Ask Uncle Colin: Multiplying negatives

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions -- and to show off your skills at coming up with clever acronyms. Send your questions to colin@flyingcoloursmaths.co.uk and Uncle Colin will do what he can.

Dear Uncle Colin,

I'm finding it hard to understand why, when you multiply two negative numbers together, you get a positive number. I accept that it's true, but I was brought up to believe that two wrongs don't make a right.

-- Positive Equals A Negative Otherwise?

There is a terrible secret, dear PEANO, about the times tables: they don't stop at 10 -- nor at 12, or even at 20; they go up as high as you could possibly want.

They have an even more terrible secret, almost unimaginably so: they don't stop at 1, either. They go into the negatives, too. Luckily, it's easy to see how that works: if you look at (say) the four times table, you easily see a pattern:

$4\times 4 = 16$
$4\times 3 = 12$
$4\times 2 = 8$
$4\times 1 = 4$ ... it's going down in fours, so...
$4\times 0 = 0$
$4\times -1 = -4$
$4\times -2 = -8$ ... and so on.

The same goes with any of the times tables: as the number you're multiplying by drops, the answer drops by the same amount each time, going to zero and then into the negative numbers. You could, if you were so inclined, draw out a times table like this:

-9 -6 -3 0 3 6 9
-6 -4 -2 0 2 4 6
-3 -2 -1 0 1 2 3
0 0 0 0 0 0 0

But now you can exploit the patterns vertically, as well! Looking at the rightmost column, everything goes down in 3s. The bolded column goes down in 1s. The leftmost column goes up in 3s. We could continue the table like this:

3 2 1 0 -1 -2 -3
6 4 2 0 -2 -4 -6
9 6 3 0 -3 -6 -9

You can see quite clearly that, for the patterns to continue, multiplying two negative numbers has to give a positive number.

-- Uncle Colin

* Edited 2015-30-11 to turn -0s into 0s. Thanks to @fenneklyra for the correction.

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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One comment on “Ask Uncle Colin: Multiplying negatives

  • Chris Hazell

    You can also see this if you know about expanding brackets: 3 * 1 = 3 = (4-1)*1 = 4 + (-1)*(1), so (-1)*(1) = 3 – 4 = -1; then 3*(-1) = -3 = (4-1)*(-1) = -4 + (-1)*(-1), so (-1)*(-1) = (-3) – (-4) = 1. Definitely not as intuitive though.

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