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What are Real Numbers?

Learn how real numbers relate to the rest of the numbers in math.

By
Jason Marshall, PhD,
July 15, 2011
Episode #070

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In today’s article we’re going to use everything we’ve learned so far about sets and subsets and unions and intersections to help us understand how various types of numbers relate to each other. And, in the process, we’re going to discover another class of numbers that we haven’t talked about yet—they’re called “real numbers.” (And no, that’s not a joke.)

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Review: Integers, Rational, and Irrational Numbers

Think of numbers as points on an infinitely long line. At regularly spaced intervals along this line are the integers: 0 in the center; the positive integers 1, 2, 3, and so on to your right; and the negative integers -1, -2, -3, and so on to your left. If you divide the interval between 0 and 1 in half, you find the location of the fraction 1/2. Do it again to get 1/4, and again to get 1/8, and so on—you can divide these intervals an infinite number of times in an infinite number of ways to find an infinite number of fractions.

When you combine this type of fraction that has integers in both its numerator and denominator with all the integers on the number line, you get what are called the rational numbers. But there are still more numbers. As we’ve seen in previous articles, the number line is also full of numbers that aren’t rational—the irrational numbers. Of course, these are “irrational” because they can’t be made from the ratio of two integers…not because they’re a bunch of crazy numbers that are out of touch with reality.

Now that we have the various pieces to our numerical puzzle, let’s see how sets and subsets and unions and intersections help us to put them all together.

Sets of Numbers

Since a set in math is just a group of things, you can see that each of the groups of numbers we’ve just mentioned—integers, rational numbers, and irrational numbers—actually form sets. So there’s the set of all integers, the set of all rational numbers, and the set of all irrational numbers.

But not only do each of these groups of numbers form a set, each group must also be a subset of some larger group of all numbers—that’s how sets work. In other words, since integers, rational numbers, and irrational numbers are all numbers, there must be some bigger set of numbers that contains all of these numbers. So what is it? It’s called the set of real numbers.

What are Real Numbers?

The easiest way to picture the set of real numbers is just to picture the number line. Each of the infinite number of points on the number line represents a real number. In fact, the number line we’ve been talking about all along is more accurately called the “real number line.”

But why do we even bother to define the real numbers? If these are all the numbers, can’t we just call it the set of “all numbers”? Well, although these are all the numbers that we’ve talked about—and that you’re probably ever going to encounter in your daily life—it turns out that there are other numbers that aren’t real numbers. They’re called imaginary numbers. Don’t let the names trick you though—imaginary numbers aren’t any less a part of reality than real numbers…that’s just what they’re called. But that’s a story for another day. For now, let’s get back to the real numbers. In particular, how you can use sets and subsets and unions and intersections to describe the relationship between the various types of numbers.

Why You Need to Know About Sets and Subsets

But first, you might be wondering: Why do I need to know this? What’s it used for? Of course, you can’t use sets and subsets to make a cup of coffee or help you tie your shoes—sorry, they’re just not those kinds of things. But if you’re a mathematician, you can use them as the basis upon which to build almost everything else in math. Which means that—believe it or not—mathematically, this is really important stuff. And, if you’re a student who wants to get into college, this is exactly the kind of stuff you might see on tests like the SAT. In fact, let’s take a look at what I mean.

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