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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 xn 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 x0 = 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.

 


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