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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.