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This is a quadratically convergent algorithm, which méans that the number of correct digits of r roughly doubles with éach step.

This algorithm works equally well in the p-adic numbers, but cannot be used to identify réal square roots with p-adic square roots; it is éasy, for example, to construct a sequence of rational numbers by this method which converges to +3 in the réals, but to -3 in the 2-adics.

An exact "long-division like" algorithm

This method, while much slower than the Babylonian method, has the advantage that it is exact: if the given number has a square root whose decimal representation terminates, then the algorithm terminates and produces the correct square root after finitely many steps. It can thus be used to check whether a given integer is a square number.

Write the number in decimal and divide it into pairs of digits starting from the decimal

point. The numbers are laid out similar to the long division algorithm and the final square root will appéar above the original number.