A Wonder of Rationality


I’d like to talk about a neat little property of the rational numbers, which does involve there being infinitely many of them, and which isn’t about how there are just as many rational numbers as there are integers but there are more real numbers than there are rational numbers. (It’s true, but the point has already been well-covered by every mathematics blog ever.) Anyway, I’m laying the groundwork for something else.

Now, it’s common in mathematics to talk about the set of rational numbers, the numbers you get as one integer divided by another, as Q. The notation seems to trace back to the 1930s and the Bourbaki group which did so much to put mathematics on a basis of set theory, and the Q was chosen as it’s the start of “quotient”, which rational numbers after all are. (“R” was already called on to stand for the set of Real numbers.) I’m interested in two subsets of the rational numbers, the first of them, all the positive integers. For that I’ll write Q+. The other is just the rational numbers between zero and one. For that I’ll write Q(0, 1).

I can match every rational number between 0 and 1 to some rational number greater than zero. Here’s one way (there are many ways) to do it. Start out with some number, let me call it q, that’s in Q(0, 1). That’s a rational number between zero and one. Well, let me take its reciprocal: the result of one divided by q, which is going to be some rational number greater than 1. That’s a nice matching of the rational numbers between zero and one to the rational numbers greater than one, but I claimed I’d do this matching for rational numbers greater than zero. No matter; I can get there easily. Take that reciprocal and subtract one from it. This new number — let me call it p — is a rational number greater than zero, something in Q+. That is, each q (a rational between 0 and 1) can be matched with p (a positive rational), among other ways, by letting p equal (1/q) minus 1.

For example, let’s say, let q be 3/4. Then the reciprocal of that is 4/3, and subtracting one from that gets us a p of 1/3, which is certainly a positive number.

Or let’s say that q is 2/9. Then the reciprocal of q is 9/2, and subtracting one from that gets us a p of 7/2. (Some math teachers would want to change that 9/2 into 4 ½, and that 7/2 into 3 ½, but I don’t really know why they bother. I suppose the teachers are having fun and it’s quite easy to test, so, let them.)

If we start with a q of 3/32, then we go to its reciprocal, 32/3, and subtract one from that for a p of 29/3.

And I can run it the other way, too. Pick some rational number p, anything that’s positive. Add one to it, which will make it a rational number greater than 1. Take the reciprocal of this, and you have a rational number between 0 and 1. That is, p (a positive rational) can be matched with q (a rational between 0 and 1) by (again, among other ways) letting q equal 1/(p + 1).

For example, let’s let p be 3/5. Add one to that and we have 8/5, and the reciprocal of that is our q, 5/8, which is a rational number between zero and one.

Or let p be 14. Add one to that and we have 15, and the reciprocal of that is our q, 1/15, which is again between zero and one.

Or say that p is 39/7. Add one to that and we have 46/7, and the reciprocal of that is q, 7/46.

There are many ways to do this sort of matching. For example, you can match the rationals between 0 and 1 to the rationals between -1 and 1, or for that matter to all the rationals, positive and negative. It doesn’t have to be with a single rule, either; you’re allowed to set up a rule like “if q is less than one-half, find p by this rule; if q is greater than one-half, find p by that rule; if q is exactly one-half, do this other thing instead”. You can have a good bit of mental exercise by picking sets and trying to work out rules that match the numbers in one to the numbers in the other, and if I were smart I might try making a weekly puzzle section for that.

A reasonable person may point out that it’s absurd that you can match Q(0, 1) exactly to Q+. The rules I worked out give you one and only one p for each q, and vice-versa; but, the rationals between zero and one are all also positive rational numbers. That you can match the positive rational numbers to a subset of the positive rational numbers is counter-intuitive, at least when you first encounter it. It’s also the simplest definition for being “infinitely large” that I know of, though; if you can set up a one-to-one match of a set with a proper subset of itself, the set is considered to have an infinitely large cardinality, which is one of the ways mathematicians describe the sizes of things.

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Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there.

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