Handy approximation for square roots

(para espanol: Aqui por favor.)

As part of my ongoing quest to emulate Richard Feynman, I’ve taken an interest in mental math. Particularly, trying to get a “sense” of different calculations. I thought it would be cool if I had a decent method of calculating square roots in my head. They come up often enough, any time you’re trying to find the length of the Hypotenuse of a right triangle, for instance, or figuring out what the dimensions of a square-ish room might be, if you know the square footage. I came up with a decent method, which can be expressed as follows:



(The φ n part is a magic function I made up for the sake of notation. It equals the closest perfect square less than n. So φ 32 = 25, and φ 80 = 64. The part to the right of the “+” gives the whole number part of the answer. The fraction to the left gives the decimal part of the answer.)

That’s mostly for show, and is probably the most complicated possible way to express the method. I have it there because fancy symbols make me feel important. But it does explain exactly what my method is. The idea of it is, that if you memorize the perfect squares, then you can make a pretty good guess as to the square root of a number that falls between them.

X 1 2 3 4 5 6 7 8 9 10 X2 1 4 9 16 25 36 49 64 81 100

If you’re given the number 47, you can tell right away that the square root is between 6 and 7. To find the next decimal place, the key is that the closer the original number is to the next perfect square, the closer the square root is to the next value. There are 13 numbers between 36 and 49 ( 72 – 62), and 47 is 11 parts of the way there. So, we estimate that √47 is 6 and 11/13th’s, which works out to 6.846… which is actually 99.86% of the true answer (6.8556…). – Try again with 27: 52 is 25, 62 is 36. So the answer is between 5 and 6. There are 11 numbers between 25 and 36, and 27 is 2/11ths of the way there, so 5 + 2/11 = 5.1818… = 99.72% of the true answer: 5.19615…

So how accurate is this method? For numbers greater than 10, you’ll always be within 2% (and < 1% in nearly all cases). The higher the number, the more accurate the method gets. Here’s a cool chart. For each number “n”, it tells how far off your approximation of √n will be:

Doing this mentally is a bit cumbersome at first, but gets quicker and quicker with moderate amounts of practice. I’ve started finding little shortcuts for Weird fractions like 7/13ths- approximations upon approximations, but that’s what I was after I suppose.

One further note: I tried doing a string of 10 random square root problems using the calculator on my cell-phone. The buttons were awkward enough, that it took me about the same amount of time as it did to do it in my head. Of course, the cell phone was much more accurate. Still though, it could make a good bar wager, to race someone in square roots, mentally vs. calculator, to one decimal point.