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/
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