-1 times -1 = +1 ?

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:

    \[(-1) \times (-1 + 1)\]

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.

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