krenn |year=2010}}</ref> Like some earlier attacks this one requires the ability to run unprivileged code on the system performing the AES encryption, which may be achieved by malware infection far more easily than commandeering the root account.<ref>{{cite web|url=http://news.ycombinator.com/item?id=1937902 |title=Breaking AES-128 in realtime, no ciphertext required &#124; Hacker News |publisher=News.ycombinator.com |date= |accessdate=2012-12-23}}</ref><br />
<br />
== NIST/CSEC validation ==<br />
The [[CMVP|Cryptographic Module Validation Program]] (CMVP) is operated jointly by the United States Government's [[National Institute of Standards and Technology]] (NIST) Computer Security Division and the [[Communications Security Establishment]] (CSE) of the Government of Canada. The use of cryptographic modules validated to NIST [[FIPS 140-2]] is required by the United States Government for encryption of all data that has a classification of Sensitive but Unclassified (SBU) or above. From NSTISSP #11, National Policy Governing the Acquisition of Information Assurance: "Encryption products for protecting classified information will be certified by NSA, and encryption products intended for protecting sensitive information will be certified in accordance with NIST FIPS 140-2."<ref name="cnss.gov">http://www.cnss.gov/Assets/pdf/nstissp_11_fs.pdf</ref><br />
<br />
The Government of Canada also recommends the use of [[FIPS 140]] validated cryptographic modules in unclassified applications of its departments.<br />
<br />
Although NIST publication 197 ("FIPS 197") is the unique document that covers the AES algorithm, vendors typically approach the CMVP under FIPS 140 and ask to have several algorithms (such as [[Triple DES|Triple&nbsp;DES]] or [[SHA1]]) validated at the same time. Therefore, it is rare to find cryptographic modules that are uniquely FIPS 197 validated and NIST itself does not generally take the time to list FIPS 197 validated modules separately on its public web site. Instead, FIPS 197 validation is typically just listed as an "FIPS approved: AES" notation (with a specific FIPS 197 certificate number) in the current list of FIPS 140 validated cryptographic modules.<br />
<br />
The Cryptographic Algorithm Validation Program (CAVP)<ref>{{cite web|url=http://csrc.nist.gov/groups/STM/cavp/index.html |title=NIST.gov - Computer Security Division - Computer Security Resource Center |publisher=Csrc.nist.gov |date= |accessdate=2012-12-23}}</ref> allows for independent validation of the correct implementation of the AES algorithm at a reasonable cost{{Citation needed|date=December 2010}}. Successful validation results in being listed on the NIST validations page. This testing is a pre-requisite for the FIPS 140-2 module validation described below. However, successful CAVP validation in no way implies that the cryptographic module implementing the algorithm is secure. A cryptographic module lacking FIPS 140-2 validation or specific approval by the NSA is not deemed secure by the US Government and cannot be used to protect government data.<ref name="cnss.gov"/><br />
<br />
FIPS 140-2 validation is challenging to achieve both technically and fiscally.<ref name="openssl">{{cite web|author=OpenSSL, openssl@openssl.org |url=http://openssl.org/docs/fips/fipsnotes.html |title=OpenSSL's Notes about FIPS certification |publisher=Openssl.org |date= |accessdate=2012-12-23}}</ref> There is a standardized battery of tests as well as an element of source code review that must be passed over a period of a few weeks. The cost to perform these tests through an approved laboratory can be significant (e.g., well over $30,000 US)<ref name="openssl" /> and does not include the time it takes to write, test, document and prepare a module for validation. After validation, modules must be re-submitted and re-evaluated if they are changed in any way. This can vary from simple paperwork updates if the security functionality did not change to a more substantial set of re-testing if the security functionality was impacted by the change.<br />
<br />
== Test vectors ==<br />
Test vectors are a set of known ciphers for a given input and key. [[NIST]] distributes the reference of AES test vectors as [http://csrc.nist.gov/groups/STM/cavp/documents/aes/KAT_AES.zip AES Known Answer Test (KAT) Vectors (in ZIP format)].<br />
<br />
== Performance ==<br />
High speed and low RAM requirements were criteria of the AES selection process. Thus AES performs well on a wide variety of hardware, from 8-bit [[smart card]]s to high-performance computers.<br />
<br />
On a [[Pentium Pro]], AES encryption requires 18 clock cycles per byte,<ref>{{cite web |url=http://www.schneier.com/paper-aes-performance.pdf |title=Performance Comparisons of the AES submissions |last1=Schneier |first1=Bruce |last2=Kelsey |first2=John |last3=Whiting |first3=Doug |last4=Wagner |first4=David |last5=Hall |first5=Chris |last6=Ferguson |first6=Niels |date=1999-02-01 |format=PDF |accessdate=2010-12-28}}</ref> equivalent to a throughput of about 11&nbsp;MB/s for a 200&nbsp;MHz processor. On a 1.7&nbsp;GHz [[Pentium M]] throughput is about 60&nbsp;MB/s.<br />
<br />
On Intel [[Core i3]]/[[Core i5|i5]]/[[Core i7|i7]] CPUs supporting [[AES instruction set|AES-NI instruction set]] extensions, throughput can be over 700&nbsp;MB/s per thread.<ref>{{cite web |url=http://grantmcwilliams.com/tech/technology/item/532-hardware-aes-showdown-via-padlock-vs-intel-aes-ni-vs-amd-hexacore |title=Hardware AES Showdown - VIA Padlock vs Intel AES-NI vs AMD Hexacore |last=McWilliams |first=Grant |date=6 July 2011 |accessdate=2013-08-28}}</ref><br />
<br />
== Implementations ==<br />
<br />
{{Main|AES implementations}}<br />
<br />
== See also ==<br />
*[[Disk encryption]]<br />
*[[Multiscale Electrophysiology Format]] (MEF)<br />
*[[Whirlpool (cryptography)|Whirlpool]] – hash function created by Vincent Rijmen and Paulo S. L. M. Barreto<br />
<br />
== Notes ==<br />
{{Reflist|30em}}<br />
<br />
== References ==<br />
* Nicolas Courtois, Josef Pieprzyk, "Cryptanalysis of Block Ciphers with Overdefined Systems of Equations". pp267–287, [[ASIACRYPT]] 2002.<br />
* Joan Daemen, Vincent Rijmen, "The Design of Rijndael: AES – The Advanced Encryption Standard." Springer, 2002. ISBN 3-540-42580-2.<br />
* Christof Paar, Jan Pelzl, [http://wiki.crypto.rub.de/Buch/sample_chapters.php "The Advanced Encryption Standard"], Chapter 4 of "Understanding Cryptography, A Textbook for Students and Practitioners". (companion web site contains online lectures on AES), Springer, 2009.<br />
<br />
== External links ==<br />
* [http://embeddedsw.net/Cipher_Reference_Home.html 256bit Ciphers - AES Reference implementation and derived code]<br />
* [http://csrc.nist.gov/publications/fips/fips197/fips-197.pdf FIPS PUB 197: the official AES standard] ([[Portable Document Format|PDF]] file)<br />
* [http://csrc.nist.gov/archive/aes/rijndael/wsdindex.html AES algorithm archive information – (old, unmaintained)]<br />
* [http://webstore.iec.ch/preview/info_isoiec18033-3%7Bed2.0%7Den.pdf Preview of ISO/IEC 18033-3]<br />
* [http://www.formaestudio.com/rijndaelinspector/archivos/Rijndael_Animation_v4_eng.swf Animation of Rijndael]<br />
* [http://www.theinquirer.net/inquirer/news/2102435/aes-encryption-cracked/ AES encryption is cracked]<br />
{{Cryptography navbox | block}}<br />
<br />
[[Category:Advanced Encryption Standard| ]]<br />
[[Category:Broken block ciphers]]</div>Adminhttps://en.formulasearchengine.com/index.php?title=Amide&diff=218567Amide2014-07-31T13:44:30Z<p>Admin: Reverted edits by MediaWiki spam cleanup (talk) to last revision by ThunderSkunk</p>
<hr />
<div>{{Distinguish|Imide}}<br />
{{Use dmy dates|date=July 2012}}<br />
[[File:AmideTypes.png|thumb|320px|Structures of three kinds of amides: an organic amide, a sulfonamide, and a phosphoramide.]]<br />
An '''amide''', also known as an [[Fatty_acid_amide|'-acid amide']], is a compound with the functional group R<sub>n</sub>E(O)<sub>x</sub>NR'<sub>2</sub> (R and R' refer to H or organic groups). Most common are "organic amides" (n = 1, E = C, x = 1), but many other important types of amides are known including phosphor amides (n = 2, E = P, x = 1 and many related formulas) and [[Sulfonamide (chemistry)|sulfonamide]]s (E = S, x= 2).<ref>{{goldbookref|file=A00266|title=amides}}</ref> The term amide refers both to ''classes of compounds'' and to the ''[[functional group]]'' (R<sub>n</sub>E(O)<sub>x</sub>NR'<sub>2</sub>) within those compounds.<br />
<br />
Amide can ''also'' refer to the [[conjugate base]] of [[ammonia]] (the anion H<sub>2</sub>N<sup>−</sup>) or of an organic [[amine]] (an anion R<sub>2</sub>N<sup>−</sup>). For discussion of these "[[anionic]] amides", see [[Metal amides#Alkali metal amides]].<br />
<br />
The remainder of this article is about the [[carbonyl]]-[[nitrogen]] sense of ''amide''.<br />
<br />
==Structure and bonding==<br />
The simplest amides are derivatives of ammonia wherein one hydrogen atom has been replaced by an [[acyl]] group. The ensemble is generally represented as RC(O)NH<sub>2</sub>. Closely related and even more numerous are amides derived from primary amines (R'NH<sub>2</sub>) with the formula RC(O)NHR'. Amides are also commonly derived from [[secondary amine]]s (R'R<nowiki>''</nowiki>NH) with the formula RC(O)NR'R<nowiki>''</nowiki>. Amide are usually regarded as derivatives of [[carboxylic acid]]s in which the [[hydroxyl]] group has been replaced by an amine or ammonia.<br />
:[[Image:AmideResonance.png|300px|Amide resonance:]]<br />
The lone pair of [[electron]]s on the [[nitrogen]] is delocalized into the carbonyl, thus forming a partial [[double bond]] between N and the [[carbonyl]] [[carbon]]. Consequently the nitrogen in amides is not pyramidal. It is estimated that acetamide is described by [[resonance structure]] A for 62% and by B for 28%<ref name = Kemnitz>{{Cite journal|doi=10.1021/ja0663024|title="Amide Resonance" Correlates with a Breadth of C−N Rotation Barriers|year=2007|last1=Kemnitz|first1=Carl R.|last2=Loewen|first2=Mark J.|journal=Journal of the American Chemical Society|volume=129|issue=9|pages=2521–8|pmid=17295481}}</ref><br />
<!--needs:<br />
barrier from DMF, and a more general ref than this JACS rpt<br />
comment on syn and anti secondary amides--><br />
[[Image:Formamide-MO-3D-balls.png|thumb|right|180px|Amides possess a [[conjugated system]] spread over the O, C and N atoms, consisting of [[molecular orbital]]s occupied by [[delocalized electron]]s. One of the [[pi bond|''π'' molecular orbitals]] in [[formamide]] is shown above.]]<br />
<br />
==Nomenclature==<br />
{{Main|IUPAC nomenclature of organic chemistry#Amines and amides}}<br />
In the usual nomenclature, one adds the term "amide" to the stem of the parent acid's name. For instance, the amide derived from [[acetic acid]] is named [[acetamide]] (CH<sub>3</sub>CONH<sub>2</sub>). IUPAC recommends [[ethanamide]], but this and related formal names are rarely encountered. When the amide is derived from a primary or secondary amine, the substitutents on nitrogen are indicated first in the name. Thus, the amide formed from [[dimethylamine]] and [[acetic acid]] is ''N,N''-dimethylacetamide (CH<sub>3</sub>CONMe<sub>2</sub>, where Me = CH<sub>3</sub>). Usually even this name is simplified to [[dimethylacetamide]]. Cyclic amides are called [[lactam]]s; they are necessarily secondary or tertiary amides. Functional groups consisting of -P(O)NR<sub>2</sub> and -SO<sub>2</sub>NR<sub>2</sub> are [[phosphonamides]] and [[sulfonamide (chemistry)|sulfonamide]]s, respectively.<ref>Organic Chemistry IUPAC Gnomenclature. Rules C-821. Amides http://www.acdlabs.com/iupac/nomenclature/79/r79_540.htm</ref><br />
<br />
===Pronunciation===<br />
Some chemists make a pronunciation distinction between the two, saying {{IPAc-en|ə|ˈ|m|iː|d}} for the [[carbonyl]]-[[nitrogen]] compound and {{IPAc-en|audio=En-uk-amide.ogg|ˈ|æ|m|aɪ|d}} {{contradict-inline|date=August 2012}} for the [[anion]]. Others substitute one of these with {{IPAc-en|ˈ|æ|m|ɨ|d}}, while still others pronounce both {{IPAc-en|ˈ|æ|m|ɨ|d}}, making them [[homonym]]s.<br />
<br />
==Properties==<br />
<br />
===Basicity===<br />
Compared to [[amine]]s, amides are very weak [[Base (chemistry)|base]]s. While the [[conjugate acid]] of an [[amine]] has a [[pKa]] of about 9.5, the [[conjugate acid]] of an amide has a pKa around −0.5. Therefore amides don't have as clearly noticeable [[acid-base]] properties in [[water]]. This relative lack of basicity is explained by the [[electron]]-withdrawing nature of the [[carbonyl group]] where the lone pair of [[electron]]s on the [[nitrogen]] is delocalized by [[resonance (chemistry)|resonance]]. On the other hand, amides are much stronger [[Base (chemistry)|base]]s than [[carboxylic acid]]s, [[ester]]s, [[aldehyde]]s, and [[ketone]]s (conjugated acid pKa between −6 and −10). It is estimated [[in silico]] that [[acetamide]] is represented by [[resonance structure]] A for 62% and by B for 28%.<ref name = Kemnitz/> Resonance is largely prevented in the very strained [[quinuclidone]].<br />
<br />
Because of the greater electronegativity of oxygen, the carbonyl (C=O) is a stronger dipole than the N–C dipole. The presence of a C=O dipole and, to a lesser extent a N–C dipole, allows amides to act as H-bond acceptors. In primary and secondary amides, the presence of N–H dipoles allows amides to function as H-bond donors as well. Thus amides can participate in hydrogen bonding with water and other protic solvents; the oxygen atom can accept hydrogen bonds from water and the N–H hydrogen atoms can donate H-bonds. As a result of interactions such as these, the water solubility of amides is greater than that of corresponding hydrocarbons.<br />
<br />
The proton of a primary or secondary amide does not dissociate readily under normal conditions; its p''K<sub>a</sub>'' is usually well above 15. Conversely, under extremely acidic conditions, the carbonyl [[oxygen]] can become protonated with a p''K<sub>a</sub>'' of roughly −1.<br />
<br />
===Solubility===<br />
The solubilities of amides and esters are roughly comparable. Typically amides are less soluble than comparable amines and carboxylic acids since these compounds can both donate and accept hydrogen bonds. Tertiary amides, with the important exception of N,N-dimethylformamide, exhibit low solubility in water.<br />
<br />
==Characterization==<br />
The presence of the functional group is generally easily established, at least in small molecules. They are the most common non-basic functional group. They can be distinguished from nitro and cyano groups by their [[IR spectroscopy|IR spectra]]. Amides exhibit a moderately intense ν<sub>CO</sub> band near 1650&nbsp;cm<sup>−1</sup>. By <sup>1</sup>H [[NMR spectroscopy]], CON''H''R signals occur at low fields. In X-ray crystallography, the C(O)N center together with the three immediately adjacent atoms characteristically define a plane.<br />
<br />
==Applications and occurrence==<br />
Amides are pervasive in nature and technology as structural materials. The amide linkage is easily formed, confers structural rigidity, and resists [[hydrolysis]]. Nylons are polyamides, as are the very resilient materials [[Aramid]], [[Twaron]], and [[Kevlar]]. Amide linkages constitute a defining molecular feature of [[protein]]s, the [[secondary structure]] of which is due in part to the [[hydrogen bonding]] abilities of amides. Amide linkages in a [[biochemistry|biochemical]] context are called [[peptide bond]]s when they occur in the main chain of a protein and [[isopeptide bond]]s when they occur to a side-chain of the protein. Proteins can have structural roles, such as in [[hair]] or [[spider silk]], but also nearly all [[enzyme]]s are proteins. Low molecular weight amides, such as [[dimethylformamide]] (HC(O)N(CH<sub>3</sub>)<sub>2</sub>), are common solvents. Many drugs are amides, including [[penicillin]] and [[LSD]]. Moreover, plant N-alkylamides have a wide range of biological functionalities.<ref>{{Cite journal|doi=10.1016/j.jep.2012.05.038|title=Alkamid database: Chemistry, occurrence and functionality of plant N-alkylamides|year=2012|last1=Boonen|first1=Jente|last2=Bronselaer|first2=Antoon|last3=Nielandt|first3=Joachim|last4=Veryser|first4=Lieselotte|last5=De Tré|first5=Guy|last6=De Spiegeleer|first6=Bart|journal=Journal of Ethnopharmacology|volume=142|issue=3|pages=563–90|pmid=22659196}}</ref><br />
<br />
==Amide synthesis==<!-- This section is linked from [[Organic reaction]] --><br />
Amides are commonly formed via reactions of a [[carboxylic acid]] with an [[amine]]. Many methods are known for driving the unfavorable equilibrium to the right:<br />
:RCO<sub>2</sub>H + R'R"NH <math>\overrightarrow{\leftarrow}</math> RC(O)NR'R" + H<sub>2</sub>O<br />
For the most part, these reactions involve "activating" the carboxylic acid and the best known method, the [[Schotten-Baumann reaction]], which involves conversion of the acid to the [[acid chloride]]s:<br />
:[[Image:SimpleAmideFormationByCondensation.png|400px|Amide bond formation]]<br />
<!--to include: *In [[solid phase peptide synthesis]]--><br />
<br />
{| class="wikitable sortable" style="background-color:white;float: center; border-collapse: collapse; margin: 0em 1em;" border="1" cellpadding="2" cellspacing="0"<br />
<br />
! width=200px|Reaction name !! Substrate !! class="unsortable" | Details<br />
|-<br />
|valign=top | '''[[Beckmann rearrangement]]'''<br />
|valign=top|Cyclic ketone<br />
| Reagent: [[hydroxylamine]] and acid<br />
|-<br />
|valign=top| '''[[Schmidt reaction]]'''<br />
|valign=top|Ketones<br />
| Reagent: hydrazoic acid<br />
|-<br />
|valign=top| '''Nitrile hydrolysis'''<br />
|valign=top|Nitrile<br />
| Reagent: water; acid catalyst<br />
|-<br />
| '''[[Willgerodt-Kindler reaction]]'''<br />
| Aryl alkyl ketones<br />
| Sulfur and morpholine<br />
|-<br />
|'''[[Passerini reaction]]'''<br />
| Carboxylic acid, ketone or aldehyde<br />
|<br />
|-<br />
|'''[[Ugi reaction]]'''<br />
| Isocyanide, carboxylic acid, ketone, primary amine<br />
|<br />
|-<br />
|'''Bodroux reaction'''<ref>{{Cite journal|author=Bodroux F.|journal=Bull. Soc. Chim. France|year= 1905|volume= 33|pages= 831}}</ref><ref>{{cite web|title=Bodroux reaction |publisher= Institute of Chemistry, Skopje, Macedonia|url=http://www.pmf.ukim.edu.mk/PMF/Chemistry/reactions/bodroux1.htm}}</ref><br />
| [[Carboxylic acid]], [[Grignard reagent]] with an [[aniline]] derivative ArNHR'<br />
| [[Image:Bodroux reaction.png|400px|Bodroux reaction]]<br />
|-<br />
|'''Chapman rearrangement'''<ref>{{Cite journal|author=Schulenberg, J. W.; Archer, S. |title=The Chapman Rearrangement|journal=[[Organic Reactions|Org. React.]]|year=1965|volume=14|doi=10.1002/0471264180.or014.01}}</ref><ref>{{Cite journal|doi=10.1039/CT9252701992|title=CCLXIX.—Imino-aryl ethers. Part III. The molecular rearrangement of ''N''-phenylbenziminophenyl ether |year=1925|last1=Chapman|first1=Arthur William|journal=Journal of the Chemical Society, Transactions|volume=127|pages=1992}}</ref><br />
|Aryl [[imidate|imino ether]]<br />
|For ''N,N''-diaryl amides. The [[reaction mechanism]] is based on a [[nucleophilic aromatic substitution]].<ref>{{Cite book|title=Advanced organic Chemistry, Reactions, mechanisms and structure|edition= 3rd |author=March, Jerry |isbn= 0-471-85472-7}}</ref> [[Image:Chapman Rearrangement.png|Left|300px|Chapman Rearrangement]]<br />
|-<br />
| '''[[List of organic reactions|Leuckart amide synthesis]]'''<ref>{{Cite journal|author= [[Rudolf Leuckart (chemist)|Leuckart, R. ]] |journal=[[Berichte der deutschen chemischen Gesellschaft]]|doi=10.1002/cber.188501801182|title= Ueber einige Reaktionen der aromatischen Cyanate|year= 1885|volume= 18|pages= 873–877}}</ref><br />
| [[Isocyanate]]<br />
| Reaction of arene with isocyanate catalysed by [[aluminium trichloride]], formation of aromatic amide.<br />
|-<br />
| '''[[Photochemistry|Photolytic]] addition of [[formamide]] to [[olefins]]'''<ref>{{cite book|last=Monson|first=Richard|title=Advanced Organic Synthesis: Methods and Techniques|date=1971|publisher=Academic Press|location=Newyork|isbn=978-0124336803|page=141|url=http://f3.tiera.ru/3/Chemistry/Organic%20chemistry/Synthesis/Monson%20R.S.%20Advanced%20organic%20synthesis.%20Methods%20and%20techniques%20%28AP,%201971%29%28T%29%28209s%29.pdf}}</ref><br />
| Terminal [[alkene]]s<br />
| A [[free radical]] [[homologation reaction]] between a terminal alkene and formamide<br />
