# the *answer*

There are lots of ways to answer this. Here is one:

If you start with a number like say 0, and then you add -3 and then add -3 again, then you have added -3 twice, or in other words 2 times -3. And where are you? You are at -6. So this is why 2 x -3 = -6.

Then, if you want to know about something like -2 x -3, you can think of -2 as -1 x 2, and so you have -1 x 2 x -3. The last part, 2 x -3, we already figured out is -6. So you have -1 x -6, which is +6.

And here is a somewhat fancier way to think about it:

You can think of addition and multiplication as "doing something" to the number line. For instance adding 4 slides the number line to the right by 4 units. If you started at 6, then after you add 4, you are at 10. If you started at -5, you will end up at -1.

(Likewise adding -4 slides to the left instead of the right.)

Multiplication, on the other hand, stretches the line out as if it's anchored at 0. Imagine the number line is elastic but pinned at 0. Multiplying by 2 stretches by a factor of 2: 1 goes to 2, 15 goes to 30, -12 goes to -24, 0 goes to 0, etc. The farther you are away from 0, the more you get moved by multiplication (unlike addition, where every number moves the same amount).

Now, multiplying by -1 is a little different: it flips the line around. (Still anchored at 0.) So 20 goes to -20, -20 goes to 20, etc.

Now, if you take a number and multiply by -3, that is like multiplying by 3 and also by -1, so you stretch by a factor of 3 and then flip the line. (Or flip first and then stretch -- the order doesn't matter.)

If you start at 2, you will end up at -6. If you start at -2, you will end up at +6. So -3 x 2 = -6, and -3 x -2 = 6.

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