Editing Square root of 2 (section)

==History==
[[File:Ybc7289-bw.jpg|right|thumb|200px|Babylonian clay tablet [[YBC 7289]] with annotations. Besides showing the square root of 2 in [[sexagesimal]] ({{nowrap|1 24 51 10}}), the tablet also gives an example where one side of the square is 30 and the diagonal then is {{nowrap|42 25 35}}. The sexagesimal digit 30 can also stand for {{nowrap|0 30}} = {{sfrac|1|2}}, in which case {{nowrap|0 42 25 35}} is approximately 0.7071065.]]
The [[Babylonia]]n clay tablet [[YBC 7289]] ({{Circa|1800}}–1600 BC) gives an approximation of <math>\sqrt{2}</math> in four [[sexagesimal]] figures, {{nowrap|1 24 51 10}}, which is accurate to about six [[decimal]] digits,<ref>{{cite journal
 | last1 = Fowler | first1 = David | author1-link = David Fowler (mathematician)
 | last2 = Robson | first2 = Eleanor | author2-link = Eleanor Robson
 | doi = 10.1006/hmat.1998.2209
 | issue = 4
 | journal = [[Historia Mathematica]]
 | mr = 1662496
 | pages = 366–378
 | title = Square root approximations in old Babylonian mathematics: YBC 7289 in context
 | volume = 25
 | year = 1998| doi-access = free
 }} See p. 368.<br />[http://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html Photograph, illustration, and description of the ''root(2)'' tablet from the Yale Babylonian Collection] {{webarchive|url=https://web.archive.org/web/20120813054036/http://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html |date=2012-08-13 }}<br />[http://www.math.ubc.ca/%7Ecass/Euclid/ybc/ybc.html High resolution photographs, descriptions, and analysis of the ''root(2)'' tablet (YBC 7289) from the Yale Babylonian Collection]</ref> and is the closest possible three-place sexagesimal representation of <math>\sqrt{2}</math>, representing a margin of error of only –0.000042%:
:<math>1 + \frac{24}{60} + \frac{51}{60^2} + \frac{10}{60^3} = \frac{305470}{216000} = 1.41421\overline{296}.</math>

Another early approximation is given in [[History of India|ancient Indian]] mathematical texts, the [[Sulba Sutras|Sulbasutras]] ({{Circa|800}}–200 BC), as follows: ''Increase the length [of the side] by its third and this third by its own fourth less the thirty-fourth part of that fourth.''<ref>{{citation |last=Henderson |first=David W. |author-link=David W. Henderson |editor-last=Gorini |editor-first=Catherine A. |date=2000 |chapter=Square roots in the Śulba Sūtras |title=Geometry At Work: Papers in Applied Geometry |series=Mathematical Association of America Notes |volume=53 |location=Washington, D.C. |publisher=[[The Mathematical Association of America]] |pages=39–45 |isbn=978-0883851647 |url=http://www.math.cornell.edu/~dwh/papers/sulba/sulba.html}}</ref> That is,
:<math>1 + \frac{1}{3} + \frac{1}{3 \times 4} - \frac{1}{3 \times4 \times 34} = \frac{577}{408} = 1.41421\overline{56862745098039}.</math>

This approximation, diverging from the actual value of <math>\sqrt{2}</math> by approximately +0.07%, is the seventh in a sequence of increasingly accurate approximations based on the sequence of [[Pell number]]s, which can be derived from the [[simple continued fraction|continued fraction expansion]] of <math>\sqrt{2}</math>. Despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation.

[[Pythagoreanism|Pythagoreans]] discovered that the diagonal of a [[square]] is incommensurable with its side, or in modern language, that the square root of two is [[irrational number|irrational]]. Little is known with certainty about the time or circumstances of this discovery, but the name of [[Hippasus]] of Metapontum is often mentioned. For a while, the Pythagoreans treated as an official secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it, though this has little to any substantial evidence in traditional historian practice.<ref>{{citation |title=The Dangerous Ratio |url=https://nrich.maths.org/2671 |access-date=2023-09-18 |website=nrich.maths.org}}</ref><ref>{{citation |last=Von Fritz |first=Kurt |date=1945 |title=The Discovery of Incommensurability by Hippasus of Metapontum  |journal=Annals of Mathematics |volume=46 |issue=2 |pages=242–264 |doi=10.2307/1969021 |jstor=1969021 |issn=0003-486X}}</ref> The square root of two is occasionally called '''Pythagoras's number'''<ref>{{citation |last1=Conway |first1=John H. |author1-link=John H. Conway |last2=Guy |first2=Richard K. |author2-link = Richard K. Guy |date=1996 |title=The Book of Numbers |title-link=The Book of Numbers (math book) |location=New York |publisher=Copernicus |page=25 |isbn=978-1461240723}}</ref> or '''Pythagoras's constant'''.

===Ancient Roman architecture===
In [[ancient Roman architecture]], [[Vitruvius]] describes the use of the square root of 2 progression or ''ad quadratum'' technique. It consists basically in a geometric, rather than arithmetic, method to double a square, in which the diagonal of the original square is equal to the side of the resulting square. Vitruvius attributes the idea to [[Plato]]. The system was employed to build pavements by creating a square [[tangent]] to the corners of the original square at 45 degrees of it. The proportion was also used to design [[Atrium (architecture)|atria]] by giving them a length equal to a diagonal taken from a square, whose sides are equivalent to the intended atrium's width.<ref>{{citation |title=Architecture and Mathematics from Antiquity to the Future: Volume I: Antiquity to the 1500s|last1=Williams|first1=Kim|author1-link=Kim Williams (architect)|last2=Ostwald|first2=Michael|publisher=Birkhäuser|year=2015|isbn=9783319001371|pages=204}}</ref>