Mind Performance Hacks

Hack #42. Estimate Square Roots

Estimate square roots and even higher-order roots by using simple processes.

It's often useful to compute the square root of a number, especially when you want to visualize areas or compute diagonal distances. There are a few methods for computing square roots on paper, some of which are not widely known. If all you have is your brain, however, it's still possible to come up with a quick estimate that's reasonably accurate.

In Action

To estimate the square root of a number, start by pairing up the digits of the number, beginning with the decimal point and moving away. So, for instance, to compute the square root of 500,000, we would pair up the digits like this:

50 00 00

Each pair of digits will, in fact, represent a digit in the square root. The leftmost pair of digits (or single digit, if there are an odd number of digits in your original number) is used to compute the leftmost (most significant) digit of the result. You find the result digit by determining the highest square that will fit into the pair without going over.

The biggest perfect square that fits into 50 is 49, which has a square root of 7. So, we know our square root will have the form 7dd—that is, a 7 followed by two digits, or "seven hundred and something." Further, since 50 is very close to 49, we can surmise that the square root will be in the low 700s. Had it been close to the next square (64, with a square root of 8), we would know that the square root would be closer to 800. Unfortunately, ...