I wont lie, I went straight for the Taylor approximation. According to wikipedia: "As an iterative method, the order of convergence is equal to the number of t

I wont lie, I went straight for the Taylor approximation. According to wikipedia: "As an iterative method, the order of convergence is equal to the number of terms used. With 2 terms, it is identical to the Babylonian method; With 3 terms, each iteration takes almost as many operations as the Bakhshali approximation, but converges more slowly. Therefore, this is not a particularly efficient way of calculation." Heh, woops. My second inkling was the way slide rules do it: exponentials and logarithms ( sqrt(n)=exp(.5*ln(n)) ) Turns out that's more on target: "Pocket calculators typically implement good routines to compute the exponential function and the natural logarithm, and then compute the square root of S"