## Mathematical and Computing Sciences Question #5271

Jackson, a 17 year old male from Imphal asks on April 26, 2011,

In my algebra, I learn that (-3 X -2) = +6 but (-3 X +2) = -6, why?

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### Aaron Abrams answered on April 27, 2011

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.

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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