My brother asked me the other day why -1 times -1 was +1. It’s the sort of rule we learn at high school and perhaps never think about again. I thought I’d add a note to prove and show why -1 x -1 = 1.

The easiest way to think about this is that a negative will negate the other term. This, -1 times +1 will negate the +1 giving -1. We can use the same argument to suggest that -1 times -1 means that the first -1 negates the second -1 leading to +1.

More formally we call also use the following proof:

Consider:

We can look at this in two ways:

1) Use the distributive law to expand the expression:

2) Evaluate the term :

Therefore the expression equals zero, that is:

If we agree that then

Note that if we were to assert that in fact , then assuming the distributive law is valid we end up with an inconsistent answer:

Assuming now that , we then obtain:

That is:

We would only get a sensible answer if we also assumed that which doesn’t seem reasonable at all.

See the MathForum for examples.