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.

Lesson Five: In Which Form Emerges.

In some ways, the integers are a fundamental example of patterns we will see repeated throughout mathematics. They can interact with each other through addition, there is a way to leave them alone denoted by 0, and anything you do can be undone with a negative.

Addition:

Math would be quite boring were there no way for numbers to interact with each other. Addition is our first example of numbers interacting, you can use addition to take two integers and produce a third. This is what is called an operation. An operation, in common usage, is something you do to something, such as when a surgeon operates on a patient, or a farmer operates a tractor. You use addition to do something to two integers, namely change them into a third integer.

Notice that if you add two integers, the result is always an integer, if you think about all the integers as being constructed from 1's and -1's you can see why this should be true. Because using addition on two integers always produces an integer, we say that the integers are closed under addition, meaning that we cannot use addition to escape the integers. Notice that the whole numbers are also closed under addition. Now we know that addition in the integers has three properties, it is associative, (1+2)+3=1+(2+3), commutative, 1+2=2+1, and closed, two integers always add to an integer.

Zero:

Zero plays a special role with respect to addition, it leaves things unchanged. This is quite useful, as sometimes you don't want to alter the number you are considering. Because n+0=n we call 0 the identity. The term identity is used because combining a number with 0 preserves the number's identity, ie it doesn't change the number, sometimes 0 is called the additive identity to denote what operation has 0 as the identity.

Negatives:

We know that 1+-1=0, so let us extend the definition of the '-' symbol to say that n+(-n)=0. This agrees with what we know so far in that 1+-1=0 and 4+-4=0, but remember n is allowed to be any integer. So -1+(-(-1))=0 must also be true, but what is -(-1)? Since addition is commutative, we know that -1+1=0 because 1+(-1)=0, so 1 might be -(-1). Could there be another?

Suppose that -1+n=0. We also know -1+1=0, so -1+n=-1+1, since they both are 0. Since they are the same number, we can add 1 to both of them and they will stay the same, so 1+(-1)+1=1+(-1)+n. Since addition is associative, I may add the first two numbers together first on both sides of the equal sign, so (1+(-1))+1=(1+(-1))+n which is the same as 0+1=0+n since 1+(-1)=0. Since 0 does not change numbers when added to them, we now know that 1=n, or that n must in fact be 1. This means that -(-1) must be 1, or that 1 is the only value which when added to -1 yields zero. One can make a similar argument for any number, replacing 1 with the number in question, so every number has a unique negative.

In general, if n is a whole number, -n will be a negative number, and -(-n) will be the number n again. This is because n+(-n)=0 also means that -n+n=0, so n must be -(-n). We call -n the inverse of n, because adding -n is the opposite of adding n, or to put it another way, adding -n undoes adding n. It is very nice for operations to have inverses, because they allow you to undo things that have been done!

Structure:

I make note of these concepts, operations, identities, and inverses, because they help define the structure of the integers, that is, how the integers relate to each other. I also mention the concepts because they are important themes underlying mathematics and shall soon come into play in a new manner.

Remember that I said the whole numbers are closed under addition. The reason we want to also include the negative numbers to "fill out" the integers is in order to have inverses. If we did not have inverses we could add 2+7 to get 9, but there would be no way to undo this, and get back to the 2 from the 9. With inverses we know that we just need to undo adding 7, which means adding -7, and we get that 9+(-7)=2.