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Why does -1 × -1 = 1 ?
By Martin Foster on Tuesday, December 18, 2001 - 07:05 pm:

I am well aware that a negative multiplied by a negative results in a positive. How can this be simply explained? If multiplication is a rapid form of addition and division a rapid form of subtraction it is simple to explain negative x positive. I have difficulty explaining negative x negative. Please help.
Martin

By Brad Rodgers on Tuesday, December 18, 2001 - 07:59 pm:

First off, here's an intuitional argument (which really makes no coherent sense): "You want the opposite of an opposite, so you get a positive!"

Don't worry if this feels like a cheat; it does to me too. There's a more slightly more advanced way to go about this, but it really gives a better understanding of the nature of (-1)×(-1).

First off, are you familiar with basic algebra? This is used in this proof, so if not, just write again.

Alright, we know that

1 + (-1) = 0.

We now multiply both sides of this by (-1):

(-1)×1 + (-1)×(-1) = 0×(-1)

Simplifying:

(-1) + (-1)×(-1) = 0

Now, we add 1 to both sides of this:

(-1) + (-1)×(-1) + 1 = 0 + 1

As the (-1) and 1 cancel each other out, we get

(-1)×(-1) = 1

Tadaa!

It's good that you're asking this. I think a lot of people just accept this as a fact (and I'd bet that a lot of teachers couldn't explain why it's true if asked), but in mathematics, you should always have an answer to "why" before just acceptng a statement and going on.

Hope this helps,

Brad

By Martin Foster on Tuesday, December 18, 2001 - 09:17 pm:

Wow, Thanks Brad.
Martin