Two minus signs make a plus

Question

Why do two minus signs make a plus sign and is there a corresponding rule for division and multiplication signs?

Answer 1

Multiplication is a scaling action, on the number line. When you multiply by a negative number, you also reflect the scaling. Reflecting again - multiplying by another negative number - bring ths scaling back to the positive direction.

Note that you can always start the process with $1$ - so for example $-3 \times -4 = 1 \times -3 \times -4$.

There is a kind of corresponding process for division. If you divide $1$ by $4$, you get $\frac{1}{4}$, then if you divide $1$ by $\frac{1}{4}$, you get $4$ again. Not exactly the same but a similar concept of double-action reverting to the initial state.

Answer 2

One solution is to use complex numbers, using $z=e^{i\theta}$. $\theta$ in this representation represents the angle through which we rotate a unit vector anti-clockwise about the origin.

As the total internal rotation angle of a circle is $2\pi$, we arrive at $e^{i\pi}=-1$.

Multiplying two complex numbers $u\times v$, where $u=e^{i\alpha}$ and $v=e^{i\beta}$, gives $e^{i\alpha}e^{i\beta}=e^{i(\alpha+\beta)}$.

Therefore $-1\times-1=e^{i\pi}\times e^{i\pi}=e^{2i\pi}=e^0=1$.