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.