Why is negative times negative = positive? Someone recently asked me why a negative $\times$ a negative is positive, and why a negative $\times$ a positive is negative, etc.
I went ahead and gave them a proof by contradiction like so:
Assume $(-x) \cdot (-y) = -xy$
Then divide both sides by $(-x)$ and you get $(-y) = y$
Since we have a contradiction, then our first assumption must be incorrect.
I'm guessing I did something wrong here. Since the conclusion of $(-x) \cdot (-y) = (xy)$ is hard to derive from what I wrote.
Is there a better way to explain this?  Is my proof incorrect?  Also, what would be an intuitive way to explain the negation concept, if there is one?

Why a negative times a negative can be reduced to the question of why
-1 x -1 = 1.
The reason for that is because it is forced upon you by the other rules of arithmetic.
1 + (-1) = 0 because of the definition of -1 as the additive inverse of 1
Now multiple both sides by -1 to get
-1(1+(-1)) = 0 because 0 times anything is 0
Use distributive law to get:
-1* 1 + (-1)x(-1) = 0
Now -1 * 1 = -1 because 1 is multiplicative identity.
So we have
-1 + (-1)x (-1) = 0
Put -1 on the other side by adding 1 to both sides to get
(-1) x (-1) = 1
So -1 x -1 = 1.
Now for any other negative numbers x, y we have
x = (-1) |x| and y= (-1) |y|
So x * y = (-1) |x| * (-1) |y| = (-1) *(-1) * |x| * |y| = |x * y| is positive.
Now that you know the reason it really doesn't make much difference in understanding. This question is not really that important. 
It's like asking why is 1 raised to the 0 power equal to 1?
Because that's forced upon you by other rules of exponents,etc.
A lot of time is wasted on this. This is not the kind of problem kids should be thinking about. 
