Editing Square root (section)

==History==
[[File:Ybc7289-bw.jpg|left|thumb|YBC 7289 clay tablet]]

The [[Yale Babylonian Collection]] clay tablet [[YBC 7289]] was created between 1800&nbsp;BC and 1600&nbsp;BC, showing <math>\sqrt{2}</math> and <math display="inline">\frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}}</math> respectively as 1;24,51,10 and 0;42,25,35 [[Sexagesimal|base 60]] numbers on a square crossed by two diagonals.<ref>{{cite web | url=http://www.math.ubc.ca/~cass/Euclid/ybc/analysis.html | title=Analysis of YBC 7289 | work=ubc.ca | access-date=19 January 2015}}</ref> (1;24,51,10) base 60 corresponds to 1.41421296, which is correct to 5 decimal places (1.41421356...).

The [[Rhind Mathematical Papyrus]] is a copy from 1650&nbsp;BC of an earlier [[Berlin Papyrus 6619|Berlin Papyrus]] and other texts{{snd}}possibly the [[Kahun Papyrus]]{{snd}}that shows how the Egyptians extracted square roots by an inverse proportion method.<ref>Anglin, W.S. (1994). ''Mathematics: A Concise History and Philosophy''. New York: Springer-Verlag.</ref>

In [[History of India|Ancient India]], the knowledge of theoretical and applied aspects of square and square root was at least as old as the ''[[Sulba Sutras]]'', dated around 800–500&nbsp;BC (possibly much earlier).<ref>{{Cite journal |last=Seidenberg |first=A. |date=1961 |title=The ritual origin of geometry |url=http://dx.doi.org/10.1007/bf00327767 |journal=Archive for History of Exact Sciences |volume=1 |issue=5 |pages=488–527 |doi=10.1007/bf00327767 |s2cid=119992603 |issn=0003-9519 |quote=Seidenberg (pp. 501-505) proposes: "It is the distinction between use and origin." [By analogy] "KEPLER needed the ellipse to describe the paths of the planets around the sun; he did not, however invent the ellipse, but made use of a curve that had been lying around for nearly 2000 years". In this manner Seidenberg argues: "Although the date of a manuscript or text cannot give us the age of the practices it discloses, nonetheless the evidence is contained in manuscripts." Seidenberg quotes Thibaut from 1875: "Regarding the time in which the Sulvasutras may have been composed, it is impossible to give more accurate information than we are able to give about the date of the Kalpasutras. But whatever the period may have been during which Kalpasutras and Sulvasutras were composed in the form now before us, we must keep in view that they only give a systematically arranged description of sacrificial rites, which had been practiced during long preceding ages." Lastly, Seidenberg summarizes: "In 1899, THIBAUT ventured to assign the fourth or the third centuries B.C. as the latest possible date for the composition of the Sulvasutras (it being understood that this refers to a codification of far older material)."}}</ref> A method for finding very good approximations to the square roots of 2 and 3 are given in the ''[[Baudhayana Sulba Sutra]]''.<ref>Joseph, ch.8.</ref> [[Apastamba]] who was dated around 600 BCE has given a strikingly accurate value for <math>\sqrt{2}</math> which is correct up to five decimal places as <math display="inline"> 1 + \frac{1}{3} + \frac{1}{3\times 4} - \frac{1}{3\times 4\times 34}</math>.<ref>{{cite journal |last1=Dutta |first1=Bibhutibhusan |date=1931 |title=On the Origin of the Hindu Terms for "Root" |url=https://www.jstor.org/stable/2300909 |journal= The American Mathematical Monthly|volume= 38|issue=7 |pages= 371–376|doi=10.2307/2300909 |jstor=2300909 |access-date= 30 March 2024}}</ref><ref>{{cite web |url=https://maa.org/press/periodicals/convergence/ancient-indian-rope-geometry-in-the-classroom-approximating-the-square-root-of-2#:~:text=The%20Śulba-sūtras%20of%20Āpastamba,is%20less%20than%200.0003%25! |title=Ancient Indian Rope Geometry in the Classroom - Approximating the Square Root of 2 |author=Cynthia J. Huffman |author2=Scott V. Thuong |date= 2015 |website=www.maa.org |publisher= |access-date=30 March 2024 |quote=Increase the measure by its third and this third by its own fourth, less the thirty-fourth part of that fourth. This is the value with a special quantity in excess.}}</ref>
<ref>{{cite web |url=https://mathshistory.st-andrews.ac.uk/Biographies/Apastamba/ |title=Apastamba |author=J J O'Connor |author2=E F Robertson |date= November 2020|website=www.mathshistory.st-andrews.ac.uk|publisher= School of Mathematics and Statistics, University of St Andrews, Scotland |access-date=30 March 2024 }}</ref> [[Aryabhata]], in the ''[[Aryabhatiya]]'' (section 2.4), has given a method for finding the square root of numbers having many digits.

