Deep understanding

When learning operations with integers you could just memorize (rote learning?) that a negative number times a negative number is positive or you could go for deep understanding as presented by Dr.Frank Wang of Saxon fame.
This is a brief excerpt of a much longer explanation:

Using the field axioms, we can prove a variety of things that you likely take for granted such as

–(–a) = a

– (a + b) = – a + (– b)

and a x 0 = 0 x a = 0

Now, we finally get to the crux of our explanation (the same explanation as above). In a field, the distributive property must hold. That is, if a, b and c are real numbers, then

a x (b + c) = a x b + a x c

and

(b + c) x a = b x a + c x a

The distributive property ties together the different operations of addition and multiplication.

Now, we replace a, b and c with -1, 1 and -1 respectively.

That is,

(-1) x (1 + (-1)) = (-1) x (1) + (-1)x(-1)

On the left hand side we see that 1 + (-1) is equal to 0 since any number plus its additive identity is equal to 0. Any number multiplied by 0 is 0 (this can be proven from the axioms for a field). Therefore, replacing the lefthand side with 0, we get

0 = (-1) x (1) + (-1) x (-1)

Since any number times 1, the multiplicative identity, is itself, we can further simplify this equation to get:

0 = –1 + (–1) x (–1)

We now need to figure out what (–1) x (–1) is. Since a number added to its additive inverse is 0, (–1) x (–1) must equal the additive inverse of –1. This is simply 1 so

(–1) x (–1) = 1

Sometimes memorization comes in handy.