I'm a negative creep

We covered operations involving negative and positive numbers in Math tonight. There was one good technique for explaining why "reversing the operation" (i.e., adding when subtracting a negative number, subtracting when adding a negative number) works.

First, realize that a positive number and its negative equivalent together add up to zero.

And of course, you can add endless zeroes to any expression or equation and not change its value.

Now, suppose you have 5 - (-2). If you recall your elementary math, you know the answer is seven, but why do you add two?

To understand this process, imagine five counters, like so:
X X X X X

Next add on zero twice, using the form X and -X.

Having done this, you now have
X X X X X X -X X -X
All you have to do then is remove the two negative counters. What remains? Seven counters! So 5 - (-2) is 7!

A way to wrap your brain around it using non-mathematical terms is this: 5 - (-2) means you are taking away the removal of two. You are removing the removal; in other words, you are putting the two back. Five and two more is seven.

What about multiplication and division using negative numbers? Well, I didn't get a nice understanding of the bizarre concept that multiplying two negative numbers gets a product that is positive, as I did with the adding. But here's a simple way to see it in action, starting with positive and negative multiplication:

5 x 3 = 15
5 x 2 = 10
5 x 1 = 5
5 x 0 = 0

See how each time you lower the value of the factor by one, you get a product that is five less? So it stands to reason that if you lower the factor by one more (to negative one), you will get five less than zero. And so on.

The same concept applies with the multiplication of two negatives. Start with one negative factor. Each time you lower the value of the factor, you will get a greater product.

3 x -4 = -12
2 x -4 = -8
1 x -4 = -4
0 x -4 = 0

So -1 x -4 must be 4, and so on.

Math is cool, even if I don't really get it.