Lesson Twelve: In Which We Are Rational

Remember that originally we had the natural numbers, which we obtained by starting with 1 and repeatedly looking at the next whole number, obtained by repeatedly adding 1. Then we saw that it would be more useful to consider the integers, so if we added something that later needed to be undone we could do so by adding a negative number. Now we come to the notion of the Rational Numbers which is all of the fractions that can be written with an integer in the numerator and a non-zero integer in the denominator. Notice that every integer is a rational number because, for any integer n, n = (n/1).

The Rationals and Addition:

Conveniently, the rationals are closed under addition, that is, the sum of two rational numbers is always another rational number. Consider (m/n)+(a/b) = (mb+na)/(nb), and since the product of two integers is always an integer, and the sum of two integers is always an integer, (mb+na) and (nb) must both be integers, so the fraction is a rational number. Furthermore, since addition and multiplication of integers are both commutative and associative, addition of rational numbers is both commutative and associative. There is an additive identity, the rational number 0 = 0/1 = 0/n, and every rational number has an additive inverse since (m/n)+(-m/n) = (m+(-m))/n = 0/n = 0.

The Rationals and Multiplication:

Even more conveniently, the rationals are closed under multiplication. This is even easier to show since (m/n)*(a/b) = (ma)/(nb) and the product of two integers is always an integer. As before, multiplication of rationals is both associative and commutative, by now do you have a firm notion of what associativity and commutativity mean respectively? There is also a multiplicative identity, since 1 = 1/1 is a rational number and (1/1)*(m/n) = (1*m)/(1*n) = m/n. Finally, every rational number EXCEPT all forms of 0 has a multiplicative inverse. Hopefully you can convince yourself that a rational number represents the number 0 exactly when its numerator is 0. Now suppose that m/n is a rational number that is not 0, so n is not zero, since m/n is a rational number so zero cannot be in the denominator, and m is not zero, since that would make m/n = 0. In this case n/m is also a rational number, and (m/n)*(n/m) = (mn)/(nm) = (mn)/(mn) = 1. This is a mathematical way of saying that we can "undo" multiplication by any rational number except for 0.

Lesson Ten: There Is No 1/0

Remember when I said that n(1/n) = 1? Well, that is only mostly true, the single exception is that we cannot let n = 0 and expect that to be true, because there is no number (1/0). You might be asking yourself, "what right does he have to tell me what number cannot exist?" Bear with me for a moment and you too will see why (1/0) simply cannot make sense.

The Reciprocal of Zero:

Keep in mind that the purpose of (1/n) is to undo multiplying by n. So if a*n = b, then b*(1/n) = a, because to get from a to be you multiplied by n, so multiplying by (1/n) undoes this and takes b back to a. For example 3*2 = 6, so 6*(1/2) = 3.

However, zero times anything is zero. So 3*0 = 0 and we expect 0 * (1/0) = 3. But 5*0 = 0 so 0*(1/0) = 5 also needs to be true. Since multiplying by zero takes everything to zero, there is no way to undo it, we lose track of where things come from and cannot "send them back."

To use a metaphor, suppose we live in a small town with an airport that has flights that come in from New York. If someone arrives at our airport, we know that they just came from New York, because there is only one place that they can come from. If we consider a larger airport which has incoming flights from multiple places, then we cannot say where an incoming passenger is coming from without knowing more, because there are multiple possibilities. Since multiplying by 0 sends everything to 0, (1/0) does not have enough information to undo multiplication by 0. Because (1/n) is defined to undo multiplication by n, (1/0) cannot exist. Since m/0 would need to be m*(1/0), m/0 cannot exist either. In short, the denominator of a fraction CANNOT be zero

There are two very important concepts in this post. The fact that (1/0), and consequentially (m/0), cannot exist is something that even advanced math students forget or gloss over. You would be surprised at the ways the value 0 can sneak up on you. However, of even deeper importance is the notion that some things cannot be undone, because too much information is lost.

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 Three: In Which Nothing Happens

Suppose people are arriving at school in the morning, and we are keeping track of how many arrive each ten minutes. For a while life is good, people arrive and we are adding them to the total number of people at school. After a while however, school starts and the arrivals stop. We eventually need to add nothing to the number of people at school!

Zero:

If we want to add nothing to our total, it would be helpful to have a numerical representation of nothing. This is of course 0. Anything plus 0 remains the same. This means that 0+1=1, so, since adding one gives us the next number, we can think about 0 as being the number before 1.

Optional Tangent:

For some rather advanced reasons, which I may get to later, 0 is actually a more philosophically sound place to start our numbers than 1. However, 1 is the starting place that I chose for three reasons: the argument for 0 is advanced, but not complicated, and I want to keep things simple for now; if I remember correctly, math education starts with the positive numbers then introduces 0 and I think that is worthy of emulation and; it makes sense historically, giving nothing its own symbol is something that came after the other numbers.