|-<br />
|}<br />
<br />
===Other methods===<br />
The seemingly simple direct reaction between an [[alcohol]] and an [[amine]] to an amide was not tried until 2007 when a special [[ruthenium]]-based [[catalyst]] was reported to be effective in a so-called dehydrogenative acylation:<ref>{{Cite journal|doi=10.1126/science.1145295|title=Direct Synthesis of Amides from Alcohols and Amines with Liberation of H<sub>2</sub>|year=2007|last1=Gunanathan|first1=C.|last2=Ben-David|first2=Y.|last3=Milstein|first3=D.|journal=Science|volume=317|issue=5839|pages=790–2|pmid=17690291}}</ref><br />
:[[Image:DehydrogenativeAmidation.svg|500px|Synthesis of Amides from Alcohols and Amines with Liberation of H2]]<br />
<br />
The generation of hydrogen gas compensates for unfavorable thermodynamics. The reaction is believed to proceed by one dehydrogenation of the alcohol to the [[aldehyde]] followed by formation of a [[hemiaminal]] and the after a second dehydrogenation to the amide. Elimination of water in the hemiaminal to the imine is not observed.<br />
<br />
Amides can also be formed from esters. Esters react slowly with amines to yield amides.<ref>{{cite book|last=Klein|first=David|title=Organic Chemistry|year=2011|publisher=John Wiley & Sons, Inc.|location=The United States of America|isbn=0471756148|pages=1003|url=http://books.google.com/?id=SsX9pbarkQkC&pg=PA1006&dq=organic+chemistry+david+%22970-1029hr.indd%22#v=onepage&q=organic%20chemistry%20david%20%22970-1029hr.indd%22&f=false}}</ref><br />
<br />
== Amide reactions ==<!-- This section is linked from [[Organic reaction]] --><br />
Amides undergo many chemical reactions, usually through an attack on the [[carbonyl]] breaking the carbonyl double bond and forming a tetrahedral intermediate. [[Thiols]], [[hydroxyl]]s and [[amines]] are all known to serve as nucleophiles. Owing to their resonance stabilization, amides are less reactive under physiological conditions than [[ester]]s. Enzymes, e.g. [[peptidase]]s or artificial catalysts, are known to accelerate the hydrolysis reactions. They can be hydrolysed in hot [[alkali]], as well as in strong [[acid]]ic conditions. Acidic conditions yield the carboxylic acid and the ammonium ion while basic hydrolysis yield the carboxylate ion and ammonia. Amides are also versatile precursors to many other [[functional group]]s.<br />
<br />
{| class="wikitable sortable" style="background-color:white;float: center; border-collapse: collapse; margin: 0em 1em;" border="1" cellpadding="2" cellspacing="0"<br />
<br />
! width=200px|Reaction name !! Product !! class="unsortable" | Comment<br />
|-<br />
|valign=top | dehydration<br />
|valign=top|nitrile<br />
| reagent: phosphorus pentoxide<br />
|-<br />
|valign=top| [[Hofmann rearrangement]]<br />
|valign=top|amine with one fewer carbon atoms<br />
|reagents: bromine and sodium hydroxide<br />
|-<br />
|valign=top| [[amide reduction]]<br />
|valign=top| amine<br />
|reagent: [[lithium aluminium hydride]]<br />
|-<br />
|[[Vilsmeier–Haack reaction]]<br />
|[[imine]]<br />
| POCl<sub>3</sub>, aromatic substrate, formamide<br />
|-<br />
|}<br />
<br />
==See also==<br />
* [[Metal amides]]<br />
<br />
==References==<br />
{{reflist}}<br />
<br />
==External links==<br />
*[http://www.organic-reaction.com/synthetic-protocols/coupling-reagents-in-amide-synthesis/ Amide synthesis (coupling reaction) – Synthetic protocols] from organic-reaction.com<br />
*[http://www.rsc.org/Chemsoc/Chembytes/IUPACGoldbook.asp IUPAC Compendium of Chemical Terminology]<br />
<br />
{{Functional Groups}}<br />
<br />
[[Category:Amides| ]]<br />
[[Category:Functional groups]]</div>Adminhttps://en.formulasearchengine.com/index.php?title=APL_(programming_language)&diff=218572APL (programming language)2014-07-31T13:43:47Z<p>Admin: Reverted edits by MediaWiki spam cleanup (talk) to last revision by 209.49.120.130</p>
<hr />
<div>{{redirects here|APL programming language family|programming languages that were influenced by APL|Category:APL programming language family}}<br />
{{infobox programming language<br />
| name = APL<br />
| paradigm = [[array programming|array]], [[functional programming|functional]], [[structured programming|structured]], [[modular programming|modular]]<br />
| year = 1964<br />
| designer = [[Kenneth E. Iverson]]<br />
| developer = [[Kenneth E. Iverson]]<br />
| latest release version =<br />
| latest release date =<br />
| typing = [[dynamic typing|dynamic]]<br />
| standards = ISO8485 (1989), ISO/IEC13751 (2001)<br />
| implementations = [http://www.dyalog.com/ Dyalog APL], [http://www-306.ibm.com/software/awdtools/apl/ IBM APL2], [http://www.apl2000.com/ APL2000], Sharp APL, [[APLX]], [http://www.nars2000.org/ NARS2000], [http://www.gnu.org/software/apl/ GNU APL]<ref>{{cite web |url=http://directory.fsf.org/wiki/GNU_APL |title=GNU APL|date= |website=directory.fsf.org |publisher=[[Free Software Directory]] |accessdate=28 September 2013}}</ref><br />
| dialects = [[A+ (programming language)|A+]], Dyalog APL, APLNext, [http://fastarray.appspot.com/ ELI], [http://www.jsoftware.com/ J]<br />
| influenced_by = [[mathematical notation]]<br />
| influenced = [[J (programming language)|J]],<ref name="jinsp">{{cite web|url=http://www.jsoftware.com/jwiki/Essays/Bibliography |title=A Bibliography of APL and J |publisher=Jsoftware.com |date= |accessdate=2010-02-03}}</ref> [[K (programming language)|K]],<ref name="kinsp">{{cite web|url=http://kx.com/Company/press-releases/arthur-interview.php |title=Kx Systems&nbsp;— An Interview with Arthur Whitney&nbsp;— Jan 2004 |publisher=Kx.com |date=2004-01-04 |accessdate=2010-02-03}}</ref> [[Mathematica]], [[MATLAB]],<ref name="mworks">{{cite web|url=http://www.mathworks.com/company/newsletters/news_notes/clevescorner/jan06.pdf |title=The Growth of MatLab&nbsp;— Cleve Moler |format=PDF |date= |accessdate=2010-02-03}}</ref> [[Nial]],<ref name="qinsp">{{cite web|url=http://www.nial.com/AboutQNial.html |title=About Q'Nial |publisher=Nial.com |date= |accessdate=2010-02-03}}</ref> [[Polymorphic Programming Language|PPL]], [[Q (programming language from Kx Systems)|Q]]<br />
}}<br />
{{APLcode}}<br />
[[File:I like APL graphic.jpg|thumb|Promotional material for APL from 1976]]<br />
'''APL''' (named after the book ''A Programming Language'')<ref name="aplbook">{{cite book | last=Iverson | first=Kenneth E. | title=A Programming Language | publisher=Wiley | year=1962 | isbn=0-471-43014-5 | url=http://www.softwarepreservation.org/projects/apl/Physics%20in%20APL2/APROGRAMMING%20LANGUAGE/view}}</ref> is a [[programming language]] developed in the 1960s by [[Kenneth E. Iverson]]. It was an important influence on the development of [[spreadsheet]]s, [[functional programming]],<ref>{{cite web | url=http://awards.acm.org/citation.cfm?id=0703524&srt=all&aw=140&ao=AMTURING | title=ACM Award Citation&nbsp;– John Backus. 1977 | publisher=Awards.acm.org | date = 1924-12-03 | accessdate=2010-02-03}}</ref> and computer math packages.<ref name="mworks" /> It has also inspired several other programming languages.<ref name="jinsp" /><ref name="kinsp" /><ref name="qinsp" /> It is still used today for certain applications.<ref>{{cite web | url=http://www.vector.org.uk/archive/v233/webber.htm | title=APLX version 4&nbsp;– from the viewpoint of an experimental physicist. Vector 23.3 | publisher=Vector.org.uk | date = 2008-05-20 | accessdate=2010-02-03 | archiveurl= http://web.archive.org/web/20100125164250/http://www.vector.org.uk/archive/v233/webber.htm | archivedate= 25 January 2010 }}</ref><ref name="prod">{{Cite journal | title = The future of APL in the insurance world | year = 1999 | journal = ACM SIGAPL APL Quote Quad | issn = 0163–6006 | volume = 30 | issue = 1 | doi = 10.1145/347194.347203 | last1 = Bergquist | first1 = Gary A. | location = New York, N.Y. | pages = 16–21 }}</ref><br />
<br />
== History ==<br />
The first incarnation of what was later to be the APL programming language was published and formalized in ''A Programming Language'',<ref name="aplbook" /> a book describing a notation invented in 1957 by [[Kenneth E. Iverson]] while at [[Harvard University]]. Iverson had developed a [[mathematical notation]] for manipulating arrays that he taught to his students.<br />
<br />
Iverson described the premise of the book in the Preface: "Applied mathematics is largely concerned with the design and analysis of explicit procedures for calculating the exact or approximate values of various functions. Such explicit procedures are called algorithms or ''programs''. Because an effective notation for the description of programs exhibits considerable syntactic structure, it is called a ''programming language''."<br />
<br />
In 1960, he began work for [[IBM]] and, working with [[Adin Falkoff]], created APL based on the notation he had developed. This notation was used inside IBM for short research reports on computer systems, such as the [[Burroughs B5000]] and its stack mechanism when stack machines versus [[register machine]]s were being evaluated by IBM for upcoming computers.<br />
<br />
Also in 1960, Iverson was already using his notation in a draft copy of Chapter 6 called "A programming language" for the book he was writing with [[Fred Brooks]], ''Automatic Data Processing'', which would later be published in 1963.<ref>Iverson, Kenneth E., [http://www.softwarepreservation.org/projects/apl/book/Iverson-AutomaticDataProcessing-color.pdf/view "Automatic Data Processing: Chapter 6: A programming language"], 1960, DRAFT copy for Brooks and Iverson 1963 book, "Automatic Data Processing".</ref><ref>[[Fred Brooks|Brooks, Fred]]; Iverson, Kenneth, (1963), ''Automatic Data Processing'', John Wiley & Sons Inc.</ref><br />
<br />
Published in 1962, the notation described in ''A Programming Language''<ref name="aplbook" /> was recognizable yet distinct from later APL.<br />
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As early as 1962, the first attempt to use the notation to describe a complete computer system happened after Falkoff discussed with Dr. William C. Carter his work in the standardization of the instruction set for the machines that later became the [[IBM System/360]] family.<br />
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In 1963, Dr. Herbert Hellerman, working at the IBM Systems Research Institute, implemented a part of the notation on an [[IBM 1620]] computer, and it was used by students in a special high school course on calculating transcendental functions by series summation. Students tested their code in Dr. Hellerman's lab. This implementation of a portion of the notation was called PAT (Personalized Array Translator).<ref>Hellerman, H., "Experimental Personalized Array Translator System", ''Communications of the ACM'', 7, 433 (July, 1964).</ref><br />
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In 1963, Falkoff, Iverson, and [[Edward H. Sussenguth Jr.]], all working at IBM, used the notation for a formal description of the [[IBM System/360]] series machine architecture and functionality, which resulted in a paper published in IBM Systems Journal in 1964. After this was published, the team turned their attention to an implementation of the notation on a computer system. One of the motivations for this focus of implementation was the interest of John L. Lawrence who had new duties with [[Science Research Associates]], an educational company bought by IBM in 1964. Lawrence asked Iverson and his group to help utilize the language as a tool for the development and use of computers in education.<ref>Falkoff, Adin D.; Iverson, Kenneth E., [http://www.jsoftware.com/papers/APLEvol.htm "The Evolution of APL"], ACM SIGPLAN Notices 13, 1978-08.</ref><br />
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After [[Lawrence M. Breed]] and [[Philip S. Abrams]] of Stanford University joined the team at IBM Research, they continued their prior work on an implementation programmed in [[FORTRAN IV]] for a portion of the notation was done for the [[IBM 7090]] computer running under the [[IBM 7090/94 IBSYS|IBSYS]] operating system. This work was finished in late 1965 and later known as IVSYS (Iverson System). The basis of this implementation was described in detail by Abrams in a Stanford University Technical Report, "An Interpreter for Iverson Notation" in 1966.<ref>Abrams, Philip S., [http://infolab.stanford.edu/TR/CS-TR-66-47.html ''An interpreter for "Iverson notation"''], Technical Report: CS-TR-66-47, Department of Computer Science, Stanford University, August 1966.</ref> Like Hellerman's PAT system earlier, this implementation did not include the APL character set but used special English reserved words for functions and operators. The system was later adapted for a time-sharing system and, by November 1966, it had been reprogrammed for the IBM/360 Model 50 computer running in a time sharing mode and was used internally at IBM.<ref>Haigh, Thomas, "Biographies: Kenneth E. Iverson", ''IEEE Annals of the History of Computing'', 2005</ref><br />
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[[File:IBM Selectric Globe Wiki.jpg|thumb|IBM typeballs (one OCR) with clip, {{€|2}} coin for scale]]<br />
<br />
A key development in the ability to use APL effectively, before the widespread use of CRT terminals, was the development of a special [[IBM Selectric typewriter]] interchangeable typeball with all the special APL characters on it. This was used on paper printing terminal workstations using the Selectric typewriter and typeball mechanism, such as the [[IBM 1050]] and [[IBM 2741]] terminal. Keycaps could be placed over the normal keys to show which APL characters would be entered and typed when that key was struck. For the first time, a programmer could actually type in and see real APL characters as used in Iverson's notation and not be forced to use awkward English keyword representations of them. Falkoff and Iverson had the special APL Selectric typeballs, 987 and 988, designed in late 1964, although no APL computer system was available to use them.<ref name="APLQQ91">Breed, Larry, [http://portal.acm.org/citation.cfm?id=138094.140933 "The First APL Terminal Session"], ''APL Quote Quad'', Association for Computing Machinery, Volume 22, Number 1, September 1991, p.2-4.</ref> Iverson cited Falkoff as the inspiration for the idea of using an IBM Selectric typeball for the APL character set.<ref>[http://www.computerhistory.org/tdih/?setdate=19/12/2009 Adin Falkoff] - Computer History Museum. "Iverson credited him for choosing the name APL and the introduction of the IBM golf-ball typewriter with the replacement typehead, which provided the famous character set to represent programs."</ref><br />
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[[File:APL-keybd2.svg|thumb|center|600px|A programmer's view of the IBM 2741 keyboard layout with the APL typeball print head inserted]]<br />
<br />
Some APL symbols, even with the APL characters on the typeball, still had to be typed in by over-striking two existing typeball characters. An example would be the "grade up" character, which had to be made from a "delta" (shift-H) and a "[[Sheffer stroke]]" (shift-M). This was necessary because the APL character set was larger than the 88 characters allowed on the Selectric typeball.<br />
<br />
The first APL interactive login and creation of an APL workspace was in 1966 by Larry Breed using an IBM 1050 terminal at the IBM Mohansic Labs near [[Thomas J. Watson Research Center]], the home of APL, in [[Yorktown Heights, New York]].<ref name="APLQQ91"/><br />
<br />
IBM was chiefly responsible for the introduction of APL to the marketplace. APL was first available in 1967 for the [[IBM 1130]] as ''APL\1130''.<ref>{{cite journal | url=http://www.vector.org.uk/archive/v223/APL_1130.htm | title=How We Got to APL\1130 | author=Larry Breed | authorlink=Larry Breed | journal=Vector (British APL Association) | volume=22 | issue=3 |date=August 2006 | issn=0955-1433 }}</ref><ref>[http://bitsavers.org/pdf/ibm/1130/lang/1130-03.3.001_APL_1130_May69.pdf APL\1130 Manual], May 1969</ref> It would run in as little as 8k 16-bit words of memory, and used a dedicated 1 megabyte hard disk.<br />
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APL gained its foothold on mainframe timesharing systems from the late 1960s through the early 1980s, in part because it would run on lower-specification systems that were not equipped with [[Dynamic Address Translation]] hardware.<ref>{{cite web|url=http://www.quadibloc.com/comp/aplint.htm |title=Remembering APL |publisher=Quadibloc.com |date= |accessdate=2013-06-17}}</ref> Additional improvements in performance for selected [[IBM System/370]] mainframe systems included the "APL Assist Microcode" in which some support for APL execution was included in the actual firmware as opposed to APL being exclusively a software product. Somewhat later, as suitably performing hardware was finally becoming available in the mid- to late-1980s, many users migrated their applications to the personal computer environment.<br />
<br />
Early IBM APL interpreters for IBM 360 and IBM 370 hardware implemented their own multi-user management instead of relying on the host services, thus they were timesharing systems in their own right. First introduced in 1966, the ''APL\360''<ref name="IBM APL\360 1968">Falkoff, Adin; Iverson, Kenneth E., [http://bitsavers.org/pdf/ibm/apl/APL_360_Users_Manual_Aug68.pdf "APL\360 Users Guide"], IBM Research, Thomas J. Watson Research Center, Yorktown Heights, NY, August 1968.</ref><ref>[http://bitsavers.org/pdf/ibm/apl/APL_360_Terminal_System_Mar67.pdf "APL\360 Terminal System"], IBM Research, Thomas J. Watson Research Center, March 1967.</ref><ref name="apl360">{{cite book | last=Pakin | first=Sandra | title=APL\360 Reference Manual | publisher=Science Research Associates, Inc. | year=1968 | isbn=0-574-16135-X}}</ref> system was a multi-user interpreter. The ability to programmatically communicate with the operating system for information and setting interpreter system variables was done through special privileged "I-beam" functions, using both monadic and dyadic operations.<ref>Falkoff, Adin D.; Iverson, Kenneth E.,[http://www.research.ibm.com/journal/rd/174/ibmrd1704F.pdf ''The Design of APL''], ''IBM Journal of Research and Development'', Volume 17, Number 4, July 1973. "These environmental defined functions were based on the use of still another class of functions—called "I-beams" because of the shape of the symbol used for them—which provide a more general facility for communication between APL programs and the less abstract parts of the system. The I-beam functions were first introduced by the system programmers to allow them to execute System/360 instructions from within APL programs, and thus use APL as a direct aid in their programming activity. The obvious convenience of functions of this kind, which appeared to be part of the language, led to the introduction of the monadic I-beam function for direct use by anyone. Various arguments to this function yielded information about the environment such as available space and time of day."</ref><br />
<br />
In 1973, IBM released ''APL.SV'', which was a continuation of the same product, but which offered [[shared variable]]s as a means to access facilities outside of the APL system, such as operating system files. In the mid-1970s, the IBM mainframe interpreter was even adapted for use on the [[IBM 5100]] desktop computer, which had a small CRT and an APL keyboard, when most other small computers of the time only offered [[BASIC]]. In the 1980s, the ''VSAPL'' program product enjoyed widespread usage with [[Conversational Monitor System|CMS]], [[Time Sharing Option|TSO]], [[VSPC]], [[MUSIC/SP]] and [[CICS]] users.<br />
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In 1973-1974, Dr. Patrick E. Hagerty directed the implementation of the University of Maryland APL interpreter for the [[UNIVAC 1100/2200 series|Sperry Univac 1100]] Series mainframe computers.<ref>{{cite web | last=Minker | first=Jack | title=Beginning of Computing and Computer Sciences at the University of Maryland | url=http://www.cs.umd.edu/department/dept-history/minker-report.pdf | publisher=University of Maryland | accessdate=23 May 2011 | location=Section 2.3.4 | page=38 | format=PDF |date=January 2004 | archiveurl= http://web.archive.org/web/20110610064807/http://www.cs.umd.edu/department/dept-history/minker-report.pdf | archivedate= 10 June 2011 }}</ref> At the time, Sperry had nothing. In 1974, student Alan Stebbens was assigned the task of implementing an internal function.<ref>{{cite web | last=Stebbens | first=Alan | title=How it all began | url=http://lathwellproductions.ca/wordpress/film-synopsis/comments-from-linkedin/#comment-15 | publisher=LinkedIn | accessdate=22 May 2011}}</ref><br />
