How to Calculate Square Roots by Hand

The long method, to obtain a correct result to any level of accuracy without resorting to estimates

In the last fifty years, the ever more widespread diffusion of increasingly cheaper and more powerful home computers has made it possible to fully delegate to machines the execution of calculations that only humans could perform until recently. Among these calculations, we can certainly include finding the square roots of whole or decimal numbers. Few people today feel the need to learn how to compute a square root by hand, preferring to entrust the tedious task to software or a pocket calculator, digital instruments that are now within reach of practically anyone.

Be that as it may, here is the method for calculating the square root of a positive number to any desired level of accuracy for all those interested in learning how to do it. (There are also simpler methods based on estimates by trial-and-error, but we will leave the discussion for a future article.)

The calculation of a square root step by step

For this example exercise, we want to calculate the square root of 66,615.61:

First, let’s write this number by separating its digits into pairs so that the decimal point doesn’t occur in the middle of a pair. Only the first digit can be left unpaired if the number we are dividing into pairs has an odd number of digits before the decimal point, as in…