A Simple Proof

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!

3 thoughts on “A Simple Proof

  1. sri

    Dear aditya, you say “the fact that 0! = 1 and not 0” is taken on faith. That’s not true.

    The proof is trivial and works off the way factorials are defined.
    By definition, factorials only work on whole numbers (and not on negative numbers).

    We can informally prove 1! is 1, “how many ways can you arrange 1 item?” — only one way clearly.

    So we have 1! = 1


    1! = 1 * 0!

    so 1 * 0! = 1



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