**
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:

- Begin with an arbitrary positive
starting value
*x*_{0}(the closer to the root, the better). - Let
*x*_{n}_{+1}be the average of*x*and_{n}*S*/*x*(using the arithmetic mean to approximate the geometric mean)._{n} - 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 *x*_{0} = 50, *x*_{0} = 1,
and *x*_{0} = −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 *x*_{0}. The number
of digits in *S* is *D*=6=2·2+2. So, *n*=2 and the rough
estimate is

Therefore,