(Archive Question of the Week)
Although high school and up probably constitute the majority of our questions, I always enjoy answering younger children. For today’s look at the archives, I thought I’d look at two such questions, both from 1999, and very memorable. The first is almost certainly the youngest “patient” we’ve ever had, and the second is a good example of a classroom question.
How “odd” are odd numbers?
Here is a question from what must have been an irrepressible four-year-old, Timothy, as transcribed by his mother:
A Four-year-old Thinks About Even and Odd Numbers (I'm typing this for my son, in HIS words, as he gets frustrated with "writing" with the keyboard. He's pre-K, but please answer if you can - I'm a housewife, not a mathematician. Some of this was asked to me personally, with a little extra banter between. I've done my best to translate.) "Mom told me that even numbers are ones that can be cut in half and both parts are the same size - like when I have to eat lunch and she has to go to the store, and she only makes me finish "half" of it. "Half" means 2 parts, and some numbers are not that way. Mom says they're "odd." It's not fair letting 2 be even all by itself! What about 3? I thought about 4, but that's just 2's twins, so it's okay. "3" has 3 parts that make the whole thing with no bits, and when we counted 9, I found out it has 3 parts made of 3. Mom says that "odd" numbers have bits left over, so why are some numbers called "odd" when they don't have bits? Mommy says "odd" numbers aren't "weird." It means something LEFT OVER, but I don't think she knows because she isn't a teacher. FOOD isn't made of numbers unless it's a "PI." When I play a game with Adam (my brother) and we're even, nobody gets mad and we both win. But when David (my other brother) says I'm "the odd-man-out" I get mad because he won't let me play then. I'm going to school when I'm bigger, and I have to know all about numbers so I can be smart. Mommy says 5 is an "odd" number, but it's bigger, and besides, it has to use 2 and 3 and 4 like step-stones to get to 20. When I'm 5, I won't cheat like it, because I have to go (counting from 6-19) to get to be old!..." (It goes on, but I think you get the gist...HELP!)
At the time my youngest son, Dan, was just about to turn 5, so I was familiar with 4-year-olds; my older son, another Timothy, was 14 (he was already an accomplished programmer, and in the last few months has helped with the creation of this new site).
I find myself imagining a video of the question, and having to pause every few seconds to ponder a new idea Timothy has introduced. I started by reinforcing what his mother had said:
It sounds like your Mom has told you many of the same things I've taught my kids about even numbers. We call a number even if it can be paired off without any extras; 5 is odd and 6 is even because they look like this: * * * * * * * * * * * 5 6
(At the bottom, I will give links to other pages, where we have discussed this definition further.)
I next looked at Timothy’s concern about the word “odd” seeming to suggest weirdness, and about “even” being used only with reference to 2 (which is thus singled out):
When you pair off people, such as to play games with two teams, the one left out is the "odd" (left over) one, which is why we also use the word "odd" for something different that doesn't fit in. It's fun thinking about how different meanings of words fit together. The number 9 is odd because it doesn't pair off evenly: * * * * * * * * * 9 But it does "triple off" evenly: * * * * * * * * * 9
So his explorations of divisibility (which is a wonderful thing for a kid to do!) discovered very useful ideas — we just don’t happen to have a single word for this. Why not? Because once you get to divisibility by a bigger number, the word you use has to convey both what you divided by, and also what remainder you get. With 2, there is only one remainder other than zero; with three, there are two.
I like to avoid talking down to a child, and in fact will often “talk up” to them — tossing advanced ideas to them as possibilities for the future (knowing that kids are masters of the art of ignoring what they aren’t ready for and grabbing anything in the environment that works, even if adults usually don’t think it will). So I gave him a little extra:
Someday you may learn about "modular arithmetic," and you'll see that it can be very useful. And before then you'll learn about prime numbers, which can't be divided evenly by anything. Maybe you can have fun by trying to find which numbers are prime - for instance, try to arrange 11 beans into rows, and you won't be able to do it (unless you call it 1 row of 11 or 11 rows of 1). Prime numbers are REALLY "odd."
I doubt that Timothy or his mother learned anything about modular arithmetic, but an exploration of prime numbers is both easy to start, and impossible to exhaust!
Even and odd fractions?
In the middle of my answer to Timothy, I mentioned this in passing:
If you talk about cutting numbers in half, you have to be careful to remember that the two "halves" have to be the same size, and that you aren't allowed to cut a pea in half to make it even. Once you talk about fractions (pieces of things), the idea of evenness no longer means anything.
This was the topic of the next question I want to look at, from Mrs. Small’s 4th and 5th grade class:
Even and Odd Fractions? 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?
I could have just said that “even” and “odd” only apply to integers; but why? I replied:
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.
So, we have a definition that we’ve applied to integers; can we apply that definition to fractions?
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.
Okay, the definition doesn’t work any more, because fractions were invented so that we can divide any integer by any other. But can we extend the definition so it will apply?
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.
Their idea was based on just looking at the numerator; but in their example, the very same number can be written in two ways, one of which would be even and the other odd. So it seems that their definition doesn’t make the number itself even or odd, just the way you write it. Moreover, we can rewrite any fraction in an infinite number of ways: not only can the “even” fraction 6/10 be written as 3/5 and become “odd”, but the “odd” fraction can still be divided by 2, yielding 3/10, so it’s wrong to say it can’t be divided by 2:
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 we have our answer: no.
But wait a minute! Why can’t we refine the definition and say that a fraction is even or odd when its numerator, when reduced to lowest terms, is even or odd? Would that work? Maybe! Math is always open-ended, because we can always ask more questions:
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.
I’ve just undone everything I said! We now have a possible definition that does apply to any fraction, and gives only one answer. I left it at this, half hoping that the class might write back with further thoughts, because there’s more to be said here. As we’ve said elsewhere, we can define anything; what matters is whether our definition is useful. With this definition, would we be able to accomplish anything new of interest? Would even and odd fractions have any useful properties? Could you still say, for example, that the sum of two even fractions is always even? That’s left for the reader to experiment with.
For further exploration
In case you are interested, here are some other discussions of even and odd numbers, with answers by Doctors Rick, Peterson (me), Jeremiah, and Twe: