Lesson Eight: In Which We Multiply With Negatives

So far our exploration into multiplication has been restricted to the non-negative numbers, that is, the whole numbers along with zero. However, it certainly seems plausible that we would want to multiply using negative numbers.

Multiplying a Non-Negative by a Negative:

For example, suppose HBGary Federal is losing fifty-thousand dollars every month, how much has it lost in a year? Since losing money can be represented as having negative money, the amount that is lost turns out to be 12*(-50,000). From how we defined multiplication using addition, we know that this is what we get if we add -50,000 to itself 12 times. So far, so good.

It turns out that we can keep our special rules for 0 and 1 even when we consider negative numbers. Since 1*n can be thought of as just one "thing" containing n, for example one moth containing -50,000 dollars, 1*n=n even if n happens to be negative. Since 0*n is still no things containing n, 0*n=0 is also true when n is negative.

Another interesting rule we can obtain is that -1*n=-n. For example, -1*7 is the same thing as 7*-1, which we know is -1 added to itself seven times, or -7. We may conclude that n+(-1)*n=0 for any integer n.

A Negative Times a Negative:

Using the previously introduced rules of multiplication, we can establish that -1*-1=1. Because anything times 0 is 0, -1*0=0. We can rewrite 0 as 1+-1, so -1*(1+(-1))=0. Using distribution we obtain that -1*1+-1*-1=0, or that -1+-1*-1=0, since anything times 1 is itself. If we add one to both sides we get 1+(-1)+(-1)*(-1)=1+0, or 0+(-1)*(-1)=1. Thus we conclude that (-1)*(-1)=1, as desired. This is further confirmation that -(-1)=1, as -1*-1 is supposed to be -(-1).

Now, if we wish to multiply together two arbitrary negative numbers, say (-7)*(-3), we may first rewrite as (-1)*7*(-1)*3 and do the positive multiplication separate from the multiplication of negative ones. Thus (-7)*(-3)=(-1)*(-1)*7*3=1*21=21. You might note that two negatives multiply together to get a positive, a fact that we are often told as children. The fundamental reason for this is the fact that (-1)*(-1)=1. We also saw that a negative times a positive was a negative, because adding together negative numbers must necessarily stay negative. It turns out that three negatives multiply together to a negative. In fact, whether you get a negative or positive depends on whether the number of negatives you multiply together is odd or even, unless you multiply by 0, in which case you will get 0 of course.