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
Methods of computing square roots that formula_211 is as small as possible. A completely different method for computing the square root is based on the CORDIC algorithm, which uses only very simple operations (addition, subtraction, with bitshift and table lookup to implement multiplication). The square root of "S" may be obtained as the output formula_197 using the hyperbolic coordinate system in vectoring mode, with the following initialization: Quadratic irrationals (numbers of the form formula_216, where "a", "b" and "c" are integers), and in particular, square roots of integers, have periodic continued fractions. Sometimes what is desired is finding not the numerical value of a square