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Several timesharing firms sprang up in the 1960s and 1970s that sold APL services using modified versions of the IBM APL\360<ref name="apl360" /> interpreter. In North America, the better-known ones were [[I. P. Sharp Associates]], [[Scientific Time Sharing Corporation|STSC]], and [[The Computer Company]] (TCC). With the advent first of less expensive mainframes such as the [[IBM 4300]] and later the personal computer, the timesharing industry had all but disappeared by the mid-1980s.<br />
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''Sharp APL'' was available from [[I. P. Sharp Associates]], first on a timesharing basis in the 1960s, and later as a program product starting around 1979. ''Sharp APL'' was an advanced APL implementation with many language extensions, such as ''packages'' (the ability to put one or more objects into a single variable), file system, nested arrays, and [[shared variable]]s.<br />
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APL interpreters were available from other mainframe and mini-computer manufacturers as well, notably [[Burroughs Corporation|Burroughs]], [[Control Data Corporation|CDC]], [[Data General]], [[Digital Equipment Corporation|DEC]], [[Harris Corporation|Harris]], [[Hewlett-Packard]], [[Siemens AG]], [[Xerox]], and others.<br />
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[[Garth Foster]] of [[Syracuse University]] sponsored regular meetings of the APL implementers' community at Syracuse's Minnowbrook Conference Center in rural upstate [[New York State|New York]]. In later years, Eugene McDonnell organized similar meetings at the [[Asilomar Conference Grounds]] near Monterey, California, and at Pajaro Dunes near Watsonville, California. The SIGAPL special interest group of the [[Association for Computing Machinery]] continues to support the APL community.<ref>{{cite web|url=http://www.sigapl.org/ |title=SIGAPL Home Page |publisher=Sigapl.org |date= |accessdate=2013-06-17}}</ref><br />
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In 1979, Iverson received the [[Turing Award]] for his work on APL.<ref>{{cite web | url=http://awards.acm.org/citation.cfm?id=9147499&srt=all&aw=140&ao=AMTURING | title=Turing Award Citation 1979 | publisher=Awards.acm.org | accessdate=2010-02-03}}</ref><br />
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=== APL2 ===<br />
Starting in the early 1980s, IBM APL development, under the leadership of Dr [[Jim Brown (Computer Scientist)|Jim Brown]], implemented a new version of the APL language that contained as its primary enhancement the concept of ''nested arrays'', where an array can contain other arrays, as well as new language features which facilitated the integration of nested arrays into program workflow. Ken Iverson, no longer in control of the development of the APL language, left IBM and joined [[I. P. Sharp Associates]], where one of his major contributions was directing the evolution of Sharp APL to be more in accordance with his vision.{{Citation needed|date=April 2010}}<br />
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As other vendors were busy developing APL interpreters for new hardware, notably [[Unix]]-based [[microcomputer]]s, APL2 was almost always the standard chosen for new APL interpreter developments. Even today, most APL vendors cite APL2 compatibility, which only approaches 100%, as a selling point for their products.{{Citation needed|date=April 2010}}<br />
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''APL2'' for IBM mainframe computers is still available as of 2013-07-29,<ref>{{cite web|title=A programming language for problem solving, visualization and database access|url=http://www-03.ibm.com/software/products/us/en/apl2|accessdate=29 July 2013}}</ref> and was first available for [[Conversational Monitor System|CMS]] and [[Time Sharing Option|TSO]] in 1984.<ref>{{cite journal | url=http://www.research.ibm.com/journal/sj/304/ibmsj3004C.pdf | title=The IBM family of APL systems | first=Adin D. | last=Falkoff | year=1991 | journal=IBM Systems Journal | volume=30 | issue=4 | pages=416–432 | publisher=[[IBM]] | format=PDF | accessdate=2009-06-13 | doi=10.1147/sj.304.0416}}</ref> The APL2 Workstation edition (Windows, OS/2, AIX, Linux, and Solaris) followed much later in the early 1990s.{{Citation needed|date=April 2010}}<br />
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=== Microcomputers ===<br />
The first microcomputer implementation of APL was on the [[Intel 8008]]-based [[MCM/70]], the first general purpose personal computer, in 1973.<br />
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IBM's own [[IBM 5100]] microcomputer (1975) offered APL as one of two built-in ROM-based interpreted languages for the computer, complete with a keyboard and display that supported all the special symbols used in the language.<br />
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In 1976 DNA Systems introduced an APL interpreter for their TSO Operating System, which ran timesharing on the IBM 1130, Digital Scientific Meta-4, General Automation GA 18/30 and Computer Hardware CHI 21/30.<br />
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The [[VideoBrain Family Computer]], released in 1977, only had one programming language available for it, and that was a dialect of APL called APL/S.<ref>[http://books.google.com/books?id=OQEAAAAAMBAJ&pg=PA133&lpg=PA133&dq=videobrain+family+computer+apl/s&source=bl&ots=_tmStYA0UG&sig=mxb5bqgWuA_NBVww1ywhpA1iNWY&hl=en&ei=rleIS8_hPN2mtgez8vi0DQ&sa=X&oi=book_result&ct=result&resnum=5&ved=0CBQQ6AEwBA#v=onepage&q=videobrain%20family%20computer%20apl%2Fs&f=false "VideoBrain Family Computer"], ''[[Popular Science]]'', November 1978, advertisement.</ref><br />
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A Small APL for the [[Intel]] 8080 called EMPL was released in 1977, and Softronics APL, with most of the functions of full APL, for 8080-based CP/M systems was released in 1979.<br />
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In 1977, the Canadian firm Telecompute Integrated Systems, Inc. released a business-oriented APL interpreter known as TIS APL, for Z80-based systems. It featured the full set of file functions for APL, plus a full screen input and switching of right and left arguments for most dyadic operators by introducing the <code>~.</code> prefix to all single character dyadic functions such as <code>-</code> or <code>/</code>.<br />
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Vanguard APL was available for [[Z80]] [[CP/M]]-based processors in the late 1970s. [[The Computer Company|TCC]] released APL.68000 in the early 1980s for Motorola [[68000]]-based processors, this system being the basis for MicroAPL Limited's [[APLX]] product. I. P. Sharp Associates released a version of their APL interpreter for the [[IBM PC]] and [[PC/370]].<ref>Higgins, Donald S., [http://portal.acm.org/citation.cfm?id=382167.383025 "PC/370 virtual machine"], ''ACM SIGSMALL/PC Notes'', Volume 11, Issue 3 (August 1985), pp.23 - 28, 1985.</ref> For the IBM PC, an [[emulator]] was written that facilitated reusing much of the IBM 370 mainframe code. Arguably, the best known APL interpreter for the IBM Personal Computer was [[Scientific Time Sharing Corporation|STSC]]'s APL*Plus/PC.{{citation needed|date=December 2012}}<br />
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The [[Commodore SuperPET]], introduced in 1981, included an APL interpreter developed by the [[University of Waterloo]].<br />
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In the early 1980s, the Analogic Corporation developed ''The APL Machine'', which was an [[vector processor|array processing]] computer designed to be programmed only in APL. There were actually three processing units, the user's workstation, an [[IBM PC]], where programs were entered and edited, a [[Motorola 68000]] processor that ran the APL interpreter, and the Analogic array processor that executed the primitives.<ref>[http://groups.yahoo.com/group/apl-l/message/8180],''Yahoo! Group APL-L'', April, 2003</ref> At the time of its introduction, The APL Machine was likely the fastest APL system available. Although a technological success, The APL Machine was a marketing failure. The initial version supported a single process at a time. At the time the project was discontinued, the design had been completed to allow multiple users. As an aside, an unusual aspect of The APL Machine was that the library of workspaces was organized such that a single function or variable that was shared by many workspaces existed only once in the library. Several of the members of The APL Machine project had previously spent a number of years with Burroughs implementing ''APL\700''.<br />
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At one stage, it was claimed by Bill Gates in his [[Open Letter to Hobbyists]], [[Microsoft Corporation]] planned to release a version of APL, but these plans never materialized.<br />
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An early 1978 publication of [[Rodnay Zaks]] from [[Sybex]] was ''A microprogrammed APL implementation'' ISBN 0-89588-005-9, which is the complete source listing for the microcode for a Digital Scientific Corporation Meta 4 microprogrammable processor implementing APL. This topic was also the subject of his PhD thesis.<ref>Zaks, Rodnay, "A Microprogrammed APL Implementation,", Ph.D. Thesis, University of California, Berkeley, June 1972.</ref><ref>Zaks, Rodnay, "Microprogrammed APL,", Fifth IEEE Computer Conference Proceedings, Sep. 1971 p 193</ref><br />
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In 1979, [[William Yerazunis]] wrote a partial version of APL in [[Prime Computer]] FORTRAN, extended it with graphics primitives, and released it. This was also the subject of his Masters thesis.<ref>{{cite web | url=http://opac.lib.rpi.edu/search~S6?/Xa:%28yerazunis%29&m=t&m=r&m=g&m=a&m=o&SORT=D&searchscope=6/Xa:%28yerazunis%29&m=t&m=r&m=g&m=a&m=o&SORT=D&searchscope=6&SUBKEY=a%3A%28yerazunis%29/1%2C14%2C14%2CB/frameset&FF=Xa:%28yerazunis%29&m=t&m=r&m=g&m=a&m=o&SORT=D&searchscope=6&2%2C2%2C | title=A Partial Implementation of APL with Graphics Primitives for PRIME Computers | author=William Yerazunis | accessdate=2013-08-14 }}</ref><br />
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=== Extensions ===<br />
Recent extensions to APL include:<br />
* [[Object-oriented programming]]<ref name="APLX : New features in Version 5">{{cite web | url=http://www.microapl.co.uk/apl/aplxv5.html | title=APLX : New features in Version 5 | publisher=Microapl.co.uk | date = 2009-04-01 | accessdate=2010-02-03 | archiveurl= http://web.archive.org/web/20100106160559/http://www.microapl.co.uk/apl/aplxv5.html | archivedate= 6 January 2010 }}</ref><br />
* Support for [[.NET Framework|.NET]],<ref name="APLX : New features in Version 5" /> ActiveX,<ref>{{cite web | url=http://www.apl2000.com/netaccess.php | title=APL2000 NetAccess | publisher=Apl2000.com | accessdate=2010-02-03}}</ref> operating system resources & connectivity<br />
* APL as a native .NET language using Visual Studio 2008<ref>{{cite web | url=http://www.visualapl.com/library/aplnext/Visual_APLNext.htm | title=Introduction to Visual APL | publisher=Visualapl.com | accessdate=2010-02-03}}</ref><br />
* Integrated charting<ref>{{cite web | url=http://www.dyalog.com/version12.html | title=Dyalog Built-in Charting | publisher=Dyalog.com | accessdate=2010-02-03 | archiveurl= http://web.archive.org/web/20100203150500/http://www.dyalog.com/version12.html | archivedate= 3 February 2010 }}</ref> and manipulation of SQL databases<br />
* XML-array conversion primitives<ref name="APLX : New features in Version 5" /><ref>{{cite web | url=http://www.dyalog.com/help/12.1/html/xml%20convert.htm | title=XML-to-APL Conversion tool in Dyalog APL and APLX | publisher=Dyalog.com | accessdate=2010-02-03}}</ref><br />
* [[Lambda expressions]]<ref>{{cite web | url=http://www.dyalog.com/download/dfns.pdf | title=Dynamic Functions in Dyalog APL&nbsp;— John Scholes. 1997 | format=PDF | accessdate=2010-02-03}}</ref><br />
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== Design ==<br />
{{unreferenced section|date=December 2012}}<br />
{{tone|section|date=January 2012}}<br />
<br />
Unlike traditionally structured programming languages, code in APL is typically structured as chains of [[unary operation|monadic]] or [[binary operation|dyadic]] [[function (programming)|functions]] and [[higher-order function|operators]] acting on [[array data type|arrays]]. As APL has many nonstandard ''primitives'' (functions and operators, indicated by a single symbol or a combination of a few symbols), it does not have function or [[operator precedence]]. Early APL implementations did not have [[control flow|control structures]] (do or while loops, if-then-else), but by using array operations, use of [[structured programming]] constructs was not necessary, as an operation was carried out on all the elements of the array in a single statement. For example, the iota (ι) function (ιN yields a one-dimensional array, or vector, 1 2 3 ... N) can replace for-loop [[iteration]]. More recent implementations of APL generally include comprehensive control structures, so that data structure and program control flow can be clearly and cleanly separated.<br />
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The APL environment is called a ''workspace''. In a workspace the user can define programs and data, i.e. the data values exist also outside the programs, and the user can manipulate the data without the necessity to define a program. For example,<br />
<br />
<pre><br />
N ← 4 5 6 7<br />
</pre><br />
<br />
assigns the [[coordinate vector|vector]] value 4 5 6 7 to N;<br />
<br />
<pre><br />
N+4<br />
</pre><br />
<br />
adds 4 to all elements of the vector (giving 8 9 10 11) and prints them (a return value not assigned at the end of a statement to a variable using the assignment arrow <math>\leftarrow </math> is displayed by the APL interpreter);<br />
<br />
<pre><br />
+/N<br />
</pre><br />
<br />
prints the sum of N, i.e. 22.<br />
<br />
These operations can then be combined in a single statement, e.g.<br />
<pre><br />
m ← +/ιn<br />
</pre><br />
<br />
assigns to m the scalar value n(n+1)/2, i.e. <math>\displaystyle\sum\limits_{i=1}^n i</math>.<br />
<br />
The user can save the workspace with all values, programs and execution status.<br />
<br />
APL is well known for its use of a set of non-[[ASCII]] symbols, which are an extension of traditional arithmetic and algebraic notation. Having single character names for [[SIMD]] vector functions is one way that APL enables compact formulation of algorithms for data transformation such as computing [[Conway's Game of Life]] in one line of code.<ref>{{cite web|url=http://catpad.net/michael/apl |title=example |publisher=Catpad.net |date= |accessdate=2013-06-17}}</ref> In nearly all versions of APL, it is theoretically possible to express any computable function in one expression, that is, in one line of code.{{citation needed|date=December 2012}}<br />
<br />
Because of the unusual [[character set]], many programmers use special [[computer keyboard|keyboards]] with APL keytops for authoring APL code. Although there are various ways to write APL code using only ASCII characters,<ref>[http://www.math.uwaterloo.ca/apl_archives/apl/translit.schemes Dickey, Lee, A list of APL Transliteration Schemes], 1993</ref> in practice, it is almost never done. (This may be thought to support Iverson's thesis about [[Linguistic relativity|notation as a tool of thought]].)<ref>Iverson K.E.,<br />
"[http://www.jsoftware.com/papers/tot.htm Notation as a Tool of Thought]", ''Communications of the ACM'', 23: 444-465 (August 1980).</ref> Most if not all modern implementations use standard keyboard layouts, with special mappings or [[input method editor]]s to access non-ASCII characters. Historically, the APL font has been distinctive, with uppercase italic alphabetic characters and upright numerals and symbols. Most vendors continue to display the APL character set in a custom font.<br />
<br />
Advocates of APL{{who|date=October 2012}} claim that the examples of so-called write-only code are almost invariably examples of poor programming practice or novice mistakes, which can occur in any language. Advocates of APL also claim that they are far more productive with APL than with more conventional computer languages, and that working software can be implemented in far less time and with far fewer programmers than using other technology. APL lets an individual solve harder problems faster. Also, being compact and terse, APL lends itself well to larger scale software development as complexity arising from a large number of lines of code can be dramatically reduced. Many APL advocates and practitioners view programming in standard programming languages, such as [[COBOL]] and [[Java (programming language)|Java]], as comparatively tedious. APL is often found where time-to-market is important, such as with trading systems. {{citation needed|date=March 2013}}<br />
<br />
Iverson later designed the [[J programming language]], which uses [[ASCII]] with [[digraph (computing)|digraphs]] instead of special symbols.<br />
<br />
== Execution ==<br />
{{unreferenced section|date=December 2012}}<br />
<br />
=== Interpreters ===<br />
APLNext (formerly APL2000) offers an advanced APL interpreter that operates under Linux, Unix, and Windows. It supports Windows automation, supports calls to operating system and user defined DLLs, has an advanced APL File System, and represents the current level of APL language development. APL2000's product is an advanced continuation of [[Scientific Time Sharing Corporation|STSC]]'s successful APL*Plus/PC and APL*Plus/386 product line.<br />
<br />
''Dyalog APL'' is an advanced APL interpreter that operates under Linux, Unix, and Windows. Dyalog has innovative extensions to the APL language, which include new [[Object-oriented programming|object-oriented]] features, numerous language enhancements, plus a consistent [[namespace]] model used for both its Microsoft Automation interface, as well as native namespaces. For the Windows platform, Dyalog APL offers tight integration with .NET, plus limited integration with the Microsoft Visual Studio development platform.<br />
<br />
IBM offers a version of IBM APL2 for IBM AIX, Linux, Sun Solaris and Windows systems. This product is a continuation of APL2 offered for IBM mainframes. IBM APL2 was arguably the most influential APL system, which provided a solid implementation standard for the next set of extensions to the language, focusing on nested arrays.<br />
<br />
NARS2000 is an open source APL interpreter written by Bob Smith, a well-known APL developer and implementor from [[STSC]] in the 1970s and 1980s. NARS2000 contains advanced features and new datatypes, runs natively under Windows (32- and 64-bit versions), and runs under [[Linux]] and Apple [[Mac OS]] with [[Wine (software)|Wine]].<br />
<br />
MicroAPL Limited offers [[APLX]], a full-featured 64 bit interpreter for [[Linux]], [[Microsoft Windows]], and [[Mac OS]] systems. The core language is closely modelled on IBM's APL2 with various enhancements. ''APLX'' includes close integration with .NET, Java, Ruby and R.<br />
<br />
[[Soliton Incorporated]] offers the SAX interpreter (Sharp APL for Unix) for Unix and Linux systems, which is a further development of I. P. Sharp Associates' Sharp APL product. Unlike most other APL interpreters, [[Kenneth E. Iverson]] had some influence in the way nested arrays were implemented in Sharp APL and SAX. Nearly all other APL implementations followed the course set by IBM with APL2, thus some important details in Sharp APL differ from other implementations.<br />
<br />
OpenAPL is an open source implementation of APL published by Branko Bratkovic, based on code by [[Ken Thompson]] of Bell Laboratories, together with contributions by others. It is licensed under the [[GNU General Public License]], and runs on Unix systems including Linux on x86, SPARC and other CPUs.<br />
