Editing Square root of 2 (section)

===Rational approximations===
A simple rational approximation {{sfrac|99|70}} (≈ '''1.4142'''857) is sometimes used. Despite having a denominator of only 70, it differs from the correct value by less than {{sfrac|1|10,000}} (approx. {{val|+0.72e-4}}). 

The next two better rational approximations are {{sfrac|140|99}} (≈ '''1.414'''1414...) with a marginally smaller error (approx. {{val|-0.72e-4}}), and {{sfrac|239|169}} (≈ '''1.4142'''012) with an error of approx {{val|-0.12e-4}}.

The rational approximation of the square root of two derived from four iterations of the Babylonian method after starting with {{math|''a''<sub>0</sub> {{=}} 1}} ({{sfrac|665,857|470,832}}) is too large by about {{val|1.6e-12}}; its square is ≈&thinsp;{{val|2.0000000000045}}.