Methods of Computing Square Roots - Babylonian Method

Babylonian Method

Perhaps the first algorithm used for approximating is known as the "Babylonian method", named after the Babylonians, or "Heron's method", named after the first-century Greek mathematician Hero of Alexandria who gave the first explicit description of the method. It can be derived from (but predates by 16 centuries) Newton's method (see below). The basic idea is that if x is an overestimate to the square root of a non-negative real number S then will be an underestimate and so the average of these two numbers may reasonably be expected to provide a better approximation (though the formal proof of that assertion depends on the inequality of arithmetic and geometric means that shows this average is always an overestimate of the square root, as noted in the article on square roots, thus assuring convergence). This is a quadratically convergent algorithm, which means that the number of correct digits of the approximation roughly doubles with each iteration. It proceeds as follows:

1. Begin with an arbitrary positive starting value x₀ (the closer to the actual square root of S, the better).
2. Let x_n+1 be the average of x_n and S / x_n (using the arithmetic mean to approximate the geometric mean).
3. Repeat step 2 until the desired accuracy is achieved.

It can also be represented as:

This algorithm works equally well in the p-adic numbers, but cannot be used to identify real square roots with p-adic square roots; it is easy, for example, to construct a sequence of rational numbers by this method that converges to +3 in the reals, but to −3 in the 2-adics.