<br />
[http://www.gnu.org/software/apl GNU APL] is a free implementation of ISO Standard 13751 and hence similar to APL2. It runs on GNU/Linux and on Windows using [[Cygwin]]. It uses Unicode internally. GNU APL was written by Jürgen Sauermann.<br />
<br />
=== Compilers ===<br />
APL programs are normally [[interpreted language|interpreted]] and less often [[compiled language|compiled]]. In reality, most APL compilers [[translated]] source APL to a lower level language such as [[C (programming language)|C]], leaving the machine-specific details to the lower level compiler. Compilation of APL programs was a frequently discussed topic in conferences. Although some of the newer enhancements to the APL language such as nested arrays have rendered the language increasingly difficult to compile, the idea of APL compilation is still under development today.<br />
<br />
In the past, APL compilation was regarded as a means to achieve execution speed comparable to other mainstream languages, especially on mainframe computers.<br />
Several APL compilers achieved some levels of success, though comparatively little of the development effort spent on APL over the years went to perfecting compilation into machine code.<br />
<br />
As is the case when moving APL programs from one vendor's APL interpreter to another, APL programs invariably will require changes to their content. Depending on the compiler, variable declarations might be needed, certain language features would need to be removed or avoided, or the APL programs would need to be cleaned up in some way. Some features of the language, such as the execute function (an expression evaluator) and the various [[reflection (computer science)|reflection]] and [[introspection (computer science)|introspection]] functions from APL, such as the ability to return a function's text or to materialize a new function from text, are simply not practical to implement in machine code compilation.<br />
<br />
A commercial compiler was brought to market by [[Scientific Time Sharing Corporation|STSC]] in the mid-1980s as an add-on to IBM's VSAPL Program Product. Unlike more modern APL compilers, this product produced machine code that would execute only in the interpreter environment, it was not possible to eliminate the interpreter component. The compiler could compile many scalar and vector operations to machine code, but it would rely on the APL interpreter's services to perform some more advanced functions, rather than attempt to compile them. However, dramatic speedups did occur, especially for heavily iterative APL code.<br />
<br />
Around the same time, the book ''[[An APL Compiler]]'' by [[Timothy Budd]] appeared in print. This book detailed the construction of an APL translator, written in [[C (programming language)|C]], which performed certain optimizations such as [[loop fusion]] specific to the needs of an array language. The source language was APL-like in that a few rules of the APL language were changed or relaxed to permit more efficient compilation. The translator would emit C code which could then be compiled and run well outside of the APL workspace.<br />
<br />
Today, execution speed is less critical and many popular languages are implemented using [[virtual machine]]s: instructions that are interpreted at runtime. The Burroughs/Unisys ''APLB'' interpreter (1982) was the first to use dynamic incremental compilation to produce code for an APL-specific virtual machine. It recompiled on-the-fly as identifiers changed their functional meanings. In addition to removing [[parsing]] and some error checking from the main execution path, such compilation also streamlines the repeated entry and exit of user-defined functional operands. This avoids the stack setup and take-down for function calls made by APL's built-in operators such as Reduce and Each.<br />
<br />
''APEX'', a research APL compiler, is available from [[Snake Island Research Inc]]. APEX compiles flat APL (a subset of ISO N8485) into [[SAC (programming language)|SAC]], a functional array language with parallel semantics, and currently runs under [[Linux]]. APEX-generated code uses [[loop fusion]] and [[array contraction]], special-case algorithms not generally available to interpreters (e.g., upgrade of [[permutation vector]]), to achieve a level of performance comparable to that of [[Fortran]].<br />
<br />
The APLNext ''VisualAPL'' system is a departure from a conventional APL system in that VisualAPL is a true .NET language which is fully interoperable with other .NET languages such as [[VB.NET]] and [[C Sharp (programming language)|C#]]. VisualAPL is inherently object oriented and Unicode-based. While VisualAPL incorporates most of the features of standard APL implementations, the VisualAPL language extends standard APL to be .NET-compliant. VisualAPL is hosted in the standard Microsoft Visual Studio IDE and as such, invokes compilation in a manner identical to that of other .NET languages. By producing [[Common Intermediate Language]] (CIL) code, it utilizes the Microsoft just-in-time compiler (JIT) to support 32-bit or 64-bit hardware. Substantial performance speed-ups over standard APL have been reported,{{Citation needed|date=September 2009}} especially when (optional) strong typing of function arguments is used.<br />
<br />
An APL to [[C Sharp (programming language)|C#]] translator is available from [[Causeway Graphical Systems]]. This product was designed to allow the APL code, translated to equivalent C#, to run completely outside of the APL environment. The Causeway compiler requires a run-time library of array functions. Some speedup, sometimes dramatic, is visible, but happens on account of the optimisations inherent in Microsoft's .NET Framework.<br />
<br />
=== Matrix optimizations ===<br />
{{unreferenced section|date=December 2012}}<br />
<br />
APL was unique in the speed with which it could perform complicated matrix operations. For example, a very large matrix multiplication would take only a few seconds on a machine that was much less powerful than those today.{{Citation needed|date=June 2010}} There were both technical and economic reasons for this advantage:<br />
* Commercial interpreters delivered highly tuned linear algebra library routines.<br />
* Very low interpretive overhead was incurred per-array—not per-element.<br />
* APL response time compared favorably to the runtimes of early optimizing compilers.<br />
* IBM provided [[microcode]] assist for APL on a number of IBM/370 mainframes.<br />
<br />
Phil Abrams' much-cited paper "An APL Machine" illustrated how APL could make effective use of [[lazy evaluation]] where calculations would not actually be performed until the results were needed and then only those calculations strictly required. An obvious (and easy to implement) lazy evaluation is the ''J-vector'': when a monadic ''iota'' is encountered in the code, it is kept as a representation instead of being expanded in memory; in future operations, a ''J-vector'''s contents are the loop's induction register, not reads from memory.<br />
<br />
Although such techniques were not widely used by commercial interpreters, they exemplify the language's best survival mechanism: not specifying the order of scalar operations or the exact contents of memory. As standardized, in 1983 by [[American National Standards Institute|ANSI]] [[working group]] X3J10, APL remains highly [[data parallelism|data-parallel]]. This gives language implementers immense freedom to schedule operations as efficiently as possible. As computer innovations such as [[cache memory]], and [[SIMD]] execution became commercially available, APL programs are ported with almost no extra effort spent re-optimizing low-level details.<br />
<br />
== Terminology ==<br />
{{unreferenced section|date=December 2012}}<br />
<br />
APL makes a clear distinction between ''functions'' and ''operators''. Functions take arrays (variables or constants or expressions) as arguments, and return arrays as results. Operators (similar to [[higher-order function]]s) take functions or arrays as arguments, and derive related functions. For example the "sum" function is derived by applying the "reduction" operator to the "addition" function. Applying the same reduction operator to the "maximum" function (which returns the larger of two numbers) derives a function which returns the largest of a group (vector) of numbers. In the J language, Iverson substituted the terms "verb" for "function" and "adverb" or "conjunction" for "operator".<br />
<br />
APL also identifies those features built into the language, and represented by a symbol, or a fixed combination of symbols, as ''primitives''. Most primitives are either functions or operators. Coding APL is largely a process of writing non-primitive functions and (in some versions of APL) operators. However a few primitives are considered to be neither functions nor operators, most noticeably assignment.<br />
<br />
Some words used in APL literature have meanings that differ from those in both mathematics and the generality of computer science.<br />
<br />
{| class="wikitable"<br />
|-<br />
! Term<br />
! Description<br />
|-<br />
! function<br />
| operation or mapping that takes zero, one (right) or two (left & right) array valued arguments and may return an array valued result. A function may be:<br />
* Primitive: built-in and represented by a single glyph;<ref name="aplxch6">{{cite web | url=http://www.microapl.co.uk/APL/apl_concepts_chapter6.html | title=APL concepts | publisher=Microapl.co.uk | accessdate=2010-02-03}}</ref><br />
* Defined: as a named and ordered collection of program statements;<ref name="aplxch6" /><br />
* Derived: as a combination of an operator with its arguments.<ref name="aplxch6" /><br />
|-<br />
! array<br />
| data valued object of zero or more [[orthogonal]] dimensions in [[row major|row-major]] order in which each item is a primitive scalar datum or another array.<ref>{{cite web | url=http://www.nial.com/ArrayTheory.html | title=Nested array theory | publisher=Nial.com | accessdate=2010-02-03}}</ref><br />
|-<br />
! niladic<br />
| not taking or requiring any arguments,<ref name="Bohman_Froberg">"Programmera i APL", Bohman, Fröberg, Studentlitteratur, ISBN 91-44-13162-3</ref><br />
|-<br />
! monadic<br />
| requiring only one argument; on the right for a function, on the left for an operator, unary<ref name="Bohman_Froberg" /><br />
|-<br />
! dyadic<br />
| requiring both a left and a right argument, binary<ref name="Bohman_Froberg" /><br />
|-<br />
! ambivalent or nomadic<br />
| capable of use in a monadic or dyadic context, permitting its left argument to be elided<ref name="aplxch6" /><br />
|-<br />
! operator<br />
| operation or mapping that takes one (left) or two (left & right) function or array valued arguments (operands) and derives a function. An operator may be:<br />
* Primitive: built-in and represented by a single glyph;<ref name="aplxch6" /><br />
* Defined: as a named and ordered collection of program statements.<ref name="aplxch6" /><br />
|}<br />
<br />
== Syntax ==<br />
{{unreferenced section|date=December 2012}}<br />
<br />
{{Main|APL syntax and symbols}}<br />
<br />
== Examples ==<br />
{{unreferenced section|date=December 2012}}<br />
<br />
This displays "Hello, world":<br />
<br />
<pre><br />
'Hello, world'<br />
</pre><br />
<br />
A design theme in APL is to define default actions in some cases that would be syntax errors in their equivalent forms in most other programming languages. APL is economical in its character usage.<br />
<br />
The 'Hello, world' string constant above displays, because display is the default action on any expression for which no action is specified explicitly (e.g. assignment, function parameter).<br />
<br />
Another example of this theme: Exponentiation in APL is of the form 2⋆3 raising 2 to the power 3. But if no base is specified, as in ⋆3, then one would have a syntax error in its equivalent form in most other programming languages. APL however assumes the missing base to be the natural logarithm constant [[e (mathematical constant)|e]] (2.71828....), so interpreting ⋆3 as 2.71828⋆3.<br />
<br />
This following immediate-mode expression generates a typical set of Pick 6 [[lottery]] numbers: six [[pseudo-random]] [[integer]]s ranging from 1 to 40, ''guaranteed non-repeating'', and displays them sorted in ascending order:<br />
<br />
<pre><br />
x[⍋x←6?40]<br />
</pre><br />
<br />
This combines the following APL functions:<br />
* The first to be executed (APL executes from right to left) is the dyadic function "?" (named "Deal" when dyadic) that returns a [[array data structure|vector]] consisting of a select number (left argument: 6 in this case) of random integers ranging from 1 to a specified maximum (right argument: 40 in this case), which, if said maximum ≥ vector length, is guaranteed to be non-repeating.<br />
* This vector is then assigned to the variable x, because it is needed later.<br />
* This vector is then sorted in ascending order by the monadic "⍋" function, which has as its right argument everything to the right of it up to the next unbalanced close-bracket or close-parenthesis. The result of ⍋ is the indices that will put its argument into ascending order.<br />
* Then the output of ⍋ is applied to the variable x, which we saved earlier, and it puts the items of x into ascending sequence.<br />
<br />
Since there is no function to the left of the left-most x to tell APL what to do with the result, it simply outputs it to the display (on a single line, separated by spaces) without needing any explicit instruction to do that.<br />
<br />
"?" also has a monadic equivalent called "Roll", which simply returns a single random integer between 1 and its sole operand [to the right of it], inclusive. Thus, a [[role-playing game]] program might use the expression "?20" to roll a twenty-sided die.<br />
<br />
The following expression finds all [[prime number]]s from 1 to R. In both time and space, the calculation complexity is <math>O(R^2)\,\!</math> (in [[Big O notation]]).<br />
<br />
<pre><br />
(~R∊R∘.×R)/R←1↓ιR<br />
</pre><br />
<br />
Executed from right to left, this means:<br />
* <code>ιR</code> creates a vector containing [[integer]]s from <code>1</code> to <code>R</code> (if <code>R = 6</code> at the beginning of the program, <code>ιR</code> is <code>1 2 3 4 5 6</code>)<br />
* Drop first element of this vector (<code>↓</code> function), i.e. <code>1</code>. So <code>1↓ιR</code> is <code>2 3 4 5 6</code><br />
* Set <code>R</code> to the new vector (<code>←</code>, assignment primitive), i.e. <code>2 3 4 5 6</code><br />
* The <code>/</code> compress function is dyadic (binary) and the interpreter first evaluates its left argument:<br />
* Generate [[outer product]] of <code>R</code> multiplied by <code>R</code>, i.e. a matrix that is the ''[[multiplication table]]'' of R by R (<code>°.×</code> function), i.e.<br />
<br />
{| class="wikitable"<br />
|-<br />
| 4<br />
| 6<br />
| 8<br />
| 10<br />
| 12<br />
|-<br />
| 6<br />
| 9<br />
| 12<br />
| 15<br />
| 18<br />
|-<br />
| 8<br />
| 12<br />
| 16<br />
| 20<br />
| 24<br />
|-<br />
| 10<br />
| 15<br />
| 20<br />
| 25<br />
| 30<br />
|-<br />
| 12<br />
| 18<br />
| 24<br />
| 30<br />
| 36<br />
|}<br />
* Build a vector the same length as <code>R</code> with <code>1</code> in each place where the corresponding number in <code>R</code> is in the outer product matrix (<code>∈</code>, set inclusion function), i.e. <code>0 0 1 0 1</code><br />
* Logically negate (not) the values in the vector (change zeros to ones and ones to zeros) (<code>∼</code>, logical not function), i.e. <code>1 1 0 1 0</code><br />
* Select the items in <code>R</code> for which the corresponding element is <code>1</code> (<code>/</code> compress function), i.e. <code>2 3 5</code><br />
(Note, this assumes the APL origin is 1, i.e., indices start with 1. APL can be set to use 0 as the origin, which is convenient for some calculations.)<br />
<br />
The following expression [[sorting|sorts]] a word list stored in matrix X according to word length:<br />
<br />
<pre><br />
X[⍋X+.≠' ';]<br />
</pre><br />
<br />
The following function "life", written in Dyalog APL, takes a boolean matrix and calculates the new generation according to [[Conway's Game of Life]]. It demonstrates the power of APL to implement a complex algorithm in very little code, but it is also very hard to follow unless one has an advanced knowledge of APL.<br />
<br />
<pre><br />
life←{↑1 ⍵∨.∧3 4=+/,¯1 0 1∘.⊖¯1 0 1∘.⌽⊂⍵}<br />
</pre><br />
<br />
In the following example, also Dyalog, the first line assigns some HTML code to a variable <code>txt</code> and then uses an APL expression to remove all the HTML tags, returning the text only as shown in the last line.<br />
<br />
txt←'{{code|lang=html4strict|<html><body><p>This is ''emphasized'' text</p></body></html>}}'<br />
⎕←{⍵/⍨~{⍵∨≠\⍵}⍵∊'<>'}txt<br />
{{color|#666666|This is emphasized text}}.<br />
<br />
== Character set ==<br />
{{unreferenced section|date=December 2012}}<br />
{{Main|APL (codepage)}}<br />
<br />
APL has always been criticized for its choice of a unique, non-standard character set. Some who learn it become ardent adherents, suggesting that there is some weight behind Iverson's idea that the notation used does make a difference. In the beginning, there were few terminal devices that could reproduce the APL character set—the most popular ones employing the [[IBM Selectric]] print mechanism along with a special APL type element. Over time, with the universal use of high-quality graphic display, printing devices and [[Unicode]] support, the APL character font problem has largely been eliminated; however, the problem of entering APL characters requires the use of [[input method editor]]s or special keyboard mappings, which may frustrate beginners accustomed to other programming languages.<br />
<br />
== Use ==<br />
APL has long had a small and fervent user base. It was and still is popular in financial and insurance applications, in simulations, and in mathematical applications. APL has been used in a wide variety of contexts and for many and varied purposes. A newsletter titled "Quote-Quad" dedicated to APL has been published since the 1970s by the SIGAPL section of the Association for Computing Machinery (Quote-Quad is the name of the APL character used for text input and output).<ref>[http://www.sigapl.org/qq.htm Quote-Quad newsletter]{{dead link|date=June 2013}}</ref><br />
<br />
Before the advent of full-screen systems and until as late as the mid-1980s, systems were written such that the user entered instructions in his own business specific vocabulary. APL [[time-sharing]] vendors delivered applications in this form. On the [[I. P. Sharp Associates|I. P. Sharp]] timesharing system, a workspace called 39 MAGIC offered access to financial and airline data plus sophisticated (for the time) graphing and reporting. Another example is the GRAPHPAK workspace supplied with IBM's APL2.<br />
<br />
Because of its matrix operations, APL was for some time quite popular for computer graphics programming, where graphic transformations could be encoded as matrix multiplications. One of the first commercial computer graphics houses, [[Digital Effects (studio)|Digital Effects]], based in New York City, produced an APL graphics product known as "Visions", which was used to create television commercials and, reportedly, animation for the 1982 film ''[[Tron]]''. Digital Effects' use of APL was informally described at a number of SIGAPL conferences in the late 1980s; examples discussed included the early UK [[Channel 4]] TV logo/ident. What is not clear is the extent to which APL was directly involved in the making of ''Tron'', and at this point in time the reference is more of an urban legend or historic curio than much else.{{synthesis-inline|date=December 2012}}<br />
<br />
Interest in APL has steadily declined since the mid-1980s. This was partially due to the lack of a smooth migration path from higher performing mainframe implementations to low-cost personal computer alternatives, as APL implementations for computers before the [[Intel 80386]] released in the late 1980s were only suitable for small applications. The growth of end-user computing tools such as [[Microsoft Excel]] and [[Microsoft Access]] also eroded into potential APL usage. These are appropriate platforms for what may have been mainframe APL applications in the 1970s and 1980s. Some APL users migrated to the [[J programming language]], which offers more advanced features. Lastly, the decline was also due in part to the growth of [[MATLAB]], [[GNU Octave]], and [[Scilab]]. These scientific computing array-oriented platforms provide an interactive computing experience similar to APL, but more resemble conventional programming languages such as Fortran, and use standard ASCII.<br />
