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{{Infobox number
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
| number =  5040
| divisor = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28,  30, 35, 36, 40, 42, 45, 48, 56,  60, 63, 70, 72, 80, 84, 90, 105,  112, 120, 126, 140, 144, 168,  180, 210, 240, 252, 280, 315,  336, 360, 420, 504, 560, 630,  720, 840, 1008, 1260, 1680,  2520, 5040
}}
'''5040''' is a [[factorial]] (7!), a [[superior highly composite number]], a [[colossally abundant number]], and the [[number]] of [[permutation]]s of 4 items out of 10 choices (10 × 9 × 8 × 7 = 5040).


==Philosophy==
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]]
[[Plato]] mentions in his [[Laws (dialogue)|Laws]] that 5040 is a convenient number to use for [[division (mathematics)|dividing]] many things (including both the citizens and the land of a state) into lesser parts.  He remarks that this number can be divided by all the [[natural numbers|(natural) numbers]] from [[1 (number)|1]] to [[12 (number)|12]] with the single exception of [[11 (number)|11]].  He rectifies this "defect" by suggesting that two families could be subtracted from the citizen body to produce the number 5038, which is [[divisible]] by 11.  Plato also took notice of the fact that 5040 can be divided by 12 twice over.  Indeed, Plato's repeated insistence on the use of 5040 for various state purposes is so evident that it is written, "Plato, writing under [[Pythagoreanism|Pythagorean]] influences, seems really to have supposed that the well-being of the city depended almost as much on the number 5040 as on [[justice]] and [[moderation]]."<ref>[http://www.gutenberg.org/files/1750/1750-h/1750-h.htm Laws], by Plato at Project Gutenberg; retrieved 7 July 2009</ref>
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


==Number Theory==
Registered users will be able to choose between the following three rendering modes:  
If <math>\sigma(n)</math> is the [[divisor function]] and <math>\gamma</math> is the [[Euler-Mascheroni constant]], then 5040 is the largest of the known numbers {{OEIS| A067698}} for which this [[Inequality (mathematics)|inequality]] holds:
:<math>\sigma(n) \geq e^\gamma n\log \log n </math>.
This is somewhat unusual, since in the [[limit (mathematics)|limit]] we have:
:<math>\limsup_{n\rightarrow\infty}\frac{\sigma(n)}{n\ \log \log n}=e^\gamma.</math>
Guy Robin showed in 1984 that the inequality fails for all larger numbers [[if and only if]] the [[Riemann hypothesis]] is true.


==Other==
'''MathML'''
▪ 5040 has exactly 60 divisors, counting itself and 1.
:<math forcemathmode="mathml">E=mc^2</math>


▪ In a [[vigesimal]] system, 5040 is represented as 12 groups of 20 and 12 groups of 20-squared (12 • 20 = 240, 12 • 20<sup>2</sup> = 4800, and 240 + 4800 = 5040).
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


▪ 5040 is the sum of 42 consecutive primes (23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 +163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211 + 223 + 227 + 229).
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


▪ 5040 is considered an important number in some systems of [[numerology]], not only because of the Plato connection, but because using round figures, the sum of the [[radius|radii]] of both the [[Earth]] and [[Moon]] (in miles) is 3960 + 1080 = 5040.<ref>''City of Revelation: On the Proportions and Symbolic Numbers of the Cosmic Temple'', by [[John Michell (writer)|John Michell]] (ISBN 0-345-23607-6), p. 61.</ref> Incidentally, the sum of their [[diameter]]s is also the number of minutes in a week (7 days × 24 hours × 60 minutes = 10,080).
<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].


▪ The [[ratio]] of the [[radius]] of the moon and the radius of the earth is 1080/3960, which simplifies to 3/11. This ratio can also be expressed as (4 - [[π]])/π, when using 22/7 as the value of π. This means that the sizes of the earth and the moon are related by a simple function of π.
==Demos==


▪ Given that the radius of the moon is 3/11 that of the earth, the sum of their radii can be broken into 3/14 (for the radius of the moon) and 11/14 (for the radius of the earth). Further, the sum of their radii in miles is 5040, which when divided by 14 is 360 (the number of degrees in a circle). This would not happen for another pair of objects with radii in the same ratio - it only happens when the sum of their radii is 5040.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


▪ 5040 is the largest factorial that is also a highly composite number. All factorials smaller than 8!=40320 are highly composite.


==Notes==
* accessibility:
<references/>
** 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]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** 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]].
** 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]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


==External links==
==Test pages ==
* [[Mathworld]] [http://mathworld.wolfram.com/PlatosNumbers.html article on Plato's numbers]


[[Category:Integers|5e03 5040]]
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
[[Category:Plato]]
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]
 
*[[Inputtypes|Inputtypes (private Wikis only)]]
*[[Url2Image|Url2Image (private Wikis only)]]
==Bug reporting==
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 .

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML

E=mc2


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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