Wednesday, October 14, 2009

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..

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


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.


Shana said...

I really tried to read all of this but Math and my brain do not Good for him for asking though. Shows he is really thinking for himself not just accepting and going along.

Deborah Ann said...

I was never any good at math...

whenpigsfly said...

Ogden Nash led in and Sherlock Holmes took it out.................the 0=-2 false equaiton in the middle is just a visualization . I deleted so man y"illustrations' as Isaac and I were trying to asnwer that not-so-simple question. It got more and mroe complicted the more someone triedto explain it. When I foudn THIS, it appealed to the visual side of my learning ability and I GOT IT!!