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:
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:
We would only get a sensible answer if we also assumed that which doesn’t seem reasonable at all.
See the MathForum for examples.