Modular arithmetic: Difference between revisions

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My name is Abe and I am studying Dramatic Literature and History and English Literature at Bydgoszcz / Poland.<br><br>My web site ... [http://www.ted.com/talks/barry_schwartz_on_our_loss_of_wisdom link schwartz]
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{{Wiktionary}}
A '''myriad''' (from [[Ancient Greek]] {{lang|grc|μυριάδες}}, ''myriades'') is technically the number [[10000 (number)|ten thousand]]; in that sense, the term is used almost exclusively in translations from Greek, Latin, or Chinese, or when talking about ancient Greek numbers. More generally, a myriad may be an [[Indefinite and fictitious numbers|indefinitely large number of things]].<ref>''[[Oxford English Dictionary]]'', Third Edition, June 2003, ''s.v.'' '[http://www.oed.com/view/Entry/124538 myriad]'</ref>
 
==History==
{{Main|Greek numerals}}
The [[Aegean numerals]] of the [[Minoans|Minoan]] and [[Mycenæan civilization]]s included a single unit to denote tens of thousands. It was written ☼.<ref name=Verdan>{{cite web|url=http://www.dma.ens.fr/culturemath/histoire%20des%20maths/htm/Verdan/Verdan.htm|title=Systèmes numéraux en Grèce ancienne: description et mise en perspective historique|author=Samuel Verdan|date=20 Mar 2007|accessdate=2 Mar 2011}} {{fr icon}}</ref>
 
In Classical [[Greek numerals]], a myriad was written as a capital [[mu (letter)|mu]]: '''Μ'''. To distinguish this numeral from letters, it was sometimes given an [[overbar]]: {{overline|M}}. Multiples were written above this sign, so that for example <math>\stackrel{\delta\phi\pi\beta}{\Mu}</math> would equal 4,582×10,000 or 45,820,000. The etymology of the word ''myriad'' itself is uncertain: it has been variously connected to [[Proto-Indo-European|PIE]] ''*meu-'' ("damp") in reference to the waves of the sea and to Greek ''myrmex'' ({{lang|grc|μύρμηξ}}, "ant") in reference to their swarms.<ref>Schwartzman, Steven. ''The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English'', [http://books.google.ca/books?id=SRw4PevE4zUC&pg=PA142 p. 142]. The Mathematical Assoc. of America, 1994.</ref>
 
The largest number named in [[Ancient Greek]] was the myriad myriad (written {{overline|MM}}) or hundred million. In his ''[[Sand Reckoner]]'', [[Archimedes|Archimedes of Syracuse]] used this quantity as the basis for a numeration system of large powers of ten, which he used to count grains of sand.
 
==Usage==
===Greek===
In [[Modern Greek]], the word "myriad" is rarely used to denote 10,000, but a million is ''ekatommyrio'' ({{lang|el|εκατομμύριο}}, ''lit.'' 'hundred myriad') and a [[1 E9|thousand million]] is ''disekatommyrio'' ({{lang|el|δισεκατομμύριο}}, ''lit.'' 'twice hundred myriad').
 
===English===
In English, "myriad" is most commonly used to mean "some large but unspecified number". It may be either an [[adjective]] or a [[noun]]: both "there are myriad people outside" and "there is a myriad of people outside" are in use.<ref name=myriamwebster>Merriam-Webster Online. "[http://www.merriam-webster.com/dictionary/myriad Myriad]". 2013. Accessed 1 November 2013.</ref> (There are small differences: the former could imply that it is a ''diverse'' group of people; the latter does not but could possibly indicate a group of exactly ten thousand.) The ''[[Merriam-Webster Dictionary]]'' notes that confusion over the use of myriad as a noun "seems to reflect a mistaken belief that the word was originally and is still properly only an adjective ... however, the noun is in fact the older form, dating to the 16th century. The noun 'myriad' has appeared in the works of such writers as [[John Milton|Milton]] (plural 'myriads') and [[Henry David Thoreau|Thoreau]] ('a myriad of'), and it continues to occur frequently in reputable English."<ref name=myriamwebster/>
 
