Product of Additive Inverses

This is a formal justification, from the ring axioms, of the formula (−a)(−b) = ab. As the article mentions, this is often phrased as "the product of two negatives is positive," but, of course, the presence of a minus sign in front of a variable does not indicate a negative number (for example, if a = −3, then −a is positive); and the formula makes sense even in a ring with no notion of positive and negative numbers.

A simple example of how this is true _even if you don't have negative numbers_:

Let's use mod 5 arithmetic. You have 5 elements in the ring -- 0,1,2,3,4

The additive inverses are as follows:

  1 + 4 = 0
  2 + 3 = 0

Which is to say that 1 is the additive inverse of 4 and 2 is the additive inverse of 3, and vice versa. 0 is the identity, of course.

So what happens if you multiply 2 * -3 (2 times the additive inverse of 3).

The additive inverse of 3 is just 2, so the answer is 2 * -3 = 2 * 2 = 4.

The other way to calculate it is to find the additive inverse of the product:

2 * -3 = -(2 * 3) = -(1) which is the additive inverse of 1: 4 again.