<br />
Notwithstanding this decline, APL finds continued use in certain fields, such as accounting research.<ref>{{cite web | url=http://www.gsb.stanford.edu/programs/phd/academic-experience/fields/accounting/requirements | title=Stanford Accounting PhD requirements | publisher=Gsb.stanford.edu | accessdate=2014-01-09}}</ref><br />
<br />
== Standardization ==<br />
APL has been standardized by the [[American National Standards Institute|ANSI]] [[working group]] X3J10 and [[International Organization for Standardization|ISO]]/[[International Electrotechnical Commission|IEC]] Joint Technical Committee 1 Subcommittee 22 Working Group 3. The Core APL language is specified in ISO 8485:1989, and the Extended APL language is specified in ISO/IEC 13751:2001.<br />
<br />
== See also ==<br />
* [[A+ (programming language)]]<br />
* [[APL Shared Variables]]<br />
* [[I. P. Sharp Associates]]<br />
* [[IBM Type-III Library]]<br />
* [[IBM 1130]]<br />
* [[Iverson Award]]<br />
* [[J (programming language)]]<br />
* [[LYaPAS]]<br />
* [[Scientific Time Sharing Corporation]]<br />
* [[Soliton Incorporated]]<br />
<br />
== References ==<br />
{{reflist | 30em}}<br />
<br />
== Further reading ==<br />
* [http://www.slac.stanford.edu/pubs/slacreports/slac-r-114.html ''An APL Machine''] (1970 Stanford doctoral dissertation by Philip Abrams)<br />
* [http://sigapl.org/Articles/MichaelMontalbanoPersonalViewOfAPL.php ''A Personal History Of APL''] (1982 article by [[Michael S. Montalbano]])<br />
* {{cite journal | url=http://www.research.ibm.com/journal/sj/304/ibmsj3004N.pdf | title=Language as an intellectual tool: From hieroglyphics to APL | year=1991 | first=Donald B. | last=McIntyre | journal=IBM Systems Journal | volume=30 | issue=4 | archiveurl=http://web.archive.org/web/20060504050437/http://www.research.ibm.com/journal/sj/304/ibmsj3004N.pdf | archivedate=May 4, 2006}}<br />
* {{cite journal | url=http://www.research.ibm.com/journal/sj/304/ibmsj3004O.pdf | title=A Personal view of APL | year=1991 | first=Kenneth E. | last=Iverson | authorlink=Kenneth E. Iverson | journal=IBM Systems Journal | volume=30 | issue=4 | archiveurl=http://web.archive.org/web/20080227012149/http://www.research.ibm.com/journal/sj/304/ibmsj3004O.pdf | archivedate=February 27, 2008}}<br />
* [http://www.softwarepreservation.org/projects/apl/Physics%20in%20APL2/APROGRAMMING%20LANGUAGE/view ''A Programming Language''] by [[Kenneth E. Iverson]]<br />
* [http://www.softwarepreservation.org/projects/apl/paper/197201_APL%20In%20Exposition_320-3010.pdf/view ''APL in Exposition''] by [[Kenneth E. Iverson]]<br />
* Brooks, Frederick P.; Kenneth Iverson (1965). ''Automatic Data Processing, System/360 Edition''. ISBN 0-471-10605-4.<br />
* {{cite book | last=Askoolum | first=Ajay | title=System Building with APL + Win | date = August 2006 | publisher=Wiley | isbn=978-0-470-03020-2}}<br />
* {{cite journal | url=http://www.research.ibm.com/journal/sj/032/falkoff.pdf | title=A Formal Description of SYSTEM/360 | first1=Adin D. | last1=Falkoff | first2=Kenneth E. | last2=Iverson | authorlink2=Kenneth E. Iverson | first3=Edward H. | last3=Sussenguth | authorlink3= Edward H. Sussenguth | journal=IBM Systems Journal | volume=3 | issue=3 | location=New York | year=1964 | archiveurl=http://web.archive.org/web/20080227012111/http://www.research.ibm.com/journal/sj/032/falkoff.pdf | archivedate=February 27, 2008}}<br />
* ''History of Programming Languages'', chapter 14{{Clarify|date=June 2010}}<br />
* {{cite book | last=Banon | first=Gerald Jean Francis | title=Bases da Computacao Grafica | publisher=Campus | location=Rio de Janeiro | year=1989 | page=141}}<br />
* {{cite book | last=LePage | first=Wilbur R. | title=Applied A.P.L. Programming | publisher=Prentice Hall | year=1978}}<br />
<br />
== External links ==<br />
{{Commons category}}<br />
* {{dmoz|Computers/Programming/Languages/APL|APL}}<br />
* [news://comp.lang.apl comp.lang.apl] [[newsgroup]] ([http://groups.google.com/group/comp.lang.apl/topics Google Groups archive])<br />
* [http://aplwiki.com/ APL Wiki]<br />
* [http://www.sigapl.org/ SIGAPL]<br />
* [http://www.fscript.org/documentation/OOPAL.pdf OOPAL: Integrating Array Programming in Object-Oriented Programming]<br />
* [http://web.archive.org/web/20080228023208/http://www.dyalog.dk/whatsnew/OO4APLERS.pdf An introduction to Object Oriented APL]<br />
* [http://www.espenhaug.com/black_scholes.html Comparison of Black-Scholes options pricing model in many languages, including APL]<br />
* [http://sourceforge.net/projects/openapl/ OpenAPL project page on Source Forge]<br />
* [http://home.earthlink.net/~swsirlin/aplcc.html The APL c compiler project (Tim Budd et al.)]<br />
* [http://www.apl2c.de/home/Links/links.html APL2C], a source of links to APL compilers<br />
* [http://www.microapl.com/apl/APL1_2.PDF A Practical Introduction to APL1 & APL2]<br />
* [http://www.computerhistory.org/atchm/computer-history-museum-software-license-agreement/ License Agreement], IBM APL\360 source code for non-commercial use<br />
<br />
[[Category:Array programming languages]]<br />
[[Category:Functional languages]]<br />
[[Category:Dynamic programming languages]]<br />
[[Category:APL programming language family]]<br />
[[Category:.NET programming languages]]<br />
[[Category:IBM software]]<br />
[[Category:Command shells]] <!-- [[IBM 5100]] as per the toggle switch on the front panel --><br />
[[Category:Programming languages created in 1964]]</div>Adminhttps://en.formulasearchengine.com/index.php?title=Kolmogorov_complexity&diff=218577Kolmogorov complexity2014-07-31T13:43:12Z<p>Admin: Reverted edits by MediaWiki spam cleanup (talk) to last revision by Jochen Burghardt</p>
<hr />
<div><br />
{| style="float:right"<br />
| [[Image:Mandelpart2 red.png|300px|right|thumb|This image illustrates part of the [[Mandelbrot set]] [[fractal]]. Simply storing the 24-bit color of each pixel in this image would require 1.62 million bits, but a small computer program can reproduce these 1.62 million bits using the definition of the Mandelbrot set and the coordinates of the corners of the image. Thus, the Kolmogorov complexity of the raw file encoding this bitmap is much less than 1.62 million.]]<br />
|}<br />
{{distinguish|descriptive complexity theory}}<br />
In [[algorithmic information theory]] (a subfield of [[computer science]] and [[mathematics]]), the '''Kolmogorov complexity''' (also known as '''descriptive complexity''', '''Kolmogorov–[[Gregory Chaitin|Chaitin]] complexity''', '''algorithmic entropy''', or '''program-size complexity''') of an object, such as a piece of text, is a measure of the [[computability]] resources needed to specify the object. It is named after [[Andrey Kolmogorov]], who first published on the subject in 1963.<ref>{{cite journal|authorlink=Andrey Kolmogorov|first=Andrey|last=Kolmogorov|year=1963|title=On Tables of Random Numbers| journal=[[Sankhya (journal)|Sankhyā]] Ser. A.|volume=25|pages=369–375|mr=178484}}</ref><ref>{{cite journal|authorlink=Andrey Kolmogorov|first=Andrey|last=Kolmogorov|year=1998|title=On Tables of Random Numbers| journal=Theoretical Computer Science|volume=207|issue=2|pages=387–395|doi=10.1016/S0304-3975(98)00075-9 |mr=1643414}}</ref><br />
<br />
For example, consider the following two [[string (computer science)|strings]] of 32 lowercase letters and digits:<br />
<br />
<pre>abababababababababababababababab</pre><br />
<pre>4c1j5b2p0cv4w1x8rx2y39umgw5q85s7</pre><br />
<br />
The first string has a short English-language description, namely "ab 16 times", which consists of '''11''' characters. The second one has no obvious simple description (using the same character set) other than writing down the string itself, which has '''32''' characters.<br />
<br />
More formally, the [[complexity]] of a string is the length of the shortest possible description of the string in some fixed [[Turing complete|universal]] description language (the sensitivity of complexity relative to the choice of description language is discussed below). It can be shown that the Kolmogorov complexity of any string cannot be more than a few bytes larger than the length of the string itself. Strings, like the ''abab'' example above, whose Kolmogorov complexity is small relative to the string's size are not considered to be complex.<br />
<br />
The notion of the Kolmogorov complexity can be used to state and prove impossibility results akin to [[Gödel's incompleteness theorem]] and [[halting problem|Turing's halting problem]].<br />
<br />
==Definition==<br />
The Kolmogorov complexity can be defined for any mathematical object, however for simplicity, it will be defined here for strings. We must first specify a description language for strings. Such a description language can be based on any computer programming language, such as [[Lisp programming language|Lisp]], [[Pascal (programming language)|Pascal]], or [[Java virtual machine]] bytecode. If '''P''' is a program which outputs a string ''x'', then '''P''' is a description of ''x''. The length of the description is just the length of '''P''' as a character string, multiplied by the number of bits in a character (e.g. 7 for [[ASCII]]).<br />
<br />
We could, alternatively, choose an encoding for [[Turing machine]]s, where an ''encoding'' is a function which associates to each Turing Machine '''M''' a bitstring <'''M'''>. If '''M''' is a Turing Machine which, on input ''w'', outputs string ''x'', then the concatenated string <'''M'''> ''w'' is a description of ''x''. For theoretical analysis, this approach is more suited for constructing detailed formal proofs and is generally preferred in the research literature. In this article, an informal approach is discussed.<br />
<br />
Any string ''s'' has at least one description, namely the program:<br />
<br />
'''function''' GenerateFixedString()<br />
'''return''' ''s''<br />
<br />
If a description of ''s'', ''d''(''s''), is of minimal length (i.e. it uses the fewest bits), it is called a '''minimal description''' of ''s''. Thus, the length of ''d''(''s'') (i.e. the number of bits in the description) is the '''Kolmogorov complexity''' of ''s'', written ''K''(''s''). Symbolically,<br />
<br />
:''K''(''s'') = |''d''(''s'')|.<br />
<br />
The length of the shortest description will depend on the choice of description language; but the effect of changing languages is bounded (a result called the ''invariance theorem'').<br />
<br />
==Invariance theorem==<br />
<br />
===Informal treatment===<br />
There are some description languages which are optimal, in the following sense: given any description of an object in a description language, I can use that description in my optimal description language with a constant overhead. The constant depends only on the languages involved, not on the description of the object, or the object being described.<br />
<br />
Here is an example of an optimal description language. A description will have two parts:<br />
<br />
* The first part describes another description language.<br />
* The second part is a description of the object in that language.<br />
<br />
In more technical terms, the first part of a description is a computer program, with the second part being the input to that computer program which produces the object as output.<br />
<br />
'''The invariance theorem follows:''' Given any description language ''L'', the optimal description language is at least as efficient as ''L'', with some constant overhead.<br />
<br />
'''Proof:''' Any description ''D'' in ''L'' can be converted into a description in the optimal language by first describing ''L'' as a computer program ''P'' (part 1), and then using the original description ''D'' as input to that program (part 2). The <br />
total length of this new description ''D''’ is (approximately):<br />
<br />
:|''D''’| = |''P''| + |''D''|<br />
<br />
The length of ''P'' is a constant that doesn't depend on ''D''. So, there is at most a constant overhead, regardless of the object described. Therefore, the optimal language is universal [[up to]] this additive constant.<br />
<br />
===A more formal treatment===<br />
'''Theorem''': If ''K''<sub>1</sub> and ''K''<sub>2</sub> are the complexity functions relative to [[Turing complete]] description languages ''L''<sub>1</sub> and ''L''<sub>2</sub>, then there is a constant ''c'' – which depends only on the languages ''L''<sub>1</sub> and ''L''<sub>2</sub> chosen – such that<br />
<br />
:∀''s''. -''c'' ≤ ''K''<sub>1</sub>(''s'') - ''K''<sub>2</sub>(''s'') ≤ ''c''.<br />
<br />
'''Proof''': By symmetry, it suffices to prove that there is some constant ''c'' such that for all strings ''s''<br />
<br />
:''K''<sub>1</sub>(''s'') ≤ ''K''<sub>2</sub>(''s'') + ''c''.<br />
<br />
Now, suppose there is a program in the language ''L''<sub>1</sub> which acts as an [[interpreter (computing)|interpreter]] for ''L''<sub>2</sub>:<br />
<br />
'''function''' InterpretLanguage('''string''' ''p'')<br />
<br />
where ''p'' is a program in ''L''<sub>2</sub>. The interpreter is characterized by the following property:<br />
<br />
: Running <code>InterpretLanguage</code> on input ''p'' returns the result of running ''p''.<br />
<br />
Thus, if '''P''' is a program in ''L''<sub>2</sub> which is a minimal description of ''s'', then <code>InterpretLanguage</code>('''P''') returns the string ''s''. The length of this description of ''s'' is the sum of<br />
<br />
# The length of the program <code>InterpretLanguage</code>, which we can take to be the constant ''c''.<br />
# The length of '''P''' which by definition is ''K''<sub>2</sub>(''s'').<br />
<br />
This proves the desired upper bound.<br />
<br />
==History and context==<br />
Algorithmic information theory is the area of computer science that studies Kolmogorov complexity and other complexity measures on strings (or other [[data structure]]s).<br />
<br />
The concept and theory of Kolmogorov Complexity is based on a crucial theorem first discovered by [[Ray Solomonoff]], who published it in 1960, describing it in "A Preliminary Report on a General Theory of Inductive Inference"<ref>{{cite journal |authorlink=Ray Solomonoff | last=Solomonoff |first= Ray | url=http://world.std.com/~rjs/rayfeb60.pdf |format=PDF | title=A Preliminary Report on a General Theory of Inductive Inference | journal= Report V-131 |publisher= Zator Co. |location= Cambridge, Ma. | date= February 4, 1960 }} [http://world.std.com/~rjs/z138.pdf revision], Nov., 1960.</ref> as part of his invention of [[algorithmic probability]]. He gave a more complete description in his 1964 publications, "A Formal Theory of Inductive Inference," Part 1 and Part 2 in ''Information and Control''.<ref>{{cite doi|10.1016/S0019-9958(64)90223-2}}</ref><ref>{{cite doi|10.1016/S0019-9958(64)90131-7 }}</ref><br />
<br />
Andrey Kolmogorov later [[multiple discovery|independently published]] this theorem in ''Problems Inform. Transmission'',<ref>{{cite journal | volume= 1| issue=1 |year=1965 | pages= 1–7 | title =Three Approaches to the Quantitative Definition of Information | url=http://www.ece.umd.edu/~abarg/ppi/contents/1-65-abstracts.html#1-65.2 | journal = Problems Inform. Transmission | first=A.N. | last=Kolmogorov }}</ref> Gregory Chaitin also presents this theorem in ''J. ACM'' – Chaitin's paper was submitted October 1966 and revised in December 1968, and cites both Solomonoff's and Kolmogorov's papers.<ref>{{cite doi | 10.1145/321526.321530}}</ref><br />
<br />
The theorem says that, among algorithms that decode strings from their descriptions (codes), there exists an optimal one. This algorithm, for all strings, allows codes as short as allowed by any other algorithm up to an additive constant that depends on the algorithms, but not on the strings themselves. Solomonoff used this algorithm, and the code lengths it allows, to define a "universal probability" of a string on which inductive inference of the subsequent digits of the string can be based. Kolmogorov used this theorem to define several functions of strings, including complexity, randomness, and information.<br />
<br />
When Kolmogorov became aware of Solomonoff's work, he acknowledged Solomonoff's priority.<ref>{{cite journal | last1=Kolmogorov | first1=A. | title=Logical basis for information theory and probability theory | journal=IEEE Transactions on Information Theory | volume=14|issue=5 | pages=662–664 | year=1968 | doi =10.1109/TIT.1968.1054210 }}</ref> For several years, Solomonoff's work was better known in the Soviet Union than in the Western World. The general consensus in the scientific community, however, was to associate this type of complexity with Kolmogorov, who was concerned with randomness of a sequence, while Algorithmic Probability became associated with Solomonoff, who focused on prediction using his invention of the universal prior probability distribution. The broader area encompassing descriptional complexity and probability is often called Kolmogorov complexity. The computer scientist Ming Li considers this an example of the [[Matthew effect (sociology)|Matthew effect]]: "... to everyone who has more will be given ..."<ref>{{Cite book<br />
| edition = 2nd<br />
| publisher = Springer<br />
| isbn = 0-387-94868-6<br />
| last = Li<br />
| first = Ming<br />
| coauthors = Paul Vitanyi<br />
| title = An Introduction to Kolmogorov Complexity and Its Applications<br />
|page=90<br />
| date = 1997-02-27<br />
}}</ref><br />
<br />
There are several other variants of Kolmogorov complexity or algorithmic information. The most widely used one is based on [[self-delimiting program]]s, and is mainly due to [[Leonid Levin]] (1974).<br />
<br />
An axiomatic approach to Kolmogorov complexity based on [[Blum axioms]] (Blum 1967) was introduced by Mark Burgin in the paper presented for publication by Andrey Kolmogorov (Burgin 1982).<br />
<br />
==Basic results==<br />
In the following discussion, let ''K''(''s'') be the complexity of the string ''s''.<br />
<br />
It is not hard to see that the minimal description of a string cannot be too much larger than the string itself - the program <code>GenerateFixedString</code> above that outputs ''s'' is a fixed amount larger than ''s''.<br />
<br />
'''Theorem''': There is a constant ''c'' such that<br />
<br />
:∀''s''. ''K''(''s'') ≤ |''s''| + ''c''.<br />
<br />
===Incomputability of Kolmogorov complexity===<br />
<br />
'''Theorem''': There exist strings of arbitrary large Kolmogorov complexity. Formally: for each ''n'' ∈ ℕ, there is a string ''s'' with ''K''(''s'') ≥ ''n''.<ref group="note">However, an ''s'' with ''K''(''s'') = ''n'' needn't exist for every ''n''. For example, if ''n'' isn't a multiple of 7 bits, no [[ASCII]] program can have a length of exactly ''n'' bits.</ref><br />
<br />
'''Proof:''' Otherwise all infinitely many possible strings could be generated by the finitely many<ref group="note">There are 1 + 2 + 2<sup>2</sup> + 2<sup>3</sup> + ... + 2<sup>''n''</sup> = 2<sup>''n''+1</sup> &minus; 1 different program texts of length up to ''n'' bits; cf. [[geometric series]]. If program lengths are to be multiples of 7 bits, even fewer program texts exist.</ref> programs with a complexity below ''n'' bits.<br />
<br />
'''Theorem''': ''K'' is not a [[computable function]]. In other words, there is no program which takes a string ''s'' as input and produces the integer ''K''(''s'') as output.<br />
<br />
The following [[indirect proof|indirect]] '''proof''' uses a simple [[Pascal (programming language)|Pascal]]-like language to denote programs; for sake of proof simplicity assume its description (i.e. an [[interpreter (computing)|interpreter]]) to have a length of 1'400'000 bits.<br />
Assume for contradiction there is a program <br />
<br />
'''function''' KolmogorovComplexity('''string''' s)<br />
<br />
which takes as input a string ''s'' and returns ''K''(''s''); for sake of proof simplicity, assume its length to be 7'000'000'000 bits.<br />
Now, consider the following program of length 1'288 bits:<br />
<br />
'''function''' GenerateComplexString()<br />
