Learn what rational numbers are, where they are located on the number line, and how they relate to the rest of the world of numbers.
The number line is a really long line. In fact, it’s infinitely long. But you don’t actually have to travel very far along that line to start discovering some interesting creatures. So far, we’ve talked a lot about integers, fractions, and their decimal companions, but there’s still a whole world of numbers left to explore. And that’s exactly what we’re going to start doing today as we answer the question: What are rational numbers?
Recap: Integers and Fractions on the Number Line
As we talked about long ago, it’s often helpful when talking about numbers to think about a number line. That is, an infinitely long line that extends out to your left and right with zero in the middle, negative numbers on the left, and positive numbers on the right. All of those evenly spaced tick marks you see are, of course, the integers: coming in from infinitely far away in the negative direction, up to –3, –2, –1, 0, 1, 2, 3, and so on extending out infinitely far into the distance in the positive direction.
We’ve also talked about the fact that the space between those integers isn’t empty—there are numbers there too (just not integers)! For example, halfway between 1 and 2 is the number 1-and-1/2. Or, in decimal form, 1.5. And midway between that and 2 is the number 1-and-3/4…and on and on. You could keep doing this divvying up forever to find more and more of what are known as fractions.
What are Rational Numbers?
Clearly these fractions and decimals are very different from integers—after all, the integers that we’re used to don’t have fractional parts. But these two types of numbers aren’t completely unrelated. In fact, if you think for a minute about how we write fractions, you’ll see that they’re made from two integers—one in the numerator (the top part) of the fraction and one in its denominator (the bottom part). Numbers like this that can be written as the ratio of two integers are called “rational” numbers. Notice that the word “ratio” is part of the word “rational”—which makes sense since we’ve just seen that rational numbers are actually ratios.
How to Write Rational Numbers
So, now you know that all fractions are also rational numbers, and that all rational numbers can therefore be written like a fraction, we can say that:
“any rational number” = m / n
What are m and n here? They’re just symbols—also known as variables in algebra—that can represent any integer values. For example, in the fraction 1/2: m=1 and n=2; in the fraction 23/32: m=23 and n=32; and so on.
What Types of Numbers are Rational?
But fractions aren’t the only types of numbers that are rational. If you think about it, you’ll see that any integer can also be written as the ratio of two integers—as long as the bottom number in that ratio is 1. In other words, the number 1 can also be written 1/1, the number 2 can also be written 2/1, and so on. It sounds kind of strange to write an integer as a ratio of two integers, but you can just think of a number like 2/1 as a fraction meaning two wholes…just as a fraction like 1/2 means one part of a half.
[[AdMiddle]So, when we put all this together, we see that the rational numbers are just a big group of numbers that include all of the fractions and all of the integers.
Pop Quiz: Which of These are Rational Numbers?
To test your understanding of this, figure out which of the following numbers are rational numbers:
54 / 7
2 / 0
The answer is all of them…except the last one! The first four are all rational because you can write them as the ratio of two integers, like this:
–10 can be written as –10 / 1
54 / 7 can be written as 54 / 7
–0.795 can be written as –795 / 1000
21.21 can be written as 2121 / 100
But 2 / 0 is not a rational number! Why not? Because a fraction can’t have zero as its denominator—the meaning of a ratio like 2/0 is not defined. We talked about why this is true way back in the episode on “What are Fractions?” If you’d like a little refresher about the reasoning, check out the Math Dude “Video Extra!” episode I posted about the subject on YouTube.
But, you might be wondering, does the fact that we’ve called this group of numbers that includes integers and fractions “rational” mean that there’s some other kind of number that’s not rational? Indeed there is. But we’re out of time for today. So the story of those numbers—logically called “irrational numbers”—is going to have to wait until next time.
In the meantime, please email your math questions and comments to firstname.lastname@example.org. You can get updates about the Math Dude podcast, the “Video Extra!” episodes on YouTube, and all my other musings about math, science, and life in general by following me on Twitter. And don’t forget to join our great community of social networking math fans by becoming a fan of the Math Dude on Facebook.
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!