## Square roots: in your head

I came across the following YouTube video a while back which uses a strange trick to accurately approximate square roots. I suggest watching at least the first minute where the presenter explains how it’s done:

Let’s do an example. Approximate to 2 decimal places:

First, YouTube tells us to find the nearest perfect square that’s less than 40, that’s 36, and take the root, giving us 6. So our answer is, obviously, 6 point something. That something is a fraction, where the numerator is the difference between 40 and 36, and the denominator is 2 times 6. So we have:

So what’s the actual answer to 2 decimals? It’s 6.32. That’s what I like to call: pretty darn close. Naturally, there are a few catches to this technique: you need to know your perfect squares, you need to know your fractions, and things tend to get hard with larger numbers.

But still, I thought this was *surprisingly* effective for such a simple piece of arcane trickery.…