There are several algorithms developed to manually calculate the square root of a number. My favorite is the method of averaging, which is probably the least complicated. In this method, the number is divided by the nearest square root estimate, and then the quotient and the square root estimate are averaged. The result is the new estimate.
Sounds complicated? Not really.
Example: Approximate .
Note: Using a calculator, .
We know that is between and . Between the two, it is nearer to . So, the first estimate is . Next, we divide by the first estimate and “average” it with the quotient .
Second estimate: (2 significant digits)
Third estimate: (4 significant digits)
Fourth estimate: (8 significant digits)
Notice that the third estimate is already correct up to 8 significant digits which means that this method is quite accurate. In the fifth, sixth, and seventh estimates, 16, 32 and 64 digits should be kept respectively. Of course, repeating the process over and over will obtain more accurate approximations.
Update: Don Cohen has a short Mathematica code for this method in his site.
Reference: Modern Elementary Mathematics