View source for YBC 7289

{{good article}}
{{short description|Babylonian mathematical clay tablet}}
[[File:YBC-7289-OBV-REV.jpg|thumb|upright=1.35|YBC 7289]]
'''YBC 7289''' is a [[Babylonia]]n [[clay tablet]] notable for containing an accurate [[sexagesimal]] approximation to the [[square root of 2]], the length of the diagonal of a [[unit square]]. This number is given to the equivalent of six decimal digits, "the greatest known computational accuracy ... in the ancient world".{{r|bs}} The tablet is believed to be the work of a student in southern [[Mesopotamia]] from some time between 1800 and 1600 BC.

== Content ==
[[File:YBC-7289-OBV-labeled.jpg|thumb|250px|right|Babylonian clay tablet YBC 7289 with annotations. The diagonal displays an approximation of the [[square root of 2]] in four [[sexagesimal]] figures, 1 24 51 10, which is good to about six [[decimal]] digits.<br />1 + 24/60 + 51/60<sup>2</sup> + 10/60<sup>3</sup> = 1.41421296... The tablet also gives an example where one side of the square is 30, and the resulting diagonal is 42 25 35 or 42.4263888...]]
The tablet depicts a square with its two diagonals. One side of the square is labeled with the sexagesimal number 30. The diagonal of the square is labeled with two sexagesimal numbers. The first of these two, 1;24,51,10 represents the fraction {{sfrac|305,470|216,000}} ≈ 1.414213, a numerical approximation of the square root of two that is off by less than one part in two million. The second of the two numbers is 42;25,35 = {{sfrac|30547|720}} ≈ 42.426. This number is the result of multiplying 30 by the given approximation to the square root of two, and approximates the length of the diagonal of a square of side length 30.{{r|fr}}

Because the Babylonian sexagesimal notation did not indicate which digit had which place value, one alternative interpretation is that the number on the side of the square is {{frac|30|60}} = {{frac|1|2}}. Under this alternative interpretation, the number on the diagonal is {{sfrac|30,547|43,200}} ≈ 0.70711, a close numerical approximation of <math>\frac{1}{\sqrt2}</math>, the length of the diagonal of a square of side length {{frac|1|2}}, that is also off by less than one part in two million. [[David Fowler (mathematician)|David Fowler]] and [[Eleanor Robson]] write, "Thus we have a reciprocal pair of numbers with a geometric interpretation…".  They point out that, while the importance of reciprocal pairs in Babylonian mathematics makes this interpretation attractive, there are reasons for skepticism.{{r|fr}}

The reverse side is partly erased, but Robson believes it contains a similar problem concerning the diagonal of a rectangle whose two sides and diagonal are in the ratio 3:4:5.{{r|robson}}

== Interpretation ==
Although YBC 7289 is frequently depicted (as in the photo) with the square oriented diagonally, the standard Babylonian conventions for drawing squares would have made the sides of the square vertical and horizontal, with the numbered side at the top.{{r|friberg}} The small round shape of the tablet, and the large writing on it, suggests that it was a "hand tablet" of a type typically used for rough work by a student who would hold it in the palm of his hand.{{r|bs|fr}} The student would likely have copied the sexagesimal value of the square root of 2 from a table of constants, but an iterative procedure for computing this value can be found in another Babylonian tablet, BM 96957 + VAT 6598.{{r|fr}} A table of constants that includes the same approximation of the square root of 2 as YBC 7289 is the tablet YBC 7243. The constant appears on line 10 of the table along with the inscription, "the diagonal of a square".{{r|fr|friberg|ns}}

The mathematical significance of this tablet was first recognized by [[Otto E. Neugebauer]] and [[Abraham Sachs]] in 1945.{{r|fr|ns}}
The tablet "demonstrates the greatest known computational accuracy obtained anywhere in the ancient world", the equivalent of six decimal digits of accuracy.{{r|bs}} Other Babylonian tablets include the computations of areas of [[hexagon]]s and [[heptagon]]s, which involve the approximation of more complicated [[algebraic number]]s such as <math>\sqrt3</math>.{{r|fr}} The same number <math>\sqrt3</math> can also be used in the interpretation of certain ancient Egyptian calculations of the dimensions of pyramids. However, the much greater numerical precision of the numbers on YBC 7289 makes it more clear that they are the result of a general procedure for calculating them, rather than merely being an estimate.{{r|rudman}}

The same sexagesimal approximation to <math>\sqrt2</math>, 1;24,51,10, was used much later by Greek mathematician [[Claudius Ptolemy]] in his ''[[Almagest]]''.{{r|neuhist|ped}} Ptolemy did not explain where this approximation came from and it may be assumed to have been well known by his time.{{r|neuhist}}

== Provenance and curation ==
It is unknown where in Mesopotamia YBC 7289 comes from, but its shape and writing style make it likely that it was created in southern Mesopotamia, sometime between 1800BC and 1600BC.{{r|bs|fr}} 

At Yale, the Institute for the Preservation of Cultural Heritage has produced a digital model of the tablet, suitable for [[3D printing]].{{r|y1|y2|renders}} The original tablet is currently kept in the [[Yale Babylonian Collection]] at Yale University.{{r|y2}}

