home school math with Isaac, brother of the chemistry dreaming sister

Ok so 13 year old Isaac asked me WHY neg x neg= pos............ we know it IS, but WHY??? My initial answer was "COZ THAT'S THE RULE!!" Not quite enough of an answer for Isaac so,
here is what we came to agree upon..
I love it!! BECAUSE IT IS THE ONLY THING THAT SEEMS TO MAKE CORRECT SENSE....read on...

"Minus times minus results in a plus,
The reason for this, we needn't discuss."
- Ogden Nash

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Why is a negative times a negative a positive?
People have suggested many ways of picturing what is going on when a negative number is multiplied by a negative number. It's not easy to do, however, and there doesn't seem to be a visualization that works for everyone.




A Mathematical Explanation

If we can agree that a negative number is just a positive number multiplied by -1, then we can always write the product of two negative numbers this way:

(-a)(-b) = (-1)(a)(-1)(b) = (-1)(-1)ab
For example,

-2 * -3 = (-1)(2)(-1)(3)

= (-1)(-1)(2)(3)

= (-1)(-1) * 6
So the real question is,

(-1)(-1) = ?
and the answer is that the following convention has been adopted:

(-1)(-1) = +1
This convention has been adopted for the simple reason that any other convention would cause something to break.

For example, if we adopted the convention that (-1)(-1) = -1, the distributive property of multiplication wouldn't work for negative numbers:

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

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

0 = -2
As Sherlock Holmes observed, "When you have excluded the impossible, whatever remains, however improbable, must be the truth."

Since everything except +1 can be excluded as impossible, it follows that, however improbable it seems, (-1)(-1) = +1.