This is a fairly specific trick, but it comes up often enough that you can dazzle people once in a while by multiplying numbers that are close together. You need to know how to square numbers (check back to this series of posts).

Here’s the trick: say you need to work out 57×63. Barely batting an eyelid, I’ll tell you that’s 3591… but that was an easy one.

The reason it works is called the ‘difference of two squares’ – to multiply the numbers, what I did was find the number midway between them (60) and square it (3600); then find the difference from the midway number to either of the originals (3) and square it (9); and simply take them away (3591).

That always works. Obviously, though, some numbers are easier to square than others: to do 22×23 (which is 506), I’d probably work out 222=484 and add 22 (506) rather than try squaring 22.5 - but it’s a handy trick in general.

Why does multiplying like this work?

Well, my little chickadee, I’m afraid that needs a little bit of algebra. What I worked out at the top there was actually (603)×(60+3). If you remember how to expand brackets1, you get 60×60+60×33×603×3=3600+1801809. The 180s in the middle cancel out and you’re left with 36009.

It works just the same way with anything you pick: let the middle number be a and the difference be b. Then you’re multiplying (ab)(a+b), which turns into a2+ababb2 and the abs disappear to leave you with a2b2 – the middle number squared minus the difference squared.

Footnotes:

1. Some people use a table, some FOIL, some a smiley face – there are several methods that do the same thing