'''for''' i = 1 '''to''' infinity:<br />
'''for each''' string s '''of''' length exactly i<br />
'''if''' KolmogorovComplexity(s) >= 8000000000<br />
'''return''' s<br />
<br />
Using <code>KolmogorovComplexity</code> as a subroutine, the program tries every string, starting with the shortest, until it returns a string with Kolmogorov complexity at least 8'000'000'000 bits,<ref group="note">By the previous theorem, such a string exists, hence the <code>for</code> loop will eventually terminate.</ref> i.e. a string that cannot be produced by any program shorter than 8'000'000'000 bits. However, the overall length of the above program that produced ''s'' is only 7'001'401'288 bits,<ref group=note>including the language interpreter and the subroutine code for <code>KolmogorovComplexity</code></ref> which is a contradiction. (If the code of <code>KolmogorovComplexity</code> is shorter, the contradiction remains. If it is longer, the constant used in <code>GenerateComplexString</code> can always be changed appropriately.)<br />
<br />
The above proof used a contradiction similar to that of the [[Berry paradox]]: "<sub>{{color|#8080ff|1}}</sub>The <sub>{{color|#8080ff|2}}</sub>smallest <sub>{{color|#8080ff|3}}</sub>positive <sub>{{color|#8080ff|4}}</sub>integer <sub>{{color|#8080ff|5}}</sub>that <sub>{{color|#8080ff|6}}</sub>cannot <sub>{{color|#8080ff|7}}</sub>be <sub>{{color|#8080ff|8}}</sub>defined <sub>{{color|#8080ff|9}}</sub>in <sub>{{color|#8080ff|10}}</sub>fewer <sub>{{color|#8080ff|11}}</sub>than <sub>{{color|#8080ff|12}}</sub>twenty <sub>{{color|#8080ff|13}}</sub>English <sub>{{color|#8080ff|14}}</sub>words". It is also possible to show the non-computability of ''K'' by reduction from the non-computability of the halting problem ''H'', since ''K'' and ''H'' are [[turing degree#Turing equivalence|Turing-equivalent]].<ref>State without proof in: [http://www.daimi.au.dk/~bromille/DC05/Kolmogorov.pdf "''Course notes for Data Compression - Kolmogorov complexity''"], 2005, P.B. Miltersen, p.7</ref><br />
<br />
There is a corollary, humorously called the "[[full employment theorem]]" in the programming language community, stating that there is no perfect size-optimizing compiler.<br />
<br />
===Chain rule for Kolmogorov complexity===<br />
{{Main| Chain rule for Kolmogorov complexity}}<br />
The chain rule for Kolmogorov complexity states that<br />
<br />
:''K''(''X'',''Y'') = ''K''(''X'') + ''K''(''Y''|''X'') + ''O''(log(''K''(''X'',''Y''))).<br />
<br />
It states that the shortest program that reproduces ''X'' and ''Y'' is [[Big-O notation|no more]] than a logarithmic term larger than a program to reproduce ''X'' and a program to reproduce ''Y'' given ''X''. Using this statement, one can define [[Mutual information#Absolute mutual information|an analogue of mutual information for Kolmogorov complexity]].<br />
<br />
==Compression==<br />
It is straightforward to compute upper bounds for ''K''(''s'') – simply [[data compression|compress]] the string ''s'' with some method, implement the corresponding decompressor in the chosen language, concatenate the decompressor to the compressed string, and measure the length of the resulting string.<br />
<br />
A string ''s'' is compressible by a number ''c'' if it has a description whose length does not exceed |''s''|&minus;''c'' bits. This is equivalent to saying that ''K''(''s'') ≤ |''s''|-''c''. Otherwise, ''s'' is incompressible by ''c''. A string incompressible by 1 is said to be simply ''incompressible'' – by the [[pigeonhole principle]], which applies because every compressed string maps to only one uncompressed string, [[incompressible string]]s must exist, since there are 2<sup>''n''</sup> bit strings of length ''n'', but only 2<sup>''n''</sup> - 1 shorter strings, that is, strings of length less than ''n'', (i.e. with length 0,1,...,''n&nbsp;&minus;&nbsp;1).<ref group=note>As there are {{nobr|1=''N''<sub>''L''</sub> = 2<sup>''L''</sup>}} strings of length ''L'', the number of strings of lengths {{nowrap|1=''L'' = 0, 1, ..., ''n'' &minus; 1}} is {{nobr|''N''<sub>0</sub> + ''N''<sub>1</sub> + ... + ''N''<sub>''n''−1</sub>}} = {{nobr|2<sup>0</sup> + 2<sup>1</sup> + ... + 2<sup>''n''−1</sup>}}, which is a finite [[geometric series]] with sum {{nobr|2<sup>0</sup> + 2<sup>1</sup> + ... + 2<sup>''n''−1</sup>}} = {{nobr|1 = 2<sup>0</sup> × (1 − 2<sup>''n''</sup>) / (1 − 2) = 2<sup>''n''</sup> − 1}}.</ref><br />
<br />
For the same reason, most strings are complex in the sense that they cannot be significantly compressed – their ''K''(''s'') is not much smaller than |''s''|, the length of ''s'' in bits. To make this precise, fix a value of ''n''. There are 2<sup>''n''</sup> bitstrings of length ''n''. The [[Uniform distribution (discrete)|uniform]] [[probability]] distribution on the space of these bitstrings assigns exactly equal weight 2<sup>-''n''</sup> to each string of length ''n''.<br />
<br />
'''Theorem''': With the uniform probability distribution on the space of bitstrings of length ''n'', the probability that a string is incompressible by ''c'' is at least 1 - 2<sup>-''c''+1</sup> + 2<sup>-''n''</sup>.<br />
<br />
To prove the theorem, note that the number of descriptions of length not exceeding ''n''-''c'' is given by the geometric series:<br />
<br />
:1 + 2 + 2<sup>2</sup> + ... + 2<sup>''n''-''c''</sup> = 2<sup>''n''-''c''+1</sup> - 1.<br />
<br />
There remain at least<br />
<br />
:2<sup>''n''</sup> - 2<sup>''n''-''c''+1</sup> + 1<br />
<br />
bitstrings of length ''n'' that are incompressible by ''c''. To determine the probability, divide by 2<sup>''n''</sup>.<br />
<br />
==Chaitin's incompleteness theorem==<br />
We know that, in the set of all possible strings, most strings are complex in the sense that they cannot be described in any significantly "compressed" way. However, it turns out that the fact that a specific string is complex cannot be formally proven, if the complexity of the string is above a certain threshold. The precise formalization is as follows. First, fix a particular [[axiomatic system]] '''S''' for the [[natural number]]s. The axiomatic system has to be powerful enough so that, to certain assertions '''A''' about complexity of strings, one can associate a formula '''F'''<sub>'''A'''</sub> in '''S'''. This association must have the following property:<br />
<br />
if '''F'''<sub>'''A'''</sub> is provable from the axioms of '''S''', then the corresponding assertion '''A''' must be true. This "formalization" can be achieved, either by an artificial encoding such as a [[Gödel numbering]], or by a formalization which more clearly respects the intended interpretation of '''S'''.<br />
<br />
'''Theorem''': There exists a constant ''L'' (which only depends on the particular axiomatic system and the choice of description language) such that there does not exist a string ''s'' for which the statement<br />
<br />
:''K''(''s'') ≥ ''L'' (as formalized in '''S''')<br />
<br />
can be proven within the axiomatic system '''S'''.<br />
<br />
Note that, by the abundance of nearly incompressible strings, the vast majority of those statements must be true.<br />
<br />
The proof of this result is modeled on a self-referential construction used in [[Berry's paradox]]. The proof is by contradiction. If the theorem were false, then<br />
<br />
:'''Assumption (X)''': For any integer ''n'' there exists a string ''s'' for which there is a proof in '''S''' of the formula "''K''(''s'')&nbsp;≥&nbsp;''n''" (which we assume can be formalized in '''S''').<br />
<br />
We can find an effective enumeration of all the formal proofs in '''S''' by some procedure<br />
<br />
'''function''' NthProof('''int''' ''n'')<br />
which takes as input ''n'' and outputs some proof. This function enumerates all proofs. Some of these are proofs for formulas we do not care about here, since every possible proof in the language of '''S''' is produced for some ''n''. Some of these are complexity formulas of the form ''K''(''s'')&nbsp;≥&nbsp;''n'' where ''s'' and ''n'' are constants in the language of '''S'''. There is a program<br />
<br />
'''function''' NthProofProvesComplexityFormula('''int''' ''n'')<br />
<br />
which determines whether the ''n''th proof actually proves a complexity formula ''K''(''s'')&nbsp;≥&nbsp;''L''. The strings ''s'', and the integer ''L'' in turn, are computable by programs:<br />
<br />
'''function''' StringNthProof('''int''' ''n'')<br />
<br />
'''function''' ComplexityLowerBoundNthProof('''int''' ''n'')<br />
<br />
Consider the following program<br />
<br />
'''function''' GenerateProvablyComplexString('''int''' ''n'')<br />
'''for''' i = 1 to infinity:<br />
'''if''' NthProofProvesComplexityFormula(i) '''and''' ComplexityLowerBoundNthProof(i) ≥ ''n''<br />
'''return''' StringNthProof(''i'')<br />
<br />
Given an ''n'', this program tries every proof until it finds a string and a proof in the [[formal system]] '''S''' of the formula ''K''(''s'')&nbsp;≥&nbsp;''L'' for some ''L''&nbsp;≥&nbsp;''n''. The program terminates by our '''Assumption (X)'''. Now, this program has a length ''U''. There is an integer ''n''<sub>0</sub> such that ''U''&nbsp;+&nbsp;log<sub>2</sub>(''n''<sub>0</sub>)&nbsp;+&nbsp;''C''&nbsp;<&nbsp;''n''<sub>0</sub>, where ''C'' is the overhead cost of<br />
<br />
'''function''' GenerateProvablyParadoxicalString()<br />
'''return''' GenerateProvablyComplexString(''n''<sub>0</sub>)<br />
<br />
(note that ''n''<sub>0</sub> is hard-coded into the above function, and the summand log<sub>2</sub>(''n''<sub>0</sub>) already allows for its encoding). The program GenerateProvablyParadoxicalString outputs a string ''s'' for which there exists an ''L'' such that ''K''(''s'')&nbsp;≥&nbsp;''L'' can be formally proved in '''S''' with ''L''&nbsp;≥&nbsp;''n''<sub>0</sub>. In particular, ''K''(''s'')&nbsp;≥&nbsp;''n''<sub>0</sub> is true. However, ''s'' is also described by a program of length ''U''&nbsp;+&nbsp;log<sub>2</sub>(''n''<sub>0</sub>)&nbsp;+&nbsp;''C'', so its complexity is less than ''n''<sub>0</sub>. This contradiction proves '''Assumption (X)''' cannot hold.<br />
<br />
Similar ideas are used to prove the properties of [[Chaitin's constant]].<br />
<br />
==Minimum message length==<br />
The [[minimum message length]] principle of statistical and inductive inference and machine learning was developed by [[Chris Wallace (computer scientist)|C.S. Wallace]] and D.M. Boulton in 1968. MML is [[Bayesian probability|Bayesian]] (i.e. it incorporates prior beliefs) and information-theoretic. It has the desirable properties of statistical invariance (i.e. the inference transforms with a re-parametrisation, such as from polar coordinates to Cartesian coordinates), statistical consistency (i.e. even for very hard problems, MML will converge to any underlying model) and efficiency (i.e. the MML model will converge to any true underlying model about as quickly as is possible). C.S. Wallace and D.L. Dowe (1999) showed a formal connection between MML and algorithmic information theory (or Kolmogorov complexity).<br />
<br />
==Kolmogorov randomness==<br />
''Kolmogorov randomness'' – also called ''algorithmic randomness'' – defines a string (usually of [[bit]]s) as being [[randomness|random]] if and only if it is shorter than any [[computer program]] that can produce that string. To make this precise, a [[universal computer]] (or universal Turing machine) must be specified, so that "program" means a program for this universal machine. A random string in this sense is "incompressible" in that it is impossible to "compress" the string into a program whose length is shorter than the length of the string itself. A [[counting argument]] is used to show that, for any universal computer, there is at least one algorithmically random string of each length. Whether any particular string is random, however, depends on the specific universal computer that is chosen. <br />
<br />
This definition can be extended to define a notion of randomness for ''infinite'' sequences from a finite alphabet. These [[algorithmically random sequence]]s can be defined in three equivalent ways. One way uses an effective analogue of [[measure theory]]; another uses effective [[Martingale (probability theory)|martingales]]. The third way defines an infinite sequence to be random if the prefix-free Kolmogorov complexity of its initial segments grows quickly enough - there must be a constant ''c'' such that the complexity of an initial segment of length ''n'' is always at least ''n''&minus;''c''. This definition, unlike the definition of randomness for a finite string, is not affected by which universal machine is used to define prefix-free Kolmogorov complexity.<br />
<ref>{{cite doi | 10.1016/S0019-9958(66)80018-9}}</ref><br />
<br />
== Relation to entropy ==<br />
For dynamical systems, entropy rate and algorithmic complexity of the trajectories are related by a theorem of Brudno, that the equality K(x;T) = h(T) holds for almost all x.<ref>{{cite journal |authors=Stefano Galatolo, Mathieu Hoyrup, Cristóbal Rojas |title=Effective symbolic dynamics, random points, statistical behavior, complexity and entropy | journal=Information and Computation | volume=208 | pages=23-41 | year=2010| url=http://www.loria.fr/~hoyrup/random_ergodic.pdf}}</ref> <br />
<br />
It can be shown<ref>{{cite journal |author=Alexei Kaltchenko |title=Algorithms for Estimating Information Distance with Application to Bioinformatics and Linguistics |journal=CoRR |volume=cs.CC/0404039 |year=2004 |url=http://arxiv.org/pdf/cs.CC/0404039 |url=http://arxiv.org/abs/cs.CC/0404039}}</ref> that for the output of [[Markov information source]]s, Kolmogorov complexity is related to the [[Entropy (information theory)|entropy]] of the information source. More precisely, the Kolmogorov complexity of the output of a Markov information source, normalized by the length of the output, converges almost surely (as the length of the output goes to infinity) to the [[Entropy (information theory)|entropy]] of the source.<br />
<br />
==See also==<br />
* [[Berry paradox]]<br />
* [[Data compression]]<br />
* [[Inductive inference]]<br />
* [[Kolmogorov structure function]]<br />
* [[List_of_important_publications_in_theoretical_computer_science#Algorithmic_information_theory|Important publications in algorithmic information theory]]<br />
* [[Levenshtein distance]]<br />
* [[Grammar induction]]<br />
<br />
==Notes==<br />
{{Reflist|group=note}}<br />
<br />
==References==<br />
{{Reflist|colwidth=30em}}<br />
<br />
* {{cite journal | authorlink=Manuel Blum|last=Blum | title=On the size of machines | journal=Information and Control |first= M. | volume=11 | issue=3 | pages=257 | year=1967 | doi = 10.1016/S0019-9958(67)90546-3 }}<br />
* Brudno, A. Entropy and the complexity of the trajectories of a dynamical system., Transactions of the Moscow Mathematical Society, 2:127{151, 1983.<br />
* Burgin, M. (1982), "Generalized Kolmogorov complexity and duality in theory of computations", ''Notices of the Russian Academy of Sciences'', v.25, No. 3, pp.&nbsp;19&ndash;23.<br />
* Cover, Thomas M. and Thomas, Joy A., ''Elements of information theory'', 1st Edition. New York: Wiley-Interscience, 1991. ISBN 0-471-06259-6. 2nd Edition. New York: Wiley-Interscience, 2006. ISBN 0-471-24195-4.<br />
* Lajos, Rónyai and Gábor, Ivanyos and Réka, Szabó, ''Algoritmusok''. TypoTeX, 1999. ISBN 963-279-014-6<br />
* Li, Ming and Vitányi, Paul, ''An Introduction to Kolmogorov Complexity and Its Applications'', Springer, 1997. [http://citeseer.ist.psu.edu/li97introduction.html Introduction chapter full-text].<br />
* Yu Manin, ''A Course in Mathematical Logic'', Springer-Verlag, 1977. ISBN 978-0-7204-2844-5<br />
* Sipser, Michael, ''Introduction to the Theory of Computation'', PWS Publishing Company, 1997. ISBN 0-534-95097-3.<br />
* [[Chris Wallace (computer scientist)|Wallace, C. S]]. and Dowe, D. L., [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.17.321 Minimum Message Length and Kolmogorov Complexity], Computer Journal, Vol. 42, No. 4, 1999).<br />
<br />
==External links==<br />
* [http://www.kolmogorov.com/ The Legacy of Andrei Nikolaevich Kolmogorov]<br />
* [http://www.cs.umaine.edu/~chaitin/ Chaitin's online publications]<br />
* [http://www.idsia.ch/~juergen/ray.html Solomonoff's IDSIA page]<br />
* [http://www.idsia.ch/~juergen/kolmogorov.html Generalizations of algorithmic information] by [[Juergen Schmidhuber|J. Schmidhuber]]<br />
* [http://homepages.cwi.nl/~paulv/kolmogorov.html Ming Li and Paul Vitanyi, An Introduction to Kolmogorov Complexity and Its Applications, 2nd Edition, Springer Verlag, 1997.]<br />
* [http://homepages.cwi.nl/~tromp/cl/cl.html Tromp's lambda calculus computer model offers a concrete definition of K()]<br />
* Universal AI based on Kolmogorov Complexity ISBN 3-540-22139-5 by [[Marcus Hutter|M. Hutter]]: ISBN 3-540-22139-5<br />
* [http://www.csse.monash.edu.au/~dld David Dowe]'s [http://www.csse.monash.edu.au/~dld/MML.html Minimum Message Length (MML)] and [http://www.csse.monash.edu.au/~dld/Occam.html Occam's razor] pages.<br />
* P. Grunwald, M. A. Pitt and I. J. Myung (ed.), [http://mitpress.mit.edu/catalog/item/default.asp?sid=4C100C6F-2255-40FF-A2ED-02FC49FEBE7C&ttype=2&tid=10478 Advances in Minimum Description Length: Theory and Applications], M.I.T. Press, April 2005, ISBN 0-262-07262-9.<br />
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{{Compression Methods}}<br />
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{{DEFAULTSORT:Kolmogorov Complexity}}<br />
[[Category:Algorithmic information theory|*]]<br />
[[Category:Information theory|*]]<br />
[[Category:Computability theory]]<br />
[[Category:Descriptive complexity]]<br />
[[Category:Measures of complexity]]</div>Adminhttps://en.formulasearchengine.com/index.php?title=Augustin-Louis_Cauchy&diff=218584Augustin-Louis Cauchy2014-07-31T13:42:34Z<p>Admin: Reverted edits by MediaWiki spam cleanup (talk) to last revision by Periglio</p>
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<div>{{Redirect|Cauchy|the lunar crater|Cauchy (crater)|the statistical distribution|Cauchy distribution|the condition on sequences|Cauchy sequence}}<br />
{{Infobox scientist<br />
| name = Augustin-Louis Cauchy<br />
| image = Augustin-Louis Cauchy 1901.jpg<br />
| caption = Cauchy around 1840. Lithography by Zéphirin Belliard after a painting by Jean Roller.<br />
| image_size = 200px<br />
| birth_date = {{Birth date|1789|8|21|df=y}}<br />
| birth_place = [[Paris]], [[France]]<br />
| death_date = {{Death date and age|1857|5|23|1789|8|21|df=y}}<br />
| death_place = [[Sceaux, Hauts-de-Seine|Sceaux]], [[France]]<br />
| residence =<br />
| nationality = [[France|French]]<br />
| field = [[Mathematics]]<br />
| occupation = [[Mathematician]]<br />
| work_institutions = [[École Centrale du Panthéon]] <br />[[École Nationale des Ponts et Chaussées]] <br /> [[École polytechnique]]<br />
| alma_mater = [[École Nationale des Ponts et Chaussées]]<br />
| doctoral_advisor =<br />
| doctoral_students = [[Francesco Faà di Bruno]]<br>[[Viktor Bunyakovsky]]<br />
| known_for = [[List of topics named after Augustin-Louis Cauchy|See list]]<br />
| prizes =<br />
| footnotes =<br />
}}<br />
[[Baron]] '''Augustin-Louis Cauchy''' ({{IPA-fr|oɡystɛ̃ lwi koʃi|lang}}; 21 August 1789 – 23 May 1857) was a [[France|French]] [[mathematician]] who was an early pioneer of [[mathematical analysis|analysis]]. He started the project of formulating and proving the theorems of [[infinitesimal calculus]] in a rigorous manner, rejecting the heuristic principle of the [[generality of algebra]] exploited by earlier authors. He defined [[continuous function|continuity]] in terms of [[infinitesimal]]s, almost singlehandedly founded [[complex analysis]] and initiated the study of [[permutation group]]s in [[abstract algebra]]. A profound mathematician, Cauchy exercised a great influence over his contemporaries and successors. His writings cover the entire range of mathematics and [[mathematical physics]].<br />
<br />
"More concepts and theorems have been named for Cauchy than for any other mathematician (in [[Elasticity (physics)|elasticity]] alone there are sixteen concepts and theorems named for Cauchy)."{{sfn|Freudenthal|2008}} Cauchy was a prolific writer; he wrote approximately eight hundred research articles and five complete textbooks. He was a devout [[Roman Catholic]], strict [[House of Bourbon|Bourbon]] royalist, and a close associate of the [[Jesuit order]].<br />
<br />