It was known to the ancient Greeks that square roots of [[Natural number|positive integers]] that are not [[Square number|perfect square]]s are always [[irrational number]]s: numbers not expressible as a [[ratio]] of two integers (that is, they cannot be written exactly as <math>\frac{m}{n}</math>, where {{mvar|m}} and {{mvar|n}} are integers). This is the theorem [[Euclid's Elements|''Euclid X, 9'']], almost certainly due to [[Theaetetus (mathematician)|Theaetetus]] dating back to {{circa|380&nbsp;BC}}.<ref>{{cite book
|first= Sir Thomas L. 
|last= Heath
|title= The Thirteen Books of The Elements, Vol. 3
|url=https://archive.org/stream/thirteenbookseu03heibgoog#page/n14/mode/1up
|year=1908
|publisher=Cambridge University Press
|page=3
}}</ref>
The discovery of irrational numbers, including the particular case of the [[square root of 2]], is widely associated with the Pythagorean school.<ref>{{cite book |title=History of Mathematics: A Supplement |author1=Craig Smorynski |edition=illustrated, annotated |publisher=Springer Science & Business Media |year=2007 |isbn=978-0-387-75480-2 |page=49 |url=https://books.google.com/books?id=_zliInaOM8UC}} [https://books.google.com/books?id=_zliInaOM8UC&pg=PA49 Extract of page 49]</ref><ref>{{cite book |title=Calculus: Single Variable, Volume 1 |author1=Brian E. Blank |author2=Steven George Krantz |edition=illustrated |publisher=Springer Science & Business Media |year=2006 |isbn=978-1-931914-59-8 |page=71 |url=https://books.google.com/books?id=hMY8lbX87Y8C}} [https://books.google.com/books?id=hMY8lbX87Y8C&pg=PA71 Extract of page 71]</ref> Although some accounts attribute the discovery to [[Hippasus]], the specific contributor remains uncertain due to the scarcity of primary sources and the secretive nature of the brotherhood.<ref>Boyer, Carl B.; Merzbach, Uta C. (2011). A History of Mathematics (3rd ed.). Hoboken, NJ: John Wiley & Sons. pp. 51–53. ISBN 978-0470525487.</ref><ref>Stillwell, John (2010). ''Mathematics and Its History'' (3rd ed.). New York, NY: Springer. pp. 14–15. ISBN 978-1441960528.</ref> It is exactly the length of the [[diagonal]] of a [[unit square|square with side length 1]].

In the Chinese mathematical work ''[[Writings on Reckoning]]'', written between 202&nbsp;BC and 186&nbsp;BC during the early [[Han dynasty]], the square root is approximated by using an "excess and deficiency" method, which says to "...combine the excess and deficiency as the divisor; (taking) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend."<ref>Dauben (2007), p. 210.</ref>

A symbol for square roots, written as an elaborate R, was invented by [[Regiomontanus]] (1436–1476). An R was also used for radix to indicate square roots in [[Gerolamo Cardano]]'s ''[[Ars Magna (Gerolamo Cardano)|Ars Magna]]''.<ref>{{cite web | url=http://nrich.maths.org/6546|title=The Development of Algebra - 2|work=maths.org|access-date=19 January 2015|url-status=live|archive-url=https://web.archive.org/web/20141124102946/http://nrich.maths.org/6546|archive-date=24 November 2014}}</ref>

According to historian of mathematics [[David Eugene Smith|D.E. Smith]], Aryabhata's method for finding the square root was first introduced in Europe by [[Pietro di Giacomo Cataneo|Cataneo]]—in 1546.

According to Jeffrey A. Oaks, Arabs used the letter ''[[Gimel#Arabic ĝīm|jīm/ĝīm]]'' ({{lang|ar|ج}}), the first letter of the word "{{lang|ar|جذر}}" (variously transliterated as ''jaḏr'', ''jiḏr'', ''ǧaḏr'' or ''ǧiḏr'', "root"), placed in its initial form ({{lang|ar|ﺟ}}) over a number to indicate its square root. The letter ''jīm'' resembles the present square root shape. Its usage goes as far as the end of the twelfth century in the works of the Moroccan mathematician [[Ibn al-Yasamin]].<ref>{{cite thesis | title=Algebraic Symbolism in Medieval Arabic Algebra | first1=Jeffrey A. | last1=Oaks | publisher=Philosophica | year=2012 | page=36 | url=http://logica.ugent.be/philosophica/fulltexts/87-2.pdf | url-status=live | archive-url=https://web.archive.org/web/20161203134229/http://logica.ugent.be/philosophica/fulltexts/87-2.pdf | archive-date=2016-12-03 }}</ref>

The symbol "√" for the square root was first used in print in 1525, in [[Christoph Rudolff]]'s ''Coss''.<ref>{{Cite book| last=Manguel|first=Alberto| chapter=Done on paper: the dual nature of numbers and the page | title=The Life of Numbers | year=2006 |publisher=Taric, S.A. | isbn=84-86882-14-1}}</ref>