Even and Odd Fractions?

Date: 01/12/99 at 14:41:25
From: Mrs. Small's 4th and 5th grade class
Subject: Odd and Even Fractions

Dear Dr. Math,

We know that an even number always has a partner - meaning that if you 
broke the number up into groups of two (partners) each number would 
have a partner. We know that odd numbers do not have partners. Our 
question is, does this apply to fractions also? For example we think 
that 6/10 is an even fraction because each part of the fraction has a 
partner (3 groups of 2). Is this correct?

Date: 01/12/99 at 17:21:26
From: Doctor Peterson
Subject: Re: Odd and Even Fractions

Hi, Class! 

This is a good question that deserves a thoughtful answer. Let's think 
about what we mean when we say a number is odd or even. There is always 
a reason for why we define something the way we do. An even number is 
one that is divisible by two. As you said, you can pair off the items 
you're counting with nothing left out.

All of this depends on the idea of divisibility: that some numbers can 
be divided and others can't. We can divide 6 by 2, but we can't divide 
7 by 2.

But once you learned about fractions, that wasn't true any more! Sure 
you can divide 7 by 2; the answer is 3 1/2. So when we talk about 
divisibility, we are restricting our thinking to integers (whole 
numbers or their negatives). The whole idea is meaningless when we 
deal with fractions, because when you work with fractions you can 
divide anything.

Take your example: you seem to be taking 6/10 as even essentially 
because its numerator is even; there is an even number of tenths. Did 
you think about the fact that 6/10 is the same as 3/5? If our 
definition of "even" depended on how we wrote the fraction, it 
wouldn't be a very good definition, would it? If a fraction changed 
from even to odd when we rewrote it, evenness wouldn't mean much.

The problem is that a fraction can be thought of as a set of "parts," 
but the "parts" can be as small as you want. Any fraction can be 
divided by 2; half of 6/10 is 3/10, and half of 3/5 is also 3/10. If 
we say 3/5 is odd because it is made of an odd number of fifths, there 
is nothing to stop us from thinking of it as an even number of tenths. 
The reason the ideas of divisibility and evenness work with whole 
numbers is that there is a smallest whole number, 1. You can't break it 
down into smaller whole numbers. But there is no smallest fraction, so 
with fractions you can break anything down!

So the answer to your question is, we simply don't define "even" except 
when we are talking about integers.

Now, I always like to ask another question after I've answered one, 
because there's always more to think about. My question is, what if we 
said that 3/5 is odd because when it is in lowest terms, as it is, the 
numerator is odd, and 4/5 is even because in lowest terms its numerator 
is even? Then there would be a difference between odd and even 
fractions: an odd one COULD be written as an odd number of pieces 
(which are unit fractions, whose numerator is 1), but an even one 
COULDN'T, no matter how hard you tried. Would that definition work? It 
might, but I don't think it would have any use. I'll have to think 
about that a little more! Maybe you can think more about it too, and 
let me know what you decide.

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