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 Seven: Multiplication and Addition

Now we know how to combine two numbers with multiplication, and we know how to combine two numbers with addition, but how do multiplication and addition interact?

Multiplication First:

It is an unfortunate truth that multiplication and addition do not get along as well with each other as addition gets along with addition and multiplication gets along with multiplication. Since both are associative 2+3+4 is unambiguously 9, and 2*3*4 is unambiguously 24. However 2+3*4 takes different values depending on if you interpret it as (2+3)*4, which is 20, or 2+(3*4), which is 14. So, to resolve this ambiguity, we decree that multiplication occurs before addition. This is a common convention upon which we agree, rather than a mathematical truth which we discover, but we should stick with it nonetheless. So, things within parenthesis happen before things outside of them and multiplication happens before addition, these are conventions by which we must agree to live.

Distribution

The other interesting way in which addition and multiplication interact is called distribution. Consider 2(3+4), which is 2*7 or 14. This is the same value as 2*3+2*4 which, since we know to multiply before we add, is 6+8 or still 14. This is not a coincidence, when we switch from doing addition first to doing multiplication first, we must make sure that we multiply both values in the sum by 2. Consider if you had two bank accounts, into one you deposited a certain amount of money each month to save up for a vacation, and into the other you deposited a different amount of money to save for a rainy day. If, after three months, you wished to know how much money was in the accounts total, you could find out in two different ways. You could first figure out how much you deposited each month, by adding the amounts going into each account together, then multiply this monthly deposit size by 3, this way you add first. You could also figure out how much money was in each account individually after three months, by multiplying the deposit to each account by 3 separately, then add these account totals together to obtain the overall total, this way you multiply first. Notice that in the second case, the amount of money you put into the first account must be multiplied by 3 and the amount you put into the second account must be multiplied by 3, we say that the 3 is distributed to each of them.

Lesson Six: In Which We Go Forth and Multiply

Now that we are familiar with addition, we might amuse ourselves by repeatedly adding a number to itself. We do this when counting by numbers larger than 1, such as listing even numbers, or counting by two, 2, 4, 6..., or when items come in packages of a fixed amount, one carton of eggs is 12, two cartons is 24, three 36, and so on. It would be convenient if there were a way to quickly add together a specific number multiple times, which is exactly the role multiplication plays.

Multiplication:

While we are often taught to represent multiplication with the 'x' symbol, the way we represent addition with '+,' this becomes confusing when 'x' eventually gets a different meaning. Instead of learning one thing then changing halfway through, let us agree to write multiplication with an asterisk, n*m is n multiplied by m, or just by writing two things next to each other, nm is also n multiplied by m. When there would otherwise be ambiguity I shall use an asterisk or parentheses to clarify, so 34 is always thirty-four, if I mean 3 times 4 I shall either write 3*4 or 3(4) . That said, what is 3*4?

If one says it out, 3*4 is 3 multiplied by 4, which means you will be adding three to itself until you have 4 of them. Thus, 3*4 = 3+3+3+3 = 12. Because multiplication can be thought of in terms of addition, the whole numbers are closed under multiplication, by which I mean two whole numbers always multiply to another whole number. It turns out that multiplication is also commutative, that is, n*m = m*n, which is something you may be able to convince yourself of by thinking of n*m as counting up m groups each with n things in them, then rearranging the things into groups of size m. It turns out that multiplication is also associative, so m(n*l) = (m*n)*l, feel free to try to convince yourself why this must be true, but I think it is a harder property to intuit than commutativity.

Multiplication With One:

If you only have one set with n things in it, then you have n things in total, so it seems reasonable that n*1 = 1*n = n. In this sense, 1 is providing the same service for multiplication that 0 did for addition, it is the multiplicative identity, that is the number which leaves every number alone when they are combined using multiplication.

Multiplication With Zero:

If you have no sets, then you have no things. If I get 100 dollars every time I win the lottery, but I never win the lottery, then I get no dollars. In fact, no matter how much the lottery pays, if I do not win, I get 0 dollars. Thus, it should not be too surprising that n*0 = 0*n = 0.