-1 times -1 = +1 ?

Originally Posted on April 21, 2013 by hsauro

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.