"Myriad" is also infrequently used in English as the specific number 10,000. Owing to the possible confusion with the generic meaning of "large quantity", however, this is generally restricted to translation of other languages like [[ancient Greek]], [[Chinese numerals|Chinese]], and [[Indian numbering system|Hindi]] where numbers may be grouped into sets of 10,000 (myriads). Such use permits the translator to remain closer to the original text and avoid repeated and unwieldy mentions of "tens of thousands": for example, "the original number of the crews supplied by the several nations I find to have been twenty-four myriads"<ref>[[Herodotus]]. ''[[The History of Herodotus]]'', VII.[http://www.sacred-texts.com/cla/hh/hh7180.htm 184]. Translation by G.C. Macaulay, 1890. Accessed 1 Nov 2013.</ref> and "What is the distance between one bridge and another? Twelve myriads of [[parasang]]s".<ref>Janowitz, Naomi. ''The Poetics of Ascent: Theories of Language in a Rabbinic Ascent Text'', [http://books.google.ca/books?id=QBmjU8WsP6IC&pg=PA118 p. 118]. SUNY Press (New York), 1989. Accessed 1 November 2013.</ref>
 
In [[British English]], a "[[Myriad (unit of area)|myriad]]" is a {{convert|100|xx|100|km|mi|sp=us|adj=on}} area, that is, 10,000 square kilometers, particularly on the [[Ordinance Survey]]'s [[British national grid reference system|National Grid]].
 
==In other languages==
 
===Europe===
Most European languages include variations of "myriad" with similar meanings to the English word.
 
Additionally, the [[metric prefix|prefix]] ''myria-'' indicating multiplication times ten thousand (×10<sup>4</sup>) was part of the original [[metric system]] adopted by [[First French Republic|France]] in 1795.<ref>''L'Histoire Du Mètre'': "[http://histoire.du.metre.free.fr/fr/Pages/Sommaire/06.htm ''La Loi Du 18 Germinal An 3'']". 2005. Accessed 1 November 2013. {{fr icon}}</ref> Although it was not retained after the 11th [[Conférence générale des poids et mesures|CGPM conference]] in 1960, "[[myriameter]]" is sometimes still encountered as a translation of the [[Scandinavian mile]] ([[Swedish language|Swedish]]&nbsp;& [[Norwegian language|Norwegian]]:&nbsp;''mil'') of {{convert|10|km|sp=us|2}}. The ''[[myriagram]]me'' was a French approximation of the [[avoirdupois]] ''quartier'' of {{convert|25|lb|sp=us}} and the "myriaton" appears in [[Isaac Asimov]]'s ''[[Foundation and Empire|Foundation]]'' trilogy.
 
===East Asia===
{{see also|Chinese numerals|Japanese numerals|Korean numerals|Vietnamese numerals}}
In [[East Asian culture|East Asia]], the traditional numeral systems of [[Chinese numerals|China]], [[Korean numerals|Korea]], and [[Japanese numerals|Japan]] are all [[decimal]]-based but grouped into ten thousands rather than thousands. The [[Chinese character|character]] for myriad is {{lang|zh|{{linktext|萬}}}} in [[traditional characters|traditional]] script and {{lang|zh|{{linktext|万}}}} in [[simplified characters|simplified]] form in both [[Simplified Chinese character#Mainland China|mainland China]] and [[Japanese script reform|Japan]]. The pronunciation varies within China and abroad: ''wàn'' ([[Mandarin Chinese|Mandarin]]), ''wan''<sup>5</sup> ([[Hakka language|Hakka]]), ''bān'' ([[Minnan language|Minnan]]), ''maan''<sup>6</sup> ([[Cantonese language|Cantonese]]), ''man'' ([[Japanese language|Japanese]] and [[Korean language|Korean]]), and ''vạn'' ([[Vietnamese language|Vietnamese]]). [[Vietnam]] is peculiar within the [[adoption of Chinese literary culture|Sinosphere]] in largely rejecting Chinese numerals in favor of [[Vietnamese numerals|its own]]: ''vạn'' is less common than the native ''mười nghìn'' ("ten thousand") and its numerals are grouped in threes.
 
