So what about the question of why two negatives multiplied give a positive?

The short answer is that if this were not so, the world would not exist.

That is, the fundamental reason that multiplication works that way is because otherwise the definition would be broken. By broken, I mean that it would lead you to nonsensical conclusions if you followed it as far as you could. Any definition multiplication in a system with negative numbers has to work pretty much that way.

On Tue, Dec 16, 2008 at 6:42 AM, wrote:

Why does a negative number times a negative number equal a positive number?
How can you show that using a manipulative?

Let's take the concrete (but abstract) example of arithmetic on a 7 hour clock. Here is the addition table:

+ 0 1 2 3 4 5 6
0 0 1 2 3 4 5 6
1 1 2 3 4 5 6 0
2 2 3 4 5 6 0 1
3 3 4 5 6 0 1 2
4 4 5 6 0 1 2 3
5 5 6 0 1 2 3 4
6 6 0 1 2 3 4 5

OK. Now we can build multiplication on top of that. I will use * instead of x to indicate multiplication because I don't have really good fonts.

* 0 1 2 3 4 5 6
0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6
2 0 2 4 6 1 3 5
3 0 3 6 2 5 1 4
4 0 4 1 5 2 6 3
5 0 5 3 1 6 4 2
6 0 6 5 4 3 2 1

OK. Not much to see here except that in the multiplication every row has all of the integers. Moreover, every row has these integers in a different order. This is a consequence of the fact that 7 is a prime number. It doesn't happen that way on the 12 hour clock.

But let's look again at the addition table. Note that 1 + 6 = 0 and that 2 + 5 = 0. Notice also that 0 only appears once in each row. That means that we can consider 6 to be a way of writing -1. Or perhaps -1 is a way to write 6. This lets us solve addition problems such as 3 + x = 5 by adding 4 (which is -3) to both sides.

But what happens when we multiply 3 * (-2)? Do we get -6?

Well, -2 = 5 so this should be the same as 3 * 5 which is 1. But 1 is also -6 !

And all of our properties like the distributive law should still work.

Thus 3 * (-2) + 3 * 4 = 1 + 5 = 6 should be 3 * (-2 + 4) which is 3 * 2 = 6. So that works.

What about (-3) * (-2) = 6?

Well, this should be (and is) the same as 4 * 5 = 6.

That's cool.

In a very limited and geeky dull kind of way.


What is more cools is that this is all just a very concrete example of how the laws or arithmetic imply a certain kind of order. We could test this out for all the different kinds of arithmetic that have the properties that we like about integers. I don't mind starting infinite tasks, but I do mind having to finish them.

SO

Let's do much better by reasoning from those properties directly.

For instance, assume that we have the following:

1. an additive unit, 0 such that x + 0 = 0 + x = x and thus 0 * x = 0
2. an additive inverse, -x = 0 - x
3. left distributive law, (a + b) * x = a * x + b * x
4. right distributive law, x * (a + b) = x * a + x * b
   

Now take x * (0 - y). This has to be equal to (x * 0) - (x * y) = 0 - (x * y) = - (x * y)

Or (0 - x) * (0 - y) = 0 - x * (0 - y) = 0 - (0 - x * y) = x * y

This means that ANY kind of arithmetic where multiplication and addition have an additive inverse and left and right distributive laws will follow the pattern that (-x) * (-y) = x * y

OR

if that system doesn't have that, then it won't have one or all of the properties mentioned.

This works for a spinning globe, for quantum mechanics, for clocks and rubik's cubes.

And THAT is why abstraction is cool. But not why it is useful. That comes next.