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Methods of computing square roots are numerical analysis algorithms for approximating the principal, or non-negative, square root {S}}} , or S 1 / 2 {\displaystyle S^{1/2}} ] of a real number. Arithmetically, it means given S {\displaystyle S} , a procedure for finding a number which when multiplied by itself, yields S {\displaystyle S} ; algebraically, it means a procedure for finding the non-negative root of the equation x 2 − S = 0 {\displaystyle x^{2}-S=0} ; geometrically, it means given two line segments, a procedure for constructing their geometric mean.
Every real number has two square roots. The principal square root of most numbers is an irrational number with an infinite decimal expansion. As a result, the decimal expansion of any such square root can only be computed to some finite-precision approximation. However, even if we are taking the square root of a perfect square integer, so that the result does have an exact finite representation, the procedure used to compute it may only return a series of increasingly accurate approximations.
The continued fraction representation of a real number can be used instead of its decimal or binary expansion and this representation has the property that the square root of any rational number has a periodic, repeating expansion, similar to how rational numbers have repeating expansions in the decimal notation system.
The most common analytical methods are iterative and consist of two steps: finding a suitable starting value, followed by iterative refinement until some termination criterion is met. The starting value can be any number, but fewer iterations will be required the closer it is to the final result. The most familiar such method, most suited for programmatic calculation, is Newton's method, which is based on a property of the derivative in the calculus. A few methods like paper-and-pencil synthetic division and series expansion, do not require a starting value. In some applications, an integer square root is required, which is the square root rounded or truncated to the nearest integer.
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