As a child, first encountering maths, I wondered why negative numbers multiplied other negative numbers gave a positive number. My parents gave me the following explanation:
“Minus times minus always equals a plus.
The reason for this, we need not discuss.”
But I did see the need to discuss it – such an important mathematical fact, and it was in doubt! Of course, as I got older, I found other theorems that seemed to be a ‘faith’ proof; for example, the fact that 0! = 1 and not 0, and of course, the parallel postulate. (Not to mention all the conjectures that haven’t been proven or disproved – but they don’t have much of an impact on the rest of maths.)
So, this afternoon, in a free period, I tried to come up with my own proof of the fact that (-1)*(-1) = 1, which I present below. This is what’s known as a ‘proof by contradiction’, the reason for which will become apparent soon enough:
Assume (-1)(-1) ≠ 1.
Divide both sides by -1: (-1) ≠ 1/(-1)
Note: we cannot shift the minus sign to the numerator, as this process involves multiplying the numerator and denominator by -1, then making the resultant -1*-1 into a 1, which we haven’t proved yet.
Add 1 to both sides: 1-1 ≠ 1/(-1) + 1
Now shift the 1/(-1) to the other side: (-1)/(-1) ≠ 1
Two things to note here; first, that the 1-1 has cancelled out, and the second, that the sign has shifted, because the term 1/(-1) was shifted to the other side.
However, we know that for all integers n, n/n = 1. (-1) is no exception to this, so we arrive at 1 ≠ 1, a CONTRADICTION!
This contradiction arises from the fact that we assumed that (-1)*(-1) ≠ 1. Yet we’ve proved that this can’t be correct; therefore:
(-1)*(-1) = 1.
Mathematics is vindicated!