Square root of 3 billion digits. The fraction (…) for the square root of three can be used as an approximation. Despite having a denominator of only 56, it differs from the correct value by less than (approximately ). The rounded value of 1.732 is correct to within 0.01% of the actual value. Archimedes reported , accurate to (six decimal places) and (four decimal places), respectively. It can be expressed as the continued fraction , expanded on the right. It can also be expressed by generalized continued fractions such as which is evaluated at every second term. The following nested square expressions converge to
Square root of 5 of five can be used. Despite having a denominator of only 72, it differs from the correct value by less than (approx. ). As of December 2013, its numerical value in decimal has been computed to at least ten billion digits. 1. This irrationality proof for the square root of 5 uses Fermat's method of infinite descent: 2. This irrationality proof is also a proof by contradiction: It can be expressed as the continued fraction The convergents and semiconvergents of this continued fraction are as follows (the black terms are the semiconvergents): Convergents of the continued fraction are colored red;