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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) Newton's method. 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 root, the better).
2. Let xn+1 be the average of xn and S / xn (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.

Graph charting the use of the Babylonian method for approximating the square root of 100 (10) using starting values x0 = 50, x0 = 1, and x0 = −5. Note that using a negative starting value yields the negative root.

Example

To calculate, where S = 125348, to 6 significant figures, we will use the rough estimation method above to get x0. The number of digits in S is D=6=2·2+2. So, n=2 and the rough estimate is

Therefore,