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/