YBC 7289

This number is given to the equivalent of six decimal digits, "the greatest known computational accuracy ... in the ancient world".

[2] 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.

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.

[1][2] 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.

[1] Other Babylonian tablets include the computations of areas of hexagons and heptagons, which involve the approximation of more complicated algebraic numbers such as

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.

[1][2] At Yale, the Institute for the Preservation of Cultural Heritage has produced a digital model of the tablet, suitable for 3D printing.

YBC 7289
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.
1 + 24/60 + 51/60 2 + 10/60 3 = 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...