## Sunday, April 21, 2013

### -1 times -1 = +1 ?

Originally Posted on  by

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.