Because most East Asian numerals are grouped into fours, higher orders of numbers are provided by the [[power (math)|powers]] of 10,000: 10,000² was {{lang|zh|{{linktext|萬萬}}}} in ancient texts but is now written as 1,0000,0000 and called {{lang|zh|{{linktext|億}}}}; 10,000³ is 1,0000,0000,0000 or {{lang|zh|{{linktext|兆}}}}; 10,000<sup>4</sup> is 1,0000,0000,0000,0000 or {{lang|zh|{{linktext|京}}}}; and so on. Conversely, Chinese, Japanese, and Korean generally don't have native words for powers of one thousand: what is called "one&nbsp;million" in English is "one hundred ''wan''" in Chinese and "one&nbsp;billion" is "ten ''chō''" in Japanese. Unusually, Vietnam employs its former translation of {{lang|zh|兆}}, ''một triệu'', to mean 1,000,000 rather than the Chinese figure.
 
{{lang|zh|萬}} and {{lang|zh|万}} are also employed colloquially in myriad expressions, [[cliché]]s, and ''[[chengyu]]'' (idioms) in the senses of "vast", "numerous", "numberless", and "infinite". A [[skeleton key]] is a {{lang|zh|{{linktext|万|能|钥|匙}}}} ("myriad-use key");<ref>Nciku.com. "[http://www.nciku.com/search/all/examples/%E4%B8%87%E8%83%BD%E9%92%A5%E5%8C%99 万能钥匙]". Accessed 1 November 2013.</ref> the [[emperor of China|emperor]] was the "lord of myriad [[chariot]]s" ({{lang|zh|萬乘之主}});<ref>Wai Keung Chan, Timothy.<!--sic--> ''Considering the End: Mortality in Early Medieval Chinese Poetic Representation'', [http://books.google.no/books?id=iQgyAQAAQBAJ&pg=PA23 23]. Brill, 2012. Accessed 1 November 2013.</ref> [[Zhu Xi]]'s {{lang|zh|{{linktext|月|映|万|川}}}} ("the moon reflects in myriad rivers") had the sense of supporting greater [[empiricism]] in [[Chinese philosophy]];<ref>Chen Derong. ''Metaphorical Metaphysics in Chinese Philosophy'', [http://books.google.no/books?id=Xt0iFhh0pAgC&pg=PA29 p. 29]. Lexington Books ([[Lanham, Maryland|Lanham]]), 2011. Accessed 1 November 2013.</ref> and Ha Qiongwen's popular 1959 [[Propaganda in the People's Republic of China|propaganda poster]] {{lang|zh|{{linktext|毛|主席|万岁}}}}, which could be literally read as "[[Chairman of the Central Military Commission|Chairman]] [[Mao Zedong|Mao]] is 10,000 years old", in fact meant "Long live Chairman Mao".<ref>Yeh Wen-hsin & al. ''Visualizing China, 1845&ndash;1965: Moving and Still Images in Historical Narratives'', [http://books.google.no/books?id=iSEyAQAAQBAJ&pg=PA416 pp. 416 ff.] Brill, 2012. Accessed 1 November 2013.</ref>
 
==See also==
 
*[[Indian numbering system]], which includes the [[lakh]] or 10 myriad
 
==References==
{{reflist|2}}
 
[[Category:Integers]]

Latest revision as of 14:15, 8 January 2015

My name is Abe and I am studying Dramatic Literature and History and English Literature at Bydgoszcz / Poland.

My web site ... link schwartz