1 Introduction

If I were to ask you to count as high as you could, starting at zero, you'd probably think that I had lost my mind. Then, just to humor me (I could be dangerous after all), you'd say: 0, 1, 2, 3, 4, .... At some point (though it might take awhile), I'd get bored and stop you. Now, why did you choose those numbers? I mean, why did you leave out 0.1, 0.2, 0.3, 0.4, ...? For that matter, why did you leave out 0.01, 0.02, 0.03, 0.04, ...? And how about numbers like 1/3, 1/9, 1/11? And why would you leave out your old friend \(\pi\)?

Well, in the words of your elementary school teachers, you decided to use the "counting numbers" (sometimes also called the "whole numbers"). You left out all of the other numbers -- the numbers that your teachers called the "fractions" and the numbers that they called the "decimal numbers". From here on in though, we'll call them the integers, rationals and reals.

2 The Number of Integers

What's the biggest integer? There isn't one! You can always take any integer and add 1 to it. What about infinity? Well, infinity isn't an integer (or a real or a rational for that matter). We just use the term "infinity" to indicate that there is no largest integer (i.e., we say that there are an infinite number of integers).

3 The Number of Reals

Now, how about the reals? Since every integer is also real, it follows that there are at least as many reals as there are integers. But, somehow, it feels like there are a lot more reals than integers, even though there are an infinite number of integers. I mean, just think about how many reals there are between every two integers (say between 0 and 1). There are an infinite number! That means that if I could somehow start "counting" the reals at zero and worked my way up I wouldn't ever get to 1. Even worse, it's hard to imagine how I would even count the reals! Starting at 0, what's the next number you would "count"? If you said 0.1, I'd ask why you left out 0.01. If you said 0.01, I'd ask why you left out 0.001. In fact, no matter how hard you try, you can't come up with the "next" real after 0. Why not, because there isn't one!

For this reason, we like to distinguish between the number of reals and the number of integers. Specifically, we say that there are a countable infinity of integers and an uncountable infinity of reals. In other words, though you can't actually count all of the integers, you can at least describe a process for doing so. On the other hand, there are so many more real numbers than integers, that it's not even possible to describe a process for counting them all.

4 The Number of Rationals

How many rational numbers are there? Well, let's see. Every integer is a rational number (since 1 = 1/1, 2 = 2/1, 3 = 3/1, etc...), so there are at least as many rationals as there are integers. In addition, there are a whole bunch of rationals that are not integers (e.g., 1/2, 1/3, 1/4, etc...). So, there must be more rationals than integers. Right? Wrong!

Remember, the real question we need to ask is whether we can describe a process for counting the rationals. If so, there are a countable infinity of them, if not there are an uncountable infinity of them. So, can we describe a process for counting the rationals? At first glance, it seems like we can't.

First, let's put all of the rational numbers in a table. In particular, label the columns with all of the integers, label the rows with all of the integers, and let the cells be the row label divided by the column label. This is illustrated below: Though there are some duplicates, this will clearly give us all of the rationals.

Now, about counting them. The most obvious way to count would be to start at row 1, column 1 (R1C1) and start counting across. This is illustrated below:

The problem with this approach is that you'll never finish row 1. Hence, you can't possibly count all of the rationals this way.

But, it turns out that there is a way to count them. Again, start at row 1, column 1. Then, go to R1C2. After that, go to: R2C1, R3C1, R2C2, R1C3, R1C4, etc.... This is illustrated below:

At least in principle, it is possible to count the rationals this way. Thus, there are the same number of rationals as there are integers, a countable infinity.

5 Geek Humor

Uncountable Sets Courtesy of SomethingOfThatIlk.com