==See also==
{{Commons category}}
*[[Babylonian mathematics]]
*[[Plimpton 322]]
*[[IM 67118]]

==External links==
* [[Cuneiform Digital Library Initiative]], [https://cdli.mpiwg-berlin.mpg.de The CDLI Collection]
** [https://cdli.ucla.edu/P255048 YBC 7289]
** [https://cdli.ucla.edu/P255045 YBC 7243]
* [[Yale Peabody Museum]], [https://peabody.yale.edu/explore/collections/yale-babylonian-collection Babylonian Collection]
** [https://collections.peabody.yale.edu/search/Record/YPM-BC-021354 YBC 7289]
** [https://collections.peabody.yale.edu/search/Record/YPM-BC-021308 YBC 7243]


== References ==
{{reflist|30em|refs=

<ref name=robson>{{citation
 | last = Robson | first = Eleanor | author-link = Eleanor Robson
 | editor-last = Katz | editor-first = Victor J.
 | page = 143
 | publisher = Princeton University Press
 | contribution = Mesopotamian Mathematics
 | title = The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook
 | url = https://books.google.com/books?id=3ullzl036UEC
 | year = 2007| isbn = 978-0-691-11485-9}}</ref>

<ref name=friberg>{{citation
 | last = Friberg | first = Jöran
 | editor1-first = Jöran
 | editor1-last = Friberg
 | doi = 10.1007/978-0-387-48977-3
 | isbn = 978-0-387-34543-7
 | mr = 2333050
 | page = 211
 | publisher = Springer, New York
 | series = Sources and Studies in the History of Mathematics and Physical Sciences
 | title = A remarkable collection of Babylonian mathematical texts
 | year = 2007}}</ref>

<ref name=fr>{{citation
 | 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
 }}</ref>

<ref name=ns>{{citation
 | last1 = Neugebauer | first1 = O. | author1-link = Otto E. Neugebauer
 | last2 = Sachs | first2 = A. J. | author2-link = Abraham Sachs
 | mr = 0016320
 | page = 43
 | publisher = American Oriental Society and the American Schools of Oriental Research, New Haven, Conn.
 | series = American Oriental Series
 | title = Mathematical Cuneiform Texts
 | year = 1945}}</ref>

<ref name=neuhist>{{citation
 | last = Neugebauer | first = O. | author-link = Otto E. Neugebauer
 | mr = 0465672
 | pages = 22–23
 | publisher = Springer-Verlag, New York-Heidelberg
 | title = A History of Ancient Mathematical Astronomy, Part One
 | url = https://books.google.com/books?id=6tkqBAAAQBAJ&pg=PA22
 | year = 1975| isbn = 978-3-642-61910-6 }}</ref>

<ref name=ped>{{citation|url=https://books.google.com/books?id=8eaHxE9jUrwC&pg=PA57|page=57|title=A Survey of the Almagest|series=Sources and Studies in the History of Mathematics and Physical Sciences|first=Olaf|last=Pedersen|editor-first=Alexander|editor-last=Jones|publisher=Springer|year=2011|isbn=978-0-387-84826-6}}</ref>

<ref name=rudman>{{citation
 | last = Rudman | first = Peter S.
 | isbn = 978-1-59102-477-4
 | mr = 2329364
 | page = 241
 | publisher = Prometheus Books, Amherst, NY
 | title = How mathematics happened: the first 50,000 years
 | url = https://books.google.com/books?id=BtcQq4RUfkUC&pg=PA241
 | year = 2007}}</ref>

<ref name=bs>{{citation
 | last1 = Beery | first1 = Janet L. | author1-link = Janet Beery
 | last2 = Swetz | first2 = Frank J.
 | date = July 2012
 | doi = 10.4169/loci003889
 | journal = Convergence
 | publisher = Mathematical Association of America
 | title = The best known old Babylonian tablet?| doi-broken-date = 10 February 2025 | doi-access = free
 }}</ref>

<ref name=y1>{{citation|title=A 3,800-year journey from classroom to classroom|first=Patrick|last=Lynch|magazine=Yale News|date=April 11, 2016|url=https://news.yale.edu/2016/04/11/3800-year-journey-classroom-classroom|access-date=2017-10-25}}</ref>

<ref name=y2>{{citation|title=A 3D-print of ancient history: one of the most famous mathematical texts from Mesopotamia|date=January 16, 2016|url=http://ipch.yale.edu/news/3d-print-ancient-history-one-most-famous-mathematical-texts-mesopotamia|publisher=Yale Institute for the Preservation of Cultural Heritage|access-date=2017-10-25}}</ref>

<ref name=renders>{{citation
 |title=Mesopotamian tablet YBC 7289
 |last=Kwan
 |first=Alistair
 |date=April 20, 2019
 |publisher=University of Auckland
|doi = 10.17608/k6.auckland.6114425.v1}}
</ref>
}}

[[Category:Babylonian mathematics]]
[[Category:Mathematics manuscripts]]
[[Category:Clay tablets]]
[[Category:18th-century BC works]]