by Jason Marshall
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?
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. 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.
Test Taking Tips: Union and Intersection
Since real numbers include all rational numbers (which include all integers) and all irrational numbers, the set of real numbers must be the union of the sets of rational and irrational numbers (remember, the union of two or more sets is the new set containing all the unique elements from each set). There are lots of possible test questions that use the union and intersection of sets of numbers to make other sets of numbers. For example, the union of the set of positive integers and the set of negative integers is the set containing all but one integer. Which is it? Zero! Or how about this: What’s the intersection of the set of integers and the set of positive real numbers? Well, it’s the set of positive integers (remember, the intersection of sets is the new set containing only the elements that are included in all the sets).
Here are a few more for you to think about: What’s the union of the set of rational numbers and the set of integers? And how about the intersection of the set of rational numbers and the set of integers? We’ll talk about the answers in next week’s article.
Number of the Week
Before we finish up today, it’s time for this week’s featured number selected from the various numbers of the day posted to the Math Dude’s Facebook page. This week’s number is 541,000,000 miles. What’s that? Well, in honor of the final flight of NASA’s space shuttles, this number is the approximate number of total miles flown by all the space shuttles during their 135 missions. If you’d like a bit of perspective on just how far this is, the 541,000,000 miles on the combined odometers of all the shuttles are enough to have taken them from Earth to Jupiter. Which shows you just how far the shuttles have flown in their 30 years of service…but also just how far it is to Jupiter!
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Do you ever mix up “further” and “farther”? Or “supposedly” and “supposably”? Have you ever wondered what “Antebellum” means? Or who came up with the word “quixotic”? Well, you’re not alone. Which is why Mignon Fogarty—aka Grammar Girl—has written the new books 101 Misused Words You’ll Never Confuse Again and 101 Words Every High School Graduate Needs to Know. They’re both chock full of words that seem to pop-up all over the place, so brush-up on your vocabulary and pick up your copies of these books from Amazon, Barnes & Noble, or iTunes.
Okay, that’s all for today. Remember to become a fan of the Math Dude on Facebook where you’ll find a new number of the day and math puzzle posted each and every weekday. And if you’re on Twitter, please follow me there too. Finally, if you have math questions, feel free to send them my way via Facebook, Twitter, or by email at firstname.lastname@example.org.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!