information:extraction, structural:reformatting
Rational approximations

A simple rational approximation 99 / 70 (≈ 1.4142 857) is sometimes used. Despite having a denominator of only 70, it differs from the correct value by less than 1 / 10,000 (approx. +0.72 × 10).

The next two better rational approximations are 140 / 99 (≈ 1.414 1414...) with a marginally smaller error (approx. −0.72 × 10), and 239 / 169 (≈ 1.4142 012) with an error of approx −0.12 × 10.

The rational approximation of the square root of two derived from four iterations of the Babylonian method after starting with a = 1 (665,857 / 470,832) is too large by about 1.6 × 10; its square is ≈ 2.000 000 000 0045.