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
Integer square root theoretical advantages. One can prove that formula_30 is the largest possible number for which the stopping criterion ensures formula_32 in the algorithm above. In implementations which use number formats that cannot represent all rational numbers exactly (for example, floating point), a stopping constant less than one should be used to protect against roundoff errors. The traditional pen-and-paper algorithm for computing the square root formula_5 is based on working from higher digit places to lower, and as each new digit pick the largest that will still yield a square formula_34. If stopping after the one's place, the result computed will be