Lost Blog
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....
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:
$$ (-1) \times (-1 + 1) $$
We can look at this in two ways:
1) Use the distributive law to expand the expression:
$$ (-1 \times -1) + (-1 \times 1) $$
2) Evaluate the term $(-1 + 1)$:
$$ (-1) \times (0) = 0 $$
Therefore the expression $(-1) \times (-1 + 1) $ equals zero, that is:
$$ (-1 \times -1) + (-1 \times 1) = 0 $$
If we agree that $(-1 \times 1) = -1$ then
$$ (-1 \times -1) = 1 $$
Note that if we were to assert that in fact $-1 \times -1 = -1$, then assuming the distributive law is valid we end up with an inconsistent answer:
$$ (-1) \times (1 + -1) = (-1) \times (1) + (-1) \times (-1)$$
Assuming now that $-1 \times -1 = -1$, we then obtain:
$$ (-1) \times 0 = -1 + -1 = 2$$
That is:
$$ 0 = -2 $$
We would only get a sensible answer if we also assumed that $-1 \times 1 = 1$ which doesn't seem reasonable at all.
See the MathForum for examples.
