A Babylonian clay tablet pinned down the square root of 2 almost perfectly
Draw a square with sides of one unit and its diagonal measures the square root of 2, roughly 1.4142. Scribes pressed a remarkably good value into a clay tablet now called YBC 7289, dated to around 1800 to 1600 BC. It is accurate to about six decimal places, with an error of only 0.000042%.
That tablet wrote the number in base 60 as 1 24 51 10, the closest possible value with three places after the 1. Ancient Indian geometry manuals, the Sulbasutras, gave a recipe of their own: add a third to the side, then a quarter of that third, minus a thirty-fourth of that quarter. Despite using a smaller denominator, the result is only slightly less accurate than the Babylonian figure.
The square root of 2 was probably the first number recognised as irrational, meaning it cannot be written as a ratio of two whole numbers. The Pythagoreans found that a square's diagonal and its side have no common measure, and for a time they reportedly kept the finding secret. Tradition often links the discovery to Hippasus of Metapontum, and a legend says he died for revealing it, though historians find little evidence for that. The classic proof assumes the root is a fraction in lowest terms and shows that numerator and denominator would both have to be even, a contradiction.
Builders put it to use. Vitruvius, attributing the idea to Plato, described a Roman technique for doubling a square by using the original diagonal as the new side, applied to pavements and to the proportions of atria. For calculation, the standard method in many calculators is the Babylonian method: guess, then repeatedly average the guess with 2 divided by it. Each round roughly doubles the number of correct digits. Handy fractions include 99/70, which misses by less than one ten-thousandth.
Computers have taken the digits to absurd lengths. Yasumasa Kanada's team reached more than 137 billion decimal places, a home computer beat that in February 2006, and Shigeru Kondo calculated one trillion places in 2010. Such runs help test whether the digits behave randomly.
Source: Square root of 2
