### Why are positive and negative not the exact opposite of each other?

This is really an interesting phenomenon in math, and one that I don't think many people understand at a foundational level simply because it's rarely explained in detail. People just take it for granted it's a rule, with no proof as to why it always works.

I'd like to take a moment and start by proving multiplication by two negatives is really positive in mathematical terms. This should shed some light as to why:

We'll start with division, as this is more obvious and will be needed to prove multiplication. So let's divide two negative numbers and see what happens:

``````x = (-a) / (-b)

(-a) = (-1)a     AND (-b) = (-1)b   [Note: we're pulling a -1 from the terms)

Therefore: x = (-1)a/ (-1)b --> ( (-1)/(-1) ) x ( a/b )

any number divided by itself is 1: so (-1/-1) = 1 --> x = (1) x (a/b) --> x = a/b

Therefore x must be positive.
``````

For multiplication we're going to prove (-1 x -1) = 1. If this holds true, the same method for (-a x -b) will also equal a positive and can be proven the same way.

Prove (-1 x -1) = 1:

``````Let y = (-1) x (-1)

--> (-1) y = (-1)(-1 x -1) [Note: Multiplying by -1 is ok so long as it's done to both sides]

--> (-1) y = (-1) x (-1) x (-1)

--> (-1) y = (-1) (1^3 )  [We just factored the (-1), which left (-1)(1 x 1 x 1) = (-1)(1^3 )]

--> y = ( (-1) ( 1^3 ) ) / (-1)  [Divided both sides by -1]

--> y = (-1 / -1) (1^3 ) [We proved before that -1/-1 is 1]

--> y = 1^3

--> y = 1
``````

So we have now proven ( -1 x -1 ) = 1.

I hope the proofs help a little to explain what's actually going on. Basically, it all comes down to the idea that you can separate the numbers and as long as you end up with a (-1 / -1) you're going to get a positive. If we multiplied three negative numbers we'd get the following (abbreviated for brevity):

``````y = (-a) x (-b) x (-c)

--> We've proven  (-a) x (-b) = ab, so going back through it we get:

y = ab x (-c) and there we end up with a positive multiplied by a negative, which gives us a negative.
``````

If we had a fourth negative term being multiplied we'd see:

y = (-a) x (-b) x (-c) x (-d)

--> y = ab x cd --> y = abcd [all based on the above proofs]

Hope that helps clear up what was missing. It's just a couple of tricks that prove two negatives must always equal a positive.