Formal proof for $(-1) \times (-1) = 1$

Is there a formal proof for $(-1) \times (-1) = 1$? It's a fundamental formula not only in arithmetic but also in the whole of math. Is there a proof for it or is it just assumed?

Solution 1:

We use only the usual field axioms for the real numbers. First we prove an intermediate result.

$0\times 0$

$=0\times(0+0)$

$=0\times 0+0\times 0$

Subtract $0\times 0$ from each side to get $0=0\times 0$. Now we are ready for the final kill.

$0$

$=0\times 0$

$=(1-1)\times(1-1)$

$=1\times 1+1\times (-1)+(-1)\times 1+(-1)\times (-1)$

$=1+(-1)+(-1)+(-1)\times (-1)$

$=(-1)+(-1)\times (-1)$

Add $1$ to each side to get $1=(-1)\times (-1)$.

Solution 2:

The Law of Signs $\rm\: (-x)(-y) = xy\:$ isn't normally assumed as an axiom. Rather, it is derived as a consequence of more fundamental Ring axioms $ $ [esp. the distributive law $\rm\,x(y+z) = xy + xz\,$], laws which abstract the common algebraic structure shared by familiar number systems. Below are a few ways to prove the law of signs (notice that the over/underlined terms $= 0)$

$\begin{eqnarray}\rm{\bf Law\ of\ Signs}\ proof\!:\ &&\rm (-x)(-y) = (-x)(-y) + \underline{x(-y + y)} = \overline{(-x+x)(-y)} + xy = xy\\ \\ \rm Equivalently,\ evaluate &&\rm\overline{(-x)(-y)\! +} \overline{ \underline {x(-y)}} \underline{ +xy_{\phantom{.}}}\ \ \text{in two ways, over or underlined first}\\ \\ \rm More\ conceptually:\quad\, &&\rm (-x)(-y)\quad\ and\ \quad xy\ \ \ \text{are both inverses of} \ \ x(-y)\\ && \text{hence they are equal by } {\bf uniqueness\ of\ inverses}\end{eqnarray}$

Indeed, the above are special cases of an analogous proof of uniqueness of additive inverses

$$\rm {x\color{#0A0}+y} = 0 = x\color{#C00}+y' \ \ \Rightarrow\ \ y' = y'\!+(x\color{#0A0}+y) = (y'\!\color{#C00}+x)+y = y$$

Notice that the proofs use only ring laws (most notably the distributive law), so the law of signs holds true in every ring. The distributive law is at the foundation of every ring theorem that is nondegenerate, i.e. involves both addition and multiplication, since it is the only ring law that connects the additive and multiplicative structures that, combined, form the ring structure. Without the distributive law a ring would be far less interesting algebraically, reducing to a set with additive and multiplicative structure, but without any hypothesized relation between the two. Therefore, in a certain sense, the distributive law is the keystone of the ring structure.