TITLE: Formal proof for $(-1) \times (-1) = 1$
QUESTION [90 upvotes]: 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?

REPLY [3 votes]: Using the axioms of the field $\mathbb{R}$:
$$\begin{align}
(-1)\times (-1)  &= (-(1))\times (-1)\\ &= -(1\times (-1)) \\ &= -(-1)\\ &= 1.
\end{align}
$$
Here we of course use that $-1 = -(1)$ which is the "definition" of $-1$. We also use that $(-a)\times b = -(a\times b)$. This we see from 
$$
a\times b + ((-a)\times b) = (a + (-a))\times b = 0\times b = 0.
$$