==Biography==<br />
<br />
===Youth and education===<br />
Cauchy was the son of [[Louis François Cauchy]] (1760–1848) and Marie-Madeleine Desestre. Cauchy had two brothers, Alexandre Laurent Cauchy (1792–1857), who became a president of a division of the court of appeal in 1847, and a judge of the court of cassation in 1849; and Eugene François Cauchy (1802–1877), a publicist who also wrote several mathematical works.<br />
<br />
Cauchy married Aloise de Bure in 1818. She was a close relative of the publisher who published most of Cauchy's works. By her he had two daughters, Marie Françoise Alicia (1819) and Marie Mathilde (1823).<br />
<br />
Cauchy's father ([[Louis François Cauchy]]) was a high official in the Parisian Police of the New Régime. He lost his position because of the [[French Revolution]] (July 14, 1789) that broke out one month before Augustin-Louis was born.<ref>His father's dismissal is sometimes seen as the cause of the deep hatred of the French Revolution that Cauchy felt all through his life.</ref> The Cauchy family survived the revolution and the following [[Reign of Terror]] (1794) by escaping to Arcueil, where Cauchy received his first education, from his father. After the execution of [[Robespierre]] (1794), it was safe for the family to return to Paris. There Louis-François Cauchy found himself a new bureaucratic job, and quickly moved up the ranks. When [[Napoleon|Napoleon Bonaparte]] came to power (1799), Louis-François Cauchy was further promoted, and became Secretary-General of the Senate, working directly under [[Pierre-Simon Laplace|Laplace]] (who is now better known for his work on mathematical physics). The famous mathematician [[Joseph Louis Lagrange|Lagrange]] was also no stranger in the Cauchy family.<br />
<br />
On Lagrange's advice, Augustin-Louis was enrolled in the [[École Centrale du Panthéon]], the best secondary school of Paris at that time, in the fall of 1802. Most of the curriculum consisted of classical languages; the young and ambitious Cauchy, being a brilliant student, won many prizes in Latin and Humanities. In spite of these successes, Augustin-Louis chose an engineering career, and prepared himself for the entrance examination to the [[École Polytechnique]].<br />
<br />
In 1805 he placed second out of 293 applicants on this exam, and he was admitted. One of the main purposes of this school was to give future civil and military engineers a high-level scientific and mathematical education. The school functioned under military discipline, which caused the young and pious Cauchy some problems in adapting. Nevertheless, he finished the Polytechnique in 1807, at the age of 18, and went on to the [[École des Ponts et Chaussées]] (School for Bridges and Roads). He graduated in civil engineering, with the highest honors.<br />
<br />
===Engineering days===<br />
After finishing school in 1810, Cauchy accepted a job as a junior engineer in Cherbourg, where Napoleon intended to build a naval base. Here Augustin-Louis stayed for three years, and although he had an extremely busy managerial job, he still found time to prepare three mathematical manuscripts, which he submitted to the ''Première Classe'' (First Class) of the [[Institut de France]].<ref>In the revolutionary years the French Académie des Sciences was known as the "First Class" of the Institut de France.</ref> Cauchy's first two manuscripts (on [[regular polyhedron|polyhedra]]) were accepted; the third one (on directrices of [[conic sections]]) was rejected.<br />
<br />
In September 1812, now 23 years old, after becoming ill from overwork, Cauchy returned to Paris. Another reason for his return to the capital was that he was losing his interest in his engineering job, being more and more attracted to the abstract beauty of mathematics; in Paris, he would have a much better chance to find a mathematics related position. Although he formally kept his engineering position, he was transferred from the payroll of the Ministry of the Marine to the Ministry of the Interior. The next three years Augustin-Louis was mainly on unpaid sick leave, and spent his time quite fruitfully, working on mathematics (on the related topics of [[symmetric functions]], the [[symmetric group]] and the theory of higher-order algebraic equations). He attempted admission to the First Class of the Institut de France but failed on three different occasions between 1813 and 1815. In 1815 Napoleon was defeated at Waterloo, and the newly installed Bourbon king [[Louis XVIII]] took the restoration in hand. The [[Académie des Sciences]] was re-established in March 1816; [[Lazare Carnot]] and [[Gaspard Monge]] were removed from this Academy for political reasons, and the king appointed Cauchy to take the place of one of them. The reaction by Cauchy's peers was harsh; they considered his acceptance of membership of the Academy an outrage, and Cauchy thereby created many enemies in scientific circles.<br />
<br />
===Professor at École Polytechnique===<br />
In November 1815, [[Louis Poinsot]], who was an associate professor at the École Polytechnique, asked to be exempted from his teaching duties for health reasons. Cauchy was by then a rising mathematical star, who certainly merited a professorship. One of his great successes at that time was the proof of [[Pierre de Fermat|Fermat]]'s [[polygonal number theorem]]. However, the fact that Cauchy was known to be very loyal to the Bourbons, doubtless also helped him in becoming the successor of Poinsot. He finally quit his engineering job, and received a one-year contract for teaching mathematics to second-year students of the École Polytechnique. In 1816, this Bonapartist, non-religious school was reorganized, and several liberal professors were fired; the reactionary Cauchy was promoted to full professor.<br />
<br />
When Cauchy was 28 years old, he was still living with his parents. His father found it high time for his son to marry; he found him a suitable bride, Aloïse de Bure, five years his junior. The de Bure family were printers and booksellers, and published most of Cauchy's works.<ref>Bradley & Sandifer page 9</ref> Aloïse and Augustin were married on April 4, 1818, with great Roman Catholic pomp and ceremony, in the Church of Saint-Sulpice. In 1819 the couple's first daughter, Marie Françoise Alicia, was born, and in 1823 the second and last daughter, Marie Mathilde.<ref><br />
{{Cite book<br />
| last = Belhoste<br />
| first = Bruno<br />
| others = Frank Ragland (trans.)<br />
| title = Augustin-Louis Cauchy: A Biography<br />
| place = Ann Arbor, Michigan<br />
| publisher = [[Springer Science+Business Media|Springer]]-Verlag New York Inc.<br />
| year = 1991<br />
| page = 134<br />
| url = http://www.amazon.com/Augustin-Louis-Studies-Mathematics-Physical-Sciences/dp/354097220X/ref=sr_11_1?<br />
| doi =<br />
| isbn = 3-540-97220-X<br />
}}</ref> Cauchy had two brothers: Alexandre Laurent Cauchy, who became a president of a division of the court of appeal in 1847, and a judge of the court of cassation in 1849; and Eugène François Cauchy, a publicist who also wrote several mathematical works.<br />
<br />
The conservative political climate that lasted until 1830 suited Cauchy perfectly. In 1824 Louis XVIII died, and was succeeded by his even more reactionary brother [[Charles X]]. During these years Cauchy was highly productive, and published one important mathematical treatise after another. He received cross appointments at the [[Collège de France]], and the Faculté des Sciences of the University.<br />
<br />
===In exile===<br />
In July 1830 France underwent another revolution. Charles X fled the country, and was succeeded by the non-Bourbon king [[Louis-Philippe]] (of the [[House of Orléans]]). Riots, in which uniformed students of the École Polytechnique took an active part, raged close to Cauchy's home in Paris.<br />
<br />
These events marked a turning point in Cauchy's life, and a break in his mathematical productivity. Cauchy, shaken by the fall of the government, and moved by a deep hatred of the liberals who were taking power, left Paris to go abroad, leaving his family behind. He spent a short time at [[Fribourg]] in Switzerland, where he had to decide whether he would swear a required oath of allegiance to the new regime. He refused to do this, and consequently lost all his positions in Paris, except his membership of the Academy, for which an oath was not required. In 1831 Cauchy went to the Italian city of Turin, and after some time there, he accepted an offer from the [[King of Sardinia]] (who ruled Turin and the surrounding Piedmont region) for a chair of theoretical physics, which was created especially for him. He taught in Turin during 1832–1833. In 1831, he had been elected a foreign member of the [[Royal Swedish Academy of Sciences]].<br />
<br />
In August 1833 Cauchy left Turin for [[Prague]], to become the science tutor of the thirteen-year-old Duke of Bordeaux [[Henri d'Artois]] (1820–1883), the exiled Crown Prince and grandson of Charles X. As a professor of the École Polytechnique, Cauchy had been a notoriously bad lecturer, assuming levels of understanding that only a few of his best students could reach, and cramming his allotted time with too much material. The young Duke had neither taste nor talent for either mathematics or science, so student and teacher were a perfect mismatch. Although Cauchy took his mission very seriously, he did this with great clumsiness, and with surprising lack of authority over the Duke.<br />
<br />
During his civil engineering days, Cauchy once had been briefly in charge of repairing a few of the Parisian sewers, and he made the mistake of telling his pupil this; with great malice, the young Duke went about saying that Mister Cauchy started his career in the sewers of Paris. His role as tutor lasted until the Duke became eighteen years old, in September 1838. Cauchy did hardly any research during those five years, while the Duke acquired a lifelong dislike of mathematics. The only good that came out of this episode was Cauchy's promotion to [[Baron]], a title that Cauchy set great store by. In 1834, his wife and two daughters moved to Prague, and Cauchy was finally reunited with his family, after four years of exile.<br />
<br />
===Last years===<br />
Cauchy returned to Paris and his position at the Academy of Sciences late in 1838. He could not regain his teaching positions, because he still refused to swear an oath of allegiance. However, he desperately wanted to regain a formal position in Parisian science.<br />
<br />
[[File:Augustin-Louis Cauchy.jpg|thumb|left|Cauchy prior to 1857]]<br />
In August 1839 a vacancy appeared in the [[Bureau des Longitudes]]. This Bureau had some resemblance to the Academy; for instance, it had the right to co-opt its members. Further, it was believed that members of the Bureau could "forget" about the oath of allegiance, although formally, unlike the Academicians, they were obliged to take it. The Bureau des Longitudes was an organization founded in 1795 to solve the problem of determining position on sea – mainly the [[longitude|longitudinal]] coordinate, since [[latitude]] is easily determined from the position of the sun. Since it was thought that position on sea was best determined by astronomical observations, the Bureau had developed into an organization resembling an academy of astronomical sciences.<br />
<br />
In November 1839 Cauchy was elected to the Bureau, and discovered immediately that the matter of the oath was not so easily dispensed with. Without his oath, the king refused to approve his election. For four years Cauchy was in the absurd position of being elected, but not being approved; hence, he was not a formal member of the Bureau, did not receive payment, could not participate in meetings, and could not submit papers. Still Cauchy refused to take any oaths; however, he did feel loyal enough to direct his research to [[celestial mechanics]]. In 1840, he presented a dozen papers on this topic to the Academy. He also described and illustrated the [[signed-digit representation]] of numbers, an innovation presented in England in 1727 by [[John Colson]]. The confounded membership of the Bureau lasted until the end of 1843, when Cauchy was finally replaced by Poinsot.<br />
<br />
All through the nineteenth century the French educational system struggled with the question of separation of Church and State. The Catholic Church sought freedom of education; the Church found in Cauchy a staunch and illustrious ally in this struggle. He lent his prestige and knowledge to the [[École Normale Écclésiastique]], a school in Paris run by Jesuits, for training teachers for their colleges. He also took part in the founding of the [[Institut Catholique]]. The purpose of this institute was to counter the effects of the absence of Catholic university education in France. These activities did not make Cauchy popular with his colleagues who, on the whole, supported [[the Enlightenment]] ideals of the French Revolution. When a chair of mathematics became vacant at the Collège de France in 1843, Cauchy applied for it, but got just three out of 45 votes.<br />
<br />
The year 1848 was the year of revolution all over Europe; revolutions broke out in numerous countries, beginning in France. King Louis-Philippe, fearful of sharing the fate of Louis XVI, fled to England. The oath of allegiance was abolished, and the road to an academic appointment was finally clear for Cauchy. On March 1, 1849, he was reinstated at the Faculté de Sciences, as a professor of mathematical astronomy. After political turmoil all through 1848, France chose to become a Republic, under the Presidency of [[Napoleon III of France|Louis Napoleon Bonaparte]], nephew of Napoleon Bonaparte, and son of Napoleon's brother, who had been installed as the first king of Holland. Soon (early 1852) the President became the Emperor of France, and took the name [[Napoleon III]].<br />
<br />
Not unexpectedly, the idea came up in bureaucratic circles that it would be useful to require a loyalty oath from all state functionaries, including university professors. Not always does history repeat itself, however, because this time a cabinet minister was able to convince the Emperor to exempt Cauchy from the oath. Cauchy remained a professor at the University until his death at the age of 67. He received the [[Last Rites]] and died at 4&nbsp;a.m. on May 23, 1857.<br />
<br />
His name is one of the [[List of the 72 names on the Eiffel Tower|72 names inscribed on the Eiffel Tower]].<br />
<br />
==Work==<br />
<br />
===Early work===<br />
The genius of Cauchy was illustrated in his simple solution of the [[problem of Apollonius]]—describing a [[circle]] touching three given circles—which he discovered in 1805, his generalization of [[Euler characteristic|Euler's formula]] on [[polyhedra]] in 1811, and in several other elegant problems. More important is his memoir on [[wave]] propagation, which obtained the Grand Prix of the French Academy of Sciences in 1816. Cauchy's writings covered notable topics including: the theory of series, where he developed the notion of [[limit of a sequence|convergence]] and discovered many of the basic formulas for [[q-series]]. In the theory of numbers and complex quantities, he was the first to define complex numbers as pairs of real numbers. He also wrote on the theory of groups and substitutions, the theory of functions, differential equations and determinants.<br />
<br />
===Wave theory, mechanics, elasticity===<br />
In the theory of light he worked on [[Augustin-Jean Fresnel|Fresnel's]] wave theory and on the [[dispersion (optics)|dispersion]] and [[polarization (waves)|polarization]] of light. <!-- In [[optics]], he developed the wave theory, and his name is associated with the simple dispersion formula. Exactly which formula? --> He also contributed significant research in [[mechanics]], substituting the notion of the continuity of geometrical displacements for the principle of the continuity of matter. He wrote on the equilibrium of rods and elastic membranes and on waves in elastic media. He introduced<ref>Cauchy, ''De la pression ou tension dans un corps solide'', [On the pressure or tension in a solid body], Exercices de Mathématiques, vol. '''2''', p. 42 (1827)</ref> a 3 × 3 symmetric [[matrix (mathematics)|matrix]] of numbers that is now known as the [[Cauchy stress tensor]]. In [[Elasticity (physics)|elasticity]], he originated the theory of [[stress (physics)|stress]], and his results are nearly as valuable as those of [[Siméon Poisson]].<br />
<br />
===Number theory===<br />
Other significant contributions include being the first to prove the [[Fermat polygonal number theorem]].<br />
<br />
===Complex functions===<br />
Cauchy is most famous for his single-handed development of [[complex function theory]]. The first pivotal theorem proved by Cauchy, now known as ''[[Cauchy's integral theorem]]'', was the following:<br />
<br />
:<math><br />
\oint_C f(z)dz = 0,<br />
</math><br />
<br />
where ''f''(''z'') is a complex-valued function [[holomorphic function|holomorphic]] on and within the non-self-intersecting closed curve ''C'' (contour) lying in the [[complex plane]]. The ''contour integral'' is taken along the contour ''C''. The rudiments of this theorem can already be found in a paper that the 24-year-old Cauchy presented to the Académie des Sciences (then still called "First Class of the Institute") on August 11, 1814. In full form<ref>Cauchy, ''Mémoire sur les intégrales définies prises entre des limites imaginaires'' [Memorandum on definite integrals taken between imaginary limits], submitted to the Académie des Sciences on February 28, 1825</ref> the theorem was given in 1825. The 1825 paper is seen by many as Cauchy's most important contribution to mathematics.<br />
<br />
In 1826<ref>Cauchy, ''Sur un nouveau genre de calcul analogue au calcul infinitésimal'' [On a new type of calculus analogous to the infinitesimal calculus], Exercices de Mathématique, vol. '''1''', p. 11 (1826)</ref> Cauchy gave a formal definition of a [[residue (mathematics)|residue]] of a function. This concept regards functions that have [[pole (complex analysis)|pole]]s—isolated singularities, i.e., points where a function goes to positive or negative infinity. If the complex-valued function ''f''(''z'') can be expanded in the [[neighborhood]] of a singularity ''a'' as<br />
<br />
:<math><br />
f(z) = \phi(z) + \frac{B_1}{z-a} + \frac{B_2}{(z-a)^2} + \cdots + \frac{B_n}{(z-a)^n},\quad<br />
B_i, z,a \in \mathbb{C},<br />
</math><br />
<br />
where φ(''z'') is analytic (i.e., well-behaved without singularities), then ''f'' is said to have a pole of order ''n'' in the point ''a''. If ''n'' = 1, the pole is called simple.<br />
The coefficient ''B''<sub>1</sub> is called by Cauchy the residue of function ''f'' at ''a''. If ''f'' is non-singular at ''a'' then the residue of ''f'' is zero at ''a''. Clearly the residue is in the case of a simple pole equal to,<br />
:<math><br />
\underset{z=a}{\mathrm{Res}} f(z) = \lim_{z \rightarrow a} (z-a) f(z),<br />
</math><br />
where we replaced ''B''<sub>1</sub> by the modern notation of the residue.<br />
<br />
In 1831, while in Turin, Cauchy submitted two papers to the Academy of Sciences of Turin. In the first<ref>Cauchy, ''Sur la mécanique céleste et sur un nouveau calcul qui s'applique à un grande nombre de questions diverses'' [On the celestial mechanics and on a new calculus that can be applied to a great number of diverse questions], presented to the Academy of Sciences of Turin, October 11, 1831.</ref> he proposed the formula now known as [[Cauchy's integral formula]],<br />
:<math><br />
f(a) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z-a} dz,<br />
</math><br />
where ''f''(''z'') is analytic on ''C'' and within the region bounded by the contour ''C'' and the complex number ''a'' is somewhere in this region. The contour integral is taken counter-clockwise. Clearly, the integrand has a simple pole at ''z'' = ''a''. In the second paper<ref>Cauchy, ''Mémoire sur les rapports qui existent entre le calcul des Résidus et le calcul des Limites, et sur les avantages qu'offrent ces deux calculs dans la résolution des équations algébriques ou transcendantes'' Memorandum on the connections that exist between the residue calculus and the limit calculus, and on the advantages that these two calculi offer in solving algebraic and transcendental equations], presented to the Academy of Sciences of Turin, November 27, 1831.</ref> he presented the [[residue theorem]],<br />
