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 2 method of computing square roots, which is one of many methods of computing square roots. It goes as follows: First, pick a guess, ; the value of the guess affects only how many iterations are required to reach an approximation of a certain accuracy. Then, using that guess, iterate through the following recursive computation: The more iterations through the algorithm (that is, the more computations performed and the greater ""), the better approximation of the square root of 2 is achieved. Each iteration approximately doubles the number of correct digits. Starting with the next approximations are The value of was