Why a double negative makes a positive. (Mathematics)

[Tiberius]

So after a discussion with TruthWorthy in another thread (where he attempted to show that a double positive made a negative), I looked into why a double negative makes a positive.

Essentially, we must first agree that a negative number is simply a positive number multiplied by -1. In other words, to get the negative of 2, we multiply by -1. -1 * 2 = -2.

If we accept this as correct, then we can write the multiplication of two negative numbers like so:

-2 * -3 = (-1)(2)(-1)(3)
-2 * -3 = (-1)(-1)(2)(3)
-2 * -3 = (-1)(-1) * 6

So, the question is, (-1)(-1) = ?

The convention (-1)(-1) = +1 has been adopted because anything else causes the distributive property of multiplication to break for negative numbers.

For example, let's assume (-1)(-1) = -1.

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

(1 + -1) on the left equals 0, so the equation can be written:

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

Since we agreed that any positive number multiplied by negative 1 is the negative form, and that (-1)(-1) = -1, we can further simplify this:

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

So:

0 = -2???

If we place any other number (apart from +1) as the answer to the calculation (-1)(-1) = ?, the same thing happens. This is why two negatives make a positive in mathematics.

Reproduced for your enjoyment from http://mathforum.org/dr.math/faq/faq.negxneg.html