how can I prove negative times negative is positive.

Question

Well, I know the fact that negative times negative is positive. It would be interesting to question this fact and try to prove it.

Answer 1

Law of Signs proof: $\rm\,\ (-x)(-y) = (-x)(-y) + x(\overbrace{-y + y}^{\large =\,0}) = (\overbrace{-x+x}^{\large =\,0})(-y) + xy = xy$

Equivalently, evaluate $\rm\:\overline{(-x)(-y) +} \overline{ \underline {x(-y)}} \underline{ +xy_{\phantom{._.}\!\!\!}}\ $ in two ways (each over/under term $ = 0).$

Said more conceptually, $\rm\:(-x)(-y)\ $ and $\rm\:xy\:$ are both inverses of $\rm\ x(-y)\ $ so they are equal by uniqueness of inverses: $ $ if $\,a\,$ has two additive inverses $\,\color{#c00}{-a}\,$ and $\,\color{#0a0}{-a},\,$ then

$$\phantom{1^{1^{1^{1^{1^1}}}}}\color{#c00}{{-}a}\, =\, \color{#c00}{-a}+\smash[t]{\overbrace{(a+\color{#0a0}{-a})}^{\large =\,0}\, =\, \overbrace{(\color{#c00}{-a}+a)}^{\large =\,0}}+\color{#0a0}{-a}\, =\, \color{#0a0}{-a}\qquad\qquad $$

This proof of the Law of Signs uses well-known laws of positive integers (esp. the distributive law), so if we require that these laws persist in the larger system of positive and negative integers, then the Law of Signs is a logical consequence of these basic laws of positive integers.

These fundamental laws of "numbers" are axiomatized by the algebraic structure known as a ring (and various specializations thereof). Since the above proof uses only ring laws (most notably the distributive law), the Law of Signs holds true in every ring, e.g. rings of polynomials, power series, matrices, differential operators, difference operators (recurrences), etc.

In fact every nontrivial ring theorem (i.e. one that does not degenerate to a theorem about the underlying additive group or multiplicative monoid), must employ the distributive law, since that is the only law that connects the additive and multiplicative structures that combine to form the ring structure. Without the distributive law a ring degenerates to a set with two completely unrelated additive and multiplicative structures. Thus, in a certain sense, the distributive law is a keystone of the ring structure.

Remark $\ $ More generally the Law of Signs holds for any odd functions under composition, e.g. polynomials with all terms having odd power. Indeed we have

$\qquad\rm\ \ f(g)\ =\ (-f)\ (-g)\ = -(f(-g))\ \iff\ f(-g)\ = -(f(g)) $

$\qquad\rm \phantom{\ \ (-f)\ (-g)\ = -(f(-g))\ =\ f(g)\ }\!\!\overset{\ \large g(x)=x}\iff \ f(-x)\ = -f(x),\ $ ie. $\rm\:f\:$ is odd

Generally such functions enjoy only a weaker near-ring structure. In the above case of rings, distributivity implies that multiplication is linear hence odd (viewing the ring in Cayley-style as the ring of endormorphisms of its abelian additive group, i.e. representing each ring element $\rm\ r\ $ by the linear map $\rm\ x \to r\ x,\ $ i.e. as a $1$-dim matrix).

Answer 2

If $a, b < 0$, then $ab = (-1) \lvert a \rvert (-1) \lvert b \rvert = (-1)^2 \lvert ab \rvert = \lvert ab \rvert > 0$.