<Dd> 1 + 1 3 + 1 3 × 4 − 1 3 × 4 × 34 = 577 408 = 1.41421 56862745098039 _̄ . (\ displaystyle 1 + (\ frac (1) (3)) + (\ frac (1) (3 \ times 4)) - (\ frac (1) (3 \ times 4 \ times 34)) = (\ frac (577) (408)) = 1.41421 (\ overline (56862745098039)).) </Dd> <P> This approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of Pell numbers, which can be derived from the continued fraction expansion of √ 2 . Despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation . </P> <P> Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, that the square root of two is irrational . Little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned . For a while, the Pythagoreans treated as an official secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it . The square root of two is occasionally called "Pythagoras' number" or "Pythagoras' constant", for example by Conway & Guy (1996). </P> <P> There are a number of algorithms for approximating √ 2, which in expressions as a ratio of integers or as a decimal can only be approximated . The most common algorithm for this, one used as a basis in many computers and calculators, is the Babylonian method of computing square roots, which is one of many methods of computing square roots . It goes as follows: </P>

Who proved that square root of 2 is irrational