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### Babylonian method

The babylonian method allows you to approximate the value of a non-negative root *x* = √S by iterations of

##### Explanation

We assume that *x _{n}* is the over-estimated value of the square root of a non-negative real number

*S*, and that is the under-estimated value. Then you can assume that the average of these two values gives a better value, so that

##### Example 1

If we start with *x*_{0} = 1 and *S* = 2, thus for *x* = √2, the subsequent iterations give

and that is even more accurate than the value 1.414213562… that shows your calculator.