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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. The student would likely have copied the sexagesimal value of the square root of 2 from another tablet, but an iterative procedure for computing this value can be found in another
Babylonian tablet, BM 96957 + VAT 6598.
86:
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 = 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.
215:
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
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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 number 305470/216000 ≈ 1.414213, a numerical approximation of the
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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 30/60 = 1/2. Under this alternative interpretation, the number on the diagonal is 30547/43200 ≈ 0.70711, a close numerical
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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. The small round shape of the tablet, and the
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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.
154:
in 1945. The tablet "demonstrates the greatest known computational accuracy obtained anywhere in the ancient world", the equivalent of six decimal digits of accuracy. Other
Babylonian tablets include the computations of areas of
51:. This number is given to the equivalent of six decimal digits, "the greatest known computational accuracy ... in the ancient world". The tablet is believed to be the work of a student in southern
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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.
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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.
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1 + 24/60 + 51/60 + 10/60 = 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...
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473:, American Oriental Series, American Oriental Society and the American Schools of Oriental Research, New Haven, Conn., p. 43,
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At Yale, the
Institute for the Preservation of Cultural Heritage has produced a digital model of the tablet, suitable for
253:. Ptolemy did not explain where this approximation came from and it may be assumed to have been well known by his time.
122:, the length of the diagonal of a square of side length 1/2, that is also off by less than one part in two million.
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667:
682:
426:, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, New York, p. 211,
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571:, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, p. 57,
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Babylonian clay tablet YBC 7289 with annotations. The diagonal displays an approximation of the
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268:. The original tablet is currently kept in the Yale Babylonian Collection at Yale University.
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A 3D-print of ancient history: one of the most famous mathematical texts from
Mesopotamia
355:(1998), "Square root approximations in old Babylonian mathematics: YBC 7289 in context",
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623:, Yale Institute for the Preservation of Cultural Heritage, January 16, 2016
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The
Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook
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311:; Swetz, Frank J. (July 2012), "The best known old Babylonian tablet?",
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The mathematical significance of this tablet was first recognized by
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396:(2007), "Mesopotamian Mathematics", in Katz, Victor J. (ed.),
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537:, Springer-Verlag, New York-Heidelberg, pp. 22–23,
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A remarkable collection of Babylonian mathematical texts
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A History of Ancient Mathematical Astronomy, Part One
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593:"A 3,800-year journey from classroom to classroom"
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494:How mathematics happened: the first 50,000 years
75:figures, 1 24 51 10, which is good to about six
565:Pedersen, Olaf (2011), Jones, Alexander (ed.),
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497:, Prometheus Books, Amherst, NY, p. 241,
422:Friberg, Jöran (2007), Friberg, Jöran (ed.),
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640:Kwan, Alistair (April 20, 2019),
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115:{\displaystyle 1/{\sqrt {2}}}
642:Mesopotamian tablet YBC 7289
471:Mathematical Cuneiform Texts
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236:{\displaystyle {\sqrt {2}}}
208:{\displaystyle {\sqrt {3}}}
184:{\displaystyle {\sqrt {3}}}
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644:, University of Auckland,
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491:Rudman, Peter S. (2007),
432:10.1007/978-0-387-48977-3
568:A Survey of the Almagest
673:Mathematics manuscripts
257:Provenance and curation
668:Babylonian mathematics
372:10.1006/hmat.1998.2209
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191:. The same number
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148:Otto E. Neugebauer
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578:978-0-387-84826-6
544:978-3-642-61910-6
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313:Convergence
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73:sexagesimal
53:Mesopotamia
49:unit square
41:sexagesimal
37:clay tablet
662:Categories
627:2017-10-25
602:2017-10-25
289:References
216:estimate.
34:Babylonian
597:Yale News
161:heptagons
531:(1975),
469:(1945),
283:IM 67118
272:See also
250:Almagest
167:such as
157:hexagons
71:in four
30:YBC 7289
25:YBC 7289
553:0465672
513:2329364
479:0016320
450:2333050
381:1662496
247:in his
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59:Content
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664::
610:^
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549:MR
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509:MR
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377:MR
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363:25
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330:^
296:^
648::
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369::
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229:2
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177:3
108:2
102:/
98:1
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