Number Flavors Part I: Rationals


Image credit: “Watermelons for sale in Shanghai” by kris krüg on flickr.

 

Numbers come in a lot of varieties. There are numbers we use to count things, like eight baseballs or two cups of coffee. There are numbers we use to measure things, like a half a cup of sugar in a batch of chocolate chip cookies. And there are numbers that get a lot weirder. Number Flavors is a series of posts about some types of numbers and why they’re interesting.

Last year, I wrote a post called The Integer Hat Shop. The word “integer” refers to the type of number that can be used to count things, like hats and anti-hats.

Once you know what an integer is, and you know what division* is, you can look at a new type of number.

A “rational number” is an integer divided by an integer.

*Interestingly, when mathematicians get really technical, they tend to define division in terms of multiplication: (3/7) is defined as the number that, when multiplied by 7, gives you 3. But the reason for doing that deserves a full discussion in its own right.

So, since 2 is an integer and 3 is an integer, 2/3 is a rational number. So is 3/2. Similarly, 7/5, and -6/3, and 4/1 are all rational numbers too. 4 is a rational number, because it’s the same thing as 4/1. They’re all just integers divided by integers.

All of this might look suspiciously familiar. Like an old menace from seventh grade. You might be asking: is all of this just another name for fractions?

Cake fractions

Oh, the horror.


The answer is, yes and no. Yes, “rational numbers” and “fractions” pretty much describe the same set of numbers. But no, the words mean slightly different things.

Here’s what I think the difference is. “Fraction” describes a number that’s written a certain way. Here are some things that are not fractions:

3

2.5

1

Some things that are fractions:

3 / 1

5 / 2

7 / 7

What’s weird about this distinction, from a mathematical point of view, is that the first three numbers are the exact same numbers as the last three. They are literally referring to the exact same things, just written differently. Just as “gray” and “grey” are two written representations of the same color, “3” and “3/1” are two written representations of the same number.

“Rational number” describes a number itself, not the way it’s written. 3 is a rational number, even though it’s not a fraction. 3/1 is a rational number, and so is 3.0– and this shouldn’t be surprising, because those are both just different names for the same number.

Questions about fractions are often pretty boring. I don’t really care what 812/17 + 5/9 is. If I need to know, I can find out with a calculator. Nor do I want to spend my time writing the number 1.52 as a fraction, because why bother? It’s the same number. (To be fair, in certain situations, it is useful to use some particular type of written representation for numbers. So it’s good to know how to convert between them in case you need to. But at the heart of it, 1.52 and 1.520 and 152/100 and 38/25 all mean the same thing.)

The idea of “rational number” can lead to more interesting questions. If you add two rational numbers, do you always get a rational number? What if you divide them? Are all of the integers also rational numbers? Are there rational numbers that are not integers?

The answer to that last question is clearly yes: 2/3, for example, is a rational number but not an integer. And it’s not hard to show that every integer is a rational number. -17, for example, is rational, because -17 is just another name for -17/1.

Come to think of it, what kind of number would not be a rational number? It’s easy to think of real-world examples of rational numbers. I might eat 1/8 of a watermelon, or decorate a birthday present with 2.5 feet of ribbon. And any number I can write out as a decimal is rational. 9.0344, for example, is the same thing as 90344/10000. That is, it’s equal to (the exact same number as) the integer 90344 divided by the integer 10000.

So, if fractions, written-out decimals, and integers are all rational, then does a number that isn’t rational even make sense? What would not-rational numbers look like?

Let’s save that for the next post.