Do two negatives really make a positive?

Question

Why does a negative times a negative equal a positive? Does this ever happen in the real world?

Answer 1

One way to picture this is to imagine a number line. Then rotate it $180^{\circ}$. Each number will now be superimposed over its negative: $-1$ will be where $+1$ was; $+2$ will be where $-2$ was. Rotation of the number line by $180^{\circ}$ is the equivalent of multiplying by $-1$.

Now do the rotation twice. The number line is unchanged. So, multiplying by $-1$ twice is the same as multiplying by $+1$.

This approach has applications with Complex numbers. In these scenarios, the number line is rotated $90^{\circ}$ counter clockwise to multiply by $i$.

But that's another story.