Each of the infinite number of points on the number line represents a real number. The easiest way to picture the set of real numbers is just to picture the number line.
So what is it? It’s called the set of real numbers. 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 there’s the set of all integers, the set of all rational numbers, and the set of all irrational numbers.īut 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. 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. 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. 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. As we’ve seen in previous articles, the number line is also full of numbers that aren’t rational-the irrational numbers. 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. 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. If you divide the interval between 0 and 1 in half, you find the location of the fraction 1/2. 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. Think of numbers as points on an infinitely long line. Review: Integers, Rational, and Irrational Numbers SPONSOR: Get your copy of Grammar Girl’s new books 101 Misused Words You’ll Never Confuse Again and 101 Words Every High School Graduate Needs to Know. 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.) 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.
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