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.

Lesson Four: In Which Less Than Nothing Happens

Continuing our example from Lesson Three, suppose the school day ends and students begin leaving. We are now faced with the challenge of making the total number of students at the school smaller. Suppose just one student leaves, when she/he arrived we added 1 to the total number of students at the school. Now we need a number that we can add to undo adding 1 earlier. To address this problem, we introduce the negative numbers.

Negative Numbers:

Negative one, which we write -1, is the number that we add to something when we want to undo adding 1 to it. This means that 1+(-1)=0 since the -1 cancels out the 1 so nothing should happen, and the way we write nothing numerically is 0. Since 2=1+1, if we want to undo 2 we need to use (-1)+(-1) which we call -2, or negative two. Just like we could use 1 repeatedly added to itself to form the whole numbers, we can build the negative whole numbers by repeatedly adding -1 to itself. Should we encounter 3+(-2) we might pause momentarily, for there is no 2 for the -2 to undo, but if we write everything in terms of 1 and -1 things are resolves. We obtain 1+1+1+(-1)+(-1) which allows two of the 1/-1 pairs to be cancelled, leaving only 1. Since adding by 0 changes nothing, -0, which undoes addition by 0, also changes nothing. Thus -0=0, since 0 is our symbol for nothing changing.

"Subtraction":

Assuming this is not your first time learning math, you probably have heard of something called subtraction. Subtraction is a way to make numbers smaller without needing to introduce the negative numbers. Basically you have two choices on how to make things smaller, introduce negative numbers and leave everything as addition, or leave everything as whole numbers and introduce subtraction. Subtraction has two very big problems. The first is that the nice properties of associativity and commutativity that addition has are not true for subtraction. That is, 3-(2-1) is not (3-2)-1, and 3-2 is not 2-3. The second problem is that it is perfectly natural to wonder, once the notion of subtraction is on the table, what IS 2-3? Then, of course, we must introduce the negative numbers after all, making all the inconvenience we went through to keep them out of things wasted effort. This is our first real lesson from thinking about math systematically and logically, subtraction is an inconvenient lie!

Integers:

Just like the numbers you get by adding together 1's has a special name, the whole numbers, the numbers you get by adding combinations of 1's and -1's has its own name, the integers. The integers are made up of the whole numbers, zero, and the negative whole numbers. Please feel free to explore for yourself that you can make all of these by adding 1's and -1's. Slightly more tricky is that these are the only things you can make by adding 1's and -1's, but it is true, and I invite you to ponder upon it.