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 The method uses the same iterative scheme as the Newton–Raphson method yields when applied to the function y = "f"("x") = "x" − "a", using the fact that its slope at any point is "dy"/"dx" = ""("x") = 2"x", but predates it by many centuries. The algorithm is to repeat a simple calculation that results in a number closer to the actual square root each time it is repeated with its result as the new input. The motivation is that if "x" is an overestimate to the square root of a nonnegative real number "a" then "a"/"x" will be an