Secrets of the Mathematical Ninja: multiplying numbers that are close together

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 $57 \times 63$. 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 $22 \times 23$ (which is 506), I'd probably work out $22^2 = 484$ 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 $(60 - 3) \times (60 + 3)$. If you remember how to expand brackets[1], you get $60 \times 60 + 60 \times 3 - 3 \times 60 - 3 \times 3 = 3600 + 180 - 180 - 9$. The 180s in the middle cancel out and you're left with $3600 - 9$.

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 $(a - b)(a + b)$, which turns into $a^2 + ab - ab - b^2$ and the $ab$s disappear to leave you with $a^2 - b^2$ -- the middle number squared minus the difference squared.


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

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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