Associated Topics || Dr. Math Home || Search Dr. Math

### Introduction to Rational Numbers for Middle School Students

```Date: 12/06/2006 at 12:59:28
From: Tracy
Subject: rational numbers

Why do we need to know about rational numbers?  Aren't all fractions
rational numbers?  What's the difference?

```

```
Date: 12/06/2006 at 13:30:27
From: Doctor Peterson
Subject: Re: rational numbers

Hi, Tracy.

That's a good question!  Too often books seem to make it sound as if
there were some special magic about rational numbers, when they are
really just fractions.

Well, not quite.  There is something new when you talk about rational
numbers, though the books might not make it very clear.

When you learned about fractions, you were focusing on a way of
WRITING a number, with a numerator and a denominator; you learned how
to add them, multiply them, and so on.  You also learned other
representations, such as decimals.  But all that was just learning the
mechanics of working with different ways to write numbers.

Now you are getting older, and are ready to dig a little deeper.  You
are learning not about ways to write a number on paper; you are
learning about the numbers themselves!  A rational number is not just
a fraction; it is any number that CAN be written as a fraction,
regardless of how you happen to write it.  And behind this statement
lies something that is really profound: that not every number can be
written that way!

The ancient Greeks at one time assumed that any length could be
measured as a fraction--that is, you could always find a small enough
unit (say, a 16th of an inch, or a 1000th of an inch) so that the
length is a whole number of those units (like 3/16 or 187/1000).  Then
they discovered that this was not true: that there were lengths that
were not fractions.  In particular, the first one they found was the
square root of two, which is the length of the diagonal of a one-inch
square.  It can't be written as a fraction; it is not a rational
number.

So the important thing about rational numbers is that they form a
specific set that is not the same thing as the set of all numbers.
You're starting to embark on a voyage of discovery in math, finding
new kinds of numbers and new things to do with them; you start that
voyage by studying again the kinds of numbers you are already familiar
with, in order to be well prepared for what comes next.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics: