We already know how to add and subtract negative numbers: to add a negative like −4, we subtract its positive counterpart, 4. And to subtract −4, we add 4. What about multiplication?
Multiplication is repeated addition: instead of writing “3 + 3 + 3 + 3”, we can write “4 × 3”. This isn’t just a useful shorthand: it’s another form of abstraction, where we defined a new operation in terms of a simpler one. (I should also note that our starting point when we add 3s is 0 – this will be important later – so our addition should be “0 + 3 + 3 + 3 + 3”.)
If we don’t know how multiplication works for negative numbers, we can try to work our way back down from multiplication to addition, going from an abstract operation to a simpler one which we already understand.
Let’s start with: 4 × −3. This tells us to do “add −3” four times. We know that “add −3” is the same as “subtract 3”, so our answer will be three subtracted from 0 four times:
0 − 3 − 3 − 3 − 3
= 0 − 12
That’s simple, but what if we want to do −4 × 3? It’s simple easy to imagine reducing something by three, but what does it mean to perform an action −4 times?
There are two ways to answer this question. The first and more annoying way would be for me to just tell you. Instead, here’s an old enemy from maths class: the number line.
One way of looking at negative numbers is that they allow us to specify direction as well as size: 4 is the same size as −4 but headed in the opposite direction. (The official term is that 4 and −4 have the same absolute value.)
So 4 × −3 means: move to the left by 3, and repeat this 4 times, giving −12:
Except that I’ve left something out of the descriptions. I just claimed that our current set of numbers – formally called the integers – have both size and direction, but I haven’t said in what direction we are meant to ‘repeat this 4 times’. I’ve just repeated it in the same direction as the 3s, to the left.
This gives us a clue about perform an addition −4 times. If multiplication by a positive number repeats the addition in the same direction as the number we’re multiplying, multiplication by a negative number repeats that addition in the opposite direction. So doing the operation “add 3” −4 times means that we subtract 3 four times.
giving for −4 × 3 the same result we got for 4 × −3: −12
We can now get an answer for the remaining combination: what is −4 × −3?
We repeat “subtract 3” −4 times, which means that we flip it over to “add 3” and repeat 4 times, which gives us −3 × −4 = 3 × 4 = 12: the same answer for a double negative which we’d expect in grammar.
In the next post, I’ll deal with the question which started this series of posts: division. And as an apology for the all the dry drawings, here is the Number Feline: