Why does -1 * -1 = +1 ?


I have been asked this by perplexed teachers, parents and kids:

Why did them silly mathematicians decide that the multiplication of two negative numbers is positive? What’s the logic?

The first thing is to acknowledge that this is a convention. We could define -1*-1 to be whatever we want. But there is good reason to decide it should be +1.

Let’s try to calculate:

-1 * ( 1 + -1)

Well, -1 is the negative of +1, which mean, by definition:

-1 + 1 = 0

So, -1 * (1+ -1) = -1 * 0

and multiplication by zero must be zero, so we have:

-1 * (1+ -1) = -1 * 0 = 0

But, if we open the parenthesis, we have:

-1 * (1 + -1) = -1*1 + -1*-1

now, multiplication by 1 is doing nothing, so -1*1 = -1, so we get:

-1 + -1*-1 = -1*1 + -1*-1 = -1*(1+ -1) = -1*0 = 0

hence, -1 is the negative of -1*-1, so the latter has to be +1.

What have we used? We used the definition of zero and 1, we used the definition of the negative of a number, and we used the law of distributivity (by which we can open the parenthesis).

We could decide, say, that -1*-1 = -1, but then we’d have to abandon one of the principles mentioned above, and that would make arithmetic rather unpleasant.

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One thought on “Why does -1 * -1 = +1 ?

  1. Anonymous

    Alternatively:
    A very intuitive definition for multiplying by -1 is to invert the sign…
    so if it makes sense that -1*5 = -5, it should make the same sense that -1*(-1)=1
    What have we used:
    * The existence of negative numbers
    * An intuitive case where the definition of multiplying by -1 makes sense
    * What must follow to maintain consistency.

    Of course, I’m not sure “inverting the sign” is the popular formal definition for multiplying by -1
    but I don’t believe anything would become inconsistent if you arbitrarily choose to make it YOUR definition.

    Extra credits for finding the popular formal definition and proving equivalence?

    Reply

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