:<math><br />
\frac{1}{2\pi i} \oint_C f(z) dz = \sum_{k=1}^n \underset{z=a_k}{\mathrm{Res}} f(z),<br />
</math><br />
where the sum is over all the ''n'' poles of ''f''(''z'') on and within the contour ''C''. These results of Cauchy's still form the core of complex function theory as it is taught today to physicists and electrical engineers. For quite some time, contemporaries of Cauchy ignored his theory, believing it to be too complicated. Only in the 1840s the theory started to get response, with [[Pierre-Alphonse Laurent]] being the first mathematician, besides Cauchy, making a substantial contribution (his [[Laurent series]] published in 1843).<br />
<br />
===Cours d'Analyse===<br />
[[Image:Cauchy.jpg|left|thumb|The title page of a textbook by Cauchy.]] In addition to his work on complex functions, Cauchy was the first to stress the importance of rigor in analysis. His book ''[[Cours d'Analyse]]'' had a such an impact that Judith Grabiner writes Cauchy was "the man who taught rigorous analysis to all of Europe."{{harv|Grabiner|1981}} This book is frequently noted as being the first place that inequalities, and <math>\delta-\epsilon</math> arguments were introduced into Calculus. Cauchy exploited [[infinitesimal]]s and wrote in his introduction that he has been "...&nbsp;unable to dispense with making the principal qualities of infinitely small quantities known...". M. Barany claims that the École mandated the inclusion of infinitesimal methods against Cauchy's better judgement {{harv|Barany|2011}}. Gilain argued that the infinitesimal portions of the book were likely a late insertion.{{harv|Gilain|1989}} Laugwitz (1989) and Benis-Sinaceur (1973) argued that Cauchy was not forced to teach infinitesimals, pointing out that he continued to use them in his own work as late as 1853.<ref>{{citation<br />
| last1 = Katz | first1 = Karin Usadi<br />
| last2 = Katz | first2 = Mikhail G.<br />
| author2-link= Mikhail Katz<br />
| mr = 2884218<br />
| doi = 10.1162/POSC_a_00047<br />
| issue = 4<br />
| journal = [[Perspectives on Science]]<br />
| pages = 426–452<br />
| title = Cauchy's continuum<br />
| volume = 19<br />
| year = 2011}}.</ref><ref>{{citation | last1 = Borovik | first1 = Alexandre| last2 = Katz | first2 = Mikhail G. | doi = 10.1007/s10699-011-9235-x | issue = 4 | journal = [[Foundations of Science]] | pages = | title = Who gave you the Cauchy—Weierstrass tale? The dual history of rigorous calculus | volume = | year = 2011}}.</ref><br />
<br />
Cauchy gave an explicit definition of an infinitesimal in terms of a sequence tending to zero. Namely, such a null sequence "becomes" an infinitesimal in Cauchy's and [[Lazare Carnot]]'s terminology. Sources disagree if Cauchy defined his notion of infinitesimal in terms of limits. Some have argued that such a claim is ambiguous, and essentially a play of words on the term "limit". Similarly, some sources contest the claim that Cauchy anticipated [[Karl Weierstrass|Weierstrassian]] rigor, and point out internal contradictions in post-Weierstrassian Cauchy scholarship relative to Cauchy's 1853 text on the sum theorem.<ref>{{citation<br />
| last1 = Katz | first1 = Karin Usadi<br />
| last2 = Katz | first2 = Mikhail G.<br />
| doi = 10.1162/POSC_a_00047<br />
| issue = 4<br />
| journal = [[Perspectives on Science]]<br />
| pages = 426–452<br />
| title = Cauchy's continuum<br />
| volume = 19<br />
| year = 2011}}.</ref><br />
<br />
Barany<ref>Barany, M. J.: revisiting the introduction to Cauchy's Cours d'analyse. Historia Mathematica 38 (2011), no. 3, 368—388. http://dx.doi.org/10.1016/j.hm.2010.12.001</ref> recently argued that Cauchy possessed a kinetic notion of limit similar to Newton's. Regardless of how Cauchy viewed the rigor of using infinitesimal methods, these methods continued in practice long after ''Cours d'Analyse'' both by Cauchy and other mathematicians and can be justified by modern techniques.<br />
<br />
===Taylor's theorem===<br />
He was the first to prove [[Taylor's theorem]] rigorously, establishing his well-known form of the remainder. He wrote a textbook<ref>Cauchy, Cours d'Analyse de l'École Royale Polytechnique, I.<sup>re</sup> partie, Analyse Algébrique, Paris (1821)</ref> (see the illustration) for his students at the École Polytechnique in which he developed the basic theorems of mathematical analysis as rigorously as possible. In this book he gave the necessary and sufficient condition for the existence of a [[limit of a function|limit]] in the form that is still taught.<!-- which is epsilon delta? --> Also Cauchy's well-known test for [[absolute convergence]] stems from this book: [[Cauchy condensation test]]. In 1829 he defined for the first time a complex function of a complex variable in another textbook.<ref>Cauchy, Leçons sur le Calcul Différentiel, Paris (1829)</ref> In spite of these, Cauchy's own research papers often used intuitive, not rigorous, methods;<ref>Morris Kline, ''Mathematics: The Loss of Certainty'', ISBN 0-19-503085-0, p. 176</ref> thus one of his theorems was exposed to a "counter-example" by [[Niels Henrik Abel|Abel]], later fixed by the introduction of the notion of [[uniform continuity]].<br />
<br />
===Argument principle, stability===<br />
In a paper published in 1855, two years before Cauchy's death, he discussed some theorems, one of which is similar to the "[[Argument principle|Argument Principle]]" in many modern textbooks on complex analysis. In modern control theory textbooks, the [[Cauchy argument principle]] is quite frequently used to derive the [[Nyquist stability criterion]], which can be used to predict the stability of negative [[feedback amplifier]] and negative [[feedback]] control systems. Thus Cauchy's work has a strong impact on both pure mathematics and practical engineering.<br />
<br />
===Output===<br />
Cauchy was very productive, in number of papers second only to [[Leonhard Euler]]. It took almost a century to collect all his writings into 27 large volumes:<br />
* ''[http://portail.mathdoc.fr/cgi-bin/oetoc?id=OE_CAUCHY_1_8 Oeuvres complètes d'Augustin Cauchy publiées sous la direction scientifique de l'Académie des sciences et sous les auspices de M. le ministre de l'Instruction publique (27 volumes)]'' (Paris : Gauthier-Villars et fils, 1882–1974)<br />
His greatest contributions to mathematical science are enveloped in the rigorous methods which he introduced; these are mainly embodied in his three great treatises:<br />
* ''[http://mathdoc.emath.fr/cgi-bin/oeitem?id=OE_CAUCHY_2_3_P5_0 Cours d'analyse de l'École royale polytechnique] (1821)''<br />
* ''Le Calcul infinitésimal'' (1823)<br />
* ''Leçons sur les applications de calcul infinitésimal''; ''La géométrie'' (1826–1828)<br />
His other works include:<br />
* ''[http://www.archive.org/details/exercicedanaly01caucrich Exercices d'analyse et de physique mathematique (Volume 1)]''<br />
* ''[http://www.archive.org/details/exercicedanaly02caucrich Exercices d'analyse et de physique mathematique (Volume 2)]''<br />
* ''[http://www.archive.org/details/exercicedanaly03caucrich Exercices d'analyse et de physique mathematique (Volume 3)]''<br />
* ''[http://www.archive.org/details/117770570_004 Exercices d'analyse et de physique mathematique (Volume 4)]'' (Paris: Bachelier, 1840–1847)<br />
* ''[http://gallica.bnf.fr/notice?N=FRBNF35030140 Analyse algèbrique]'' (Imprimerie Royale, 1821)<br />
* ''[http://gallica.bnf.fr/notice?N=FRBNF37281629 Nouveaux exercices de mathématiques]'' (Paris : Gauthier-Villars, 1895)<br />
* ''Courses of mechanics'' (for the École Polytechnique)<br />
* ''Higher algebra'' (for the [[Faculté des Sciences]])<br />
* ''Mathematical physics'' (for the Collège de France).<br />
* ''[http://gallica.bnf.fr/ark:/12148/bpt6k90188b/f34 Mémoire sur l'emploi des equations symboliques dans le calcul infinitésimal et dans le calcul aux différences finis]'' CR Ac ad. Sci. Paris, t. XVII, 449–458 (1843) credited as originating the [[operational calculus]].<br />
<br />
==Politics and religious beliefs==<br />
Augustin-Louis Cauchy grew up in the house of a staunch royalist. This made his father flee with the family to [[Arcueil]] during the [[French Revolution]]. Their life there was apparently hard; Augustin-Louis's father, Louis François, spoke of living on rice, bread, and crackers during the period. A paragraph from an undated letter from Louis François to his mother in [[Rouen]] says:<ref>C. A. Valson. ''[http://books.google.com/books?id=vQ7tw0rVKPsC La Vie et les Travaux du baron Cauchy]'', v. 1, p. 13.</ref><br />
{{quote|We never had more than a half pound of bread — and sometimes not even that. This we supplement with little supply of hard crackers and rice that we are allotted. Otherwise, we are getting along quite well, which is the important thing and goes to show that human beings can get by with little. I should tell you that for my children's pap I still have a bit of fine flour, made from wheat that I grew on my own land. I had three bushels, and I also have a few pounds of [[potato starch]]. It is as white as snow and very good, too, especially for very young children. It, too, was grown on my own land.<ref><br />
{{Cite book<br />
| last = Belhoste<br />
| first = Bruno<br />
| others = Frank Ragland (trans.)<br />
| title = Augustin-Louis Cauchy: A Biography<br />
| place = Ann Arbor, Michigan<br />
| publisher = [[Springer Science+Business Media|Springer]]-Verlag New York Inc.<br />
| year = 1991<br />
| page = 3<br />
| url = http://www.amazon.com/Augustin-Louis-Studies-Mathematics-Physical-Sciences/dp/354097220X/ref=sr_11_1?<br />
| doi =<br />
| isbn = 3-540-97220-X<br />
}}</ref>}}<br />
In any event, he inherited his father's staunch royalism and hence refused to take oaths to any government after the overthrow of Charles X.<br />
<br />
He was an equally staunch Catholic and a member of the [[Society of Saint Vincent de Paul]].<ref>{{cite web|url=http://www.newadvent.org/cathen/03457a.htm |title=CATHOLIC ENCYCLOPEDIA: Augustin-Louis Cauchy |publisher=Newadvent.org |date=1908-11-01 |accessdate=2009-06-19}}</ref> He also had links to the [[Society of Jesus]] and defended them at the Academy when it was politically unwise to do so. His zeal for his faith may have led to his caring for [[Charles Hermite]] during his illness and leading Hermite to become a faithful Catholic. It also inspired Cauchy to plead on behalf of the Irish during the [[Great Irish Famine|Potato Famine]].<br />
<br />
His royalism and religious zeal also made him contentious, which caused difficulties with his colleagues. He felt that he was mistreated for his beliefs, but his opponents felt he intentionally provoked people by berating them over religious matters or by defending the Jesuits after they had been suppressed. [[Niels Henrik Abel]] called him a "bigoted Catholic"<ref>{{citation|title=Men of Mathematics|first=E. T.|last= Bell|authorlink=Eric Temple Bell|publisher=Simon and Schuster|year=1986|isbn= 9780671628185|page=273|url=http://books.google.com/books?id=BLFL3coT5i4C&pg=PA273}}.</ref> and added he was "mad and there is nothing that can be done about him", but at the same time praised him as a mathematician. Cauchy's views were widely unpopular among mathematicians and when [[Guglielmo Libri Carucci dalla Sommaja]] was made chair in mathematics before him he, and many others, felt his views were the cause. When Libri was accused of stealing books he was replaced by [[Joseph Liouville]] which caused a rift between him and Cauchy. Another dispute concerned Jean Marie Constant Duhamel and a claim on inelastic shocks. Cauchy was later shown, by [[Jean-Victor Poncelet]], to be wrong.<br />
<br />
==See also==<br />
{{div col|3}}<br />
* [[List of topics named after Augustin-Louis Cauchy]]<br />
* [[Cauchy–Binet formula]]<br />
* [[Cauchy boundary condition]]<br />
* [[Cauchy's convergence test]]<br />
* [[Cauchy (crater)]]<br />
* [[Cauchy determinant]]<br />
* [[Cauchy distribution]]<br />
* [[Cauchy's equation]]<br />
* [[Cauchy–Euler equation]]<br />
* [[Cauchy functional equation]]<br />
* [[Cauchy horizon]]<br />
* [[Cauchy formula for repeated integration]]<br />
* [[Cauchy–Frobenius lemma]]<br />
* [[Cauchy–Hadamard theorem]]<br />
* [[Cauchy–Kovalevskaya theorem]]<br />
* [[Cauchy momentum equation]]<br />
* [[Cauchy–Peano theorem]]<br />
* [[Cauchy principal value]]<br />
* [[Cauchy problem]]<br />
* [[Cauchy product]]<br />
* [[Cauchy's radical test]]<br />
* [[Cauchy–Rassias stability]]<br />
* [[Cauchy–Riemann equations]]<br />
* [[Cauchy–Schwarz inequality]]<br />
* [[Cauchy sequence]]<br />
* [[Cauchy surface]]<br />
* [[Cauchy's theorem (geometry)]]<br />
* [[Cauchy's theorem (group theory)]]<br />
* [[Maclaurin-Cauchy test]]<br />
<br />
{{div col end}}<br />
<br />
==Notes==<br />
{{Reflist|30em}}<!--added under references heading by script-assisted edit--><br />
<br />
==References==<br />
*{{EB1911|wstitle=Cauchy, Augustin Louis}}<br />
*{{Citizendium|title=Augustin-Louis Cauchy}}<br />
<br />
==Further reading==<br />
* {{citation|last=Barany|first=Michael|title=God, king, and geometry: revisiting the introduction to Cauchy's ''Cours d'analyse|journal=Historia Mathematica|volume=38|year=2011}}<br />
* Bradley, Robert E. and C. Edward Sandifer, ''Cauchy's Cours d'analyse: An Annotated Translation''; Springer, 2009; ISBN 1-4419-0548-0<br />
* Boyer, C.: The concepts of the calculus. Hafner Publishing Company, 1949.<br />
* Cauchy, Augustin-Louis, ''Cours d'analyse de l'Ecole Royale Polytechnique''; Imprimerie royale, 1821 (reissued by [[Cambridge University Press]], 2009; ISBN 978-1-108-00208-0)<br />
* Cauchy, Augustin-Louis, ''Oeuvres completes''; Gauthier-Villars, 1882 (reissued by [[Cambridge University Press]], 2009; ISBN 978-1-108-00317-9)<br />
*{{Cite encyclopedia<br />
| first = Hans<br />
| last = Freudenthal<br />
| title = Cauchy, Augustin-Louis.<br />
| url = http://www.encyclopedia.com/topic/Augustin-Louis_Cauchy.aspx#1<br />
| publisher = Scribner & American Council of Learned Societies<br />
| isbn = 978-0-684-10114-9<br />
| editor-last = Gillispie<br />
| editor-first = Charles<br />
| encyclopedia = [[Dictionary of Scientific Biography]]<br />
| location = New York<br />
| year = 2008<br />
| ref = harv<br />
}}<br />
*Benis-Sinaceur Hourya. Cauchy et Bolzano. In: Revue d'histoire des sciences. 1973, Tome 26 n°2. pp.&nbsp;97–112.<br />
*{{Citation|authorlink=Detlef Laugwitz|last=Laugwitz|first=D.|year=1989|title=Definite values of infinite sums: aspects of the foundations of infinitesimal analysis around 1820|journal=Arch. Hist. Exact Sci.|volume=39|issue=3|pages=195–245|doi=10.1007/BF00329867}}.<br />
* {{citation|last=Gilain|first=C.|title=Cauchy et le Course d'Analyse de l'École Polytechnique|journal=Bulletin de la Société des amis de la Bibliothèque de l'École polytechnique|volume=5|pages=3–145|year=1989}}<br />
* {{citation|last=Grabiner|first=J. V.|title=The Origins of Cauchy's Rigorous Calculus|publisher=The MIT press|location=Cambridge, MA.|year=1981}}<br />
<br />
==External links==<br />
{{Wikiquote|Augustin Louis Cauchy}}<br />
{{Wikisource1913CatholicEnc|Augustin-Louis Cauchy}}<br />
{{Commons category|Augustin Louis Cauchy}}<br />
* {{MacTutor Biography|id=Cauchy}}<br />
* [http://planetmath.org/encyclopedia/CauchyCriterionForConvergence.html Cauchy criterion for convergence]<br />
* [http://www.archive.org/details/oeuvresdaugusti01caucrich ''Œuvres complètes d'Augustin Cauchy''] Académie des sciences (France). Ministère de l'éducation nationale.<br />
* [http://portail.mathdoc.fr/cgi-bin/oetoc?id=OE_CAUCHY_1_1 Augustin-Louis Cauchy – Œuvres complètes] (in 2 series) Gallica-Math<br />
* {{MathGenealogy |id=55177}}<br />
* [http://math.berkeley.edu/~robin/Cauchy/ Augustin-Louis Cauchy – Cauchy's Life] by [[Robin Hartshorne]]<br />
* Th. M. Rassias, [http://www.worldscibooks.com/mathematics/0659.html Topics in Mathematical Analysis, A Volume Dedicated to the Memory of A. L. Cauchy''], World Scientific Co., Singapore, New Jersey, London, 1989.<br />
* {{Cite NIE|Cauchy, Augustin Louis|year=1905}}<br />
<br />
{{Infinitesimals}}<br />
<br />
{{Authority control|VIAF=73851086|GND=118519735}}<br />
<br />
<!-- Metadata: see [[Wikipedia:Persondata]] --><br />
{{Persondata<br />
| NAME = Cauchy, Augustin Louis<br />
| ALTERNATIVE NAMES =<br />
| SHORT DESCRIPTION = French mathematician<br />
| DATE OF BIRTH = 1789-08-21<br />
| PLACE OF BIRTH = [[Dijon]], [[France]]<br />
| DATE OF DEATH = 1857-05-23<br />
| PLACE OF DEATH = [[Paris]], [[France]]<br />
}}<br />
{{DEFAULTSORT:Cauchy, Augustin Louis}}<br />
[[Category:École Polytechnique alumni]]<br />
[[Category:École des Ponts ParisTech alumni]]<br />
[[Category:Corps des ponts]]<br />
[[Category:1789 births]]<br />
[[Category:1857 deaths]]<br />
[[Category:University of Turin faculty]]<br />
[[Category:19th-century French mathematicians]]<br />
[[Category:Geometers]]<br />
[[Category:History of calculus]]<br />
[[Category:Mathematical analysts]]<br />
[[Category:Foreign Members of the Royal Society]]<br />
[[Category:Members of the French Academy of Sciences]]<br />
[[Category:Members of the Royal Swedish Academy of Sciences]]<br />
[[Category:French Roman Catholics]]<br />
[[Category:Textbook writers]]<br />
<br />
{{Link FA|de}}</div>Adminhttps://en.formulasearchengine.com/index.php?title=Absolute_magnitude&diff=218608Absolute magnitude2014-07-31T13:41:26Z<p>Admin: Reverted edits by MediaWiki spam cleanup (talk) to last revision by Trappist the monk</p>
<hr />
<div>{{About|the brightness of stars|the science fiction magazine|Absolute Magnitude (magazine)}}<br />
<br />
'''Absolute magnitude''' is the measure of a celestial object's intrinsic brightness. It is the hypothetical [[apparent magnitude]] of an object at a standard [[luminosity distance]] of exactly 10.0&nbsp;[[parsecs]] or about 32.6 [[light year]]s from the [[Observation|observer]], assuming no [[Extinction (astronomy)|astronomical extinction]] of starlight. This allows the true energy output of astronomical objects to be compared without regard to their variable distances. As with all astronomical [[magnitude (astronomy)|magnitudes]], the absolute magnitude can be specified for different [[Filter (optics)#Bandpass|wavelength intervals]]; for stars the most commonly quoted absolute magnitude is the '''absolute visual magnitude''', which is the absolute magnitude in the visual (V) band of the [[UBV photometric system|UBV system]]. Also commonly used is the '''absolute bolometric magnitude''', which is the total [[luminosity]] expressed in magnitude units; it takes into account energy radiated at all wavelengths, whether observed or not.<br />
<br />
The absolute magnitude uses the same conventions as the visual magnitude: brighter objects have smaller magnitudes, and 5 magnitudes corresponds exactly to a factor of 100, so a factor of 10<sup>0.4</sup> (≈2.512) ratio of [[brightness]] corresponds to a difference of 1.0 in magnitude. The [[Milky Way]], for example, has an absolute magnitude of about −20.5, so a [[quasar]] with an absolute magnitude of −25.5 is 100 times brighter than our [[galaxy]]. If this particular quasar and our galaxy could be seen side by side at the same distance, the quasar would be 5 magnitudes (or 100 times) brighter than our galaxy. Similarly, [[Canopus]] has an absolute visual magnitude of about -5.5, while [[Ross 248]] has an absolute visual magnitude of +14.8, for a difference of slightly more than 20 magnitudes, so if the two stars were at the same distance, Canopus would be seen as about 20 magnitudes brighter; stated another way, Canopus gives off slightly more than 100 million (10<sup>8</sup>) times more visual power than Ross 248.<br />
<br />
== Stars and galaxies (''M'') ==<br />
In stellar and galactic astronomy, the standard distance is 10 parsecs (about 32.616 light years, 308.57 Petameters or 308.57 [[Orders_of_magnitude_(numbers)#1012|trillion]] kilometres).<br />
A star at 10 parsecs has a [[parallax]] of 0.1" (100 milli [[minute of arc|arc seconds]]).<br />
Galaxies (and other [[nebula|extended objects]]) are much larger than 10 parsecs, their light is radiated over an extended patch of sky, and their overall brightness cannot be directly observed from relatively short distances, but the same convention is used. A galaxy's magnitude is defined by measuring all the light radiated over the entire object, trea