This is a brief excerpt of a much longer explanation:
Using the field axioms, we can prove a variety of things that you likely take for granted such asSometimes memorization comes in handy.
–(–a) = a
– (a + b) = – a + (– b)
and a x 0 = 0 x a = 0
Now, we finally get to the crux of our explanation (the same explanation as above). In a field, the distributive property must hold. That is, if a, b and c are real numbers, then
a x (b + c) = a x b + a x c
(b + c) x a = b x a + c x a
The distributive property ties together the different operations of addition and multiplication.
Now, we replace a, b and c with -1, 1 and -1 respectively.
(-1) x (1 + (-1)) = (-1) x (1) + (-1)x(-1)
On the left hand side we see that 1 + (-1) is equal to 0 since any number plus its additive identity is equal to 0. Any number multiplied by 0 is 0 (this can be proven from the axioms for a field). Therefore, replacing the lefthand side with 0, we get
0 = (-1) x (1) + (-1) x (-1)
Since any number times 1, the multiplicative identity, is itself, we can further simplify this equation to get:
0 = –1 + (–1) x (–1)
We now need to figure out what (–1) x (–1) is. Since a number added to its additive inverse is 0, (–1) x (–1) must equal the additive inverse of –1. This is simply 1 so
(–1) x (–1) = 1