The Set Tour, Part 6: One Big One Plus Some Rubble


I have a couple of sets for this installment of the Set Tour. It’s still an unusual installment because only one of the sets is that important for my purposes here. The rest I mention because they appear a lot, even if they aren’t much used in these contexts.

I, or J, or maybe Z

The important set here is the integers. You know the integers: they’re the numbers everyone knows. They’re the numbers we count with. They’re 1 and 2 and 3 and a hundred million billion. As we get older we come to accept 0 as an integer, and even the negative integers like “negative 12” and “minus 40” and all that. The integers might be the easiest mathematical construct to know. The positive integers, anyway. The negative ones are still a little suspicious.

The set of integers has several shorthand names. I is a popular and common one. As with the real-valued numbers R and the complex-valued numbers C it gets written by hand, and typically typeset, with a double vertical stroke. And we’ll put horizontal serifs on the top and bottom of the symbol. That’s a concession to readability. You see the same effect in comic strip lettering. A capital “I” in the middle of a word will often be written without serifs, while the word by itself needs the extra visual bulk.

The next popular symbol is J, again with a double vertical stroke. This gets used if we want to reserve “I”, or the word “I”, for some other purpose. J probably gets used because it’s so very close to I, and it’s only quite recently (in historic terms) that they’ve even been seen as different letters.

The symbol that seems to come out of nowhere is Z. It comes less from nowhere than it does from German. The symbol derives from “Zahl”, meaning “number”. It seems to have got into mathematics by way of Nicolas Bourbaki, the renowned imaginary French mathematician. The Z gets written with a double diagonal stroke.

Personally, I like Z most of this set, but on trivial grounds. It’s a more fun letter to write, especially since I write it with the middle horizontal stroke that. I’ve got no good cultural or historical reason for this. I just picked it up as a kid and never set it back down.

In these Set Tour essays I’m trying to write about sets that get used often as domains and ranges for functions. The integers get used a fair bit, although not nearly as often as real numbers do. The integers are a natural way to organize sequences of numbers. If the record of a week’s temperatures (in Fahrenheit) are “58, 45, 49, 54, 58, 60, 64”, there’s an almost compelling temperature function here. f(1) = 58, f(2) = 45, f(3) = 49, f(4) = 54, f(5) = 58, f(6) = 60, f(7) = 64. This is a function that has as its domain the integers. It happens that the range here is also integers, although you might be able to imagine a day when the temperature reading was 54.5.

Sequences turn up a lot. We are almost required to measure things we are interested in in discrete samples. So mathematical work with sequences uses integers as the domain almost by default. The use of integers as a domain gets done so often that it often becomes invisible, though. Someone studying my temperature data above might write the data as f1, f2, f3, and so on. One might reasonably never even notice there’s a function there, or a domain.

And that’s fine. A tool can be so useful it disappears. Attend a play; the stage is in light and the audience in darkness. The roles the light and darkness play disappear unless the director chooses to draw attention to this choice.

And to be honest, integers are a lousy domain for functions. It’s achingly hard to prove things for functions defined just on the integers. The easiest way to do anything useful is typically to find an equivalent problem for a related function that’s got the real numbers as a domain. Then show the answer for that gives you your best-possible answer for the original question.

If all we want are the positive integers, we put a little superscript + to our symbol: I+ or J+ or Z+. That’s a popular choice if we’re using the integers as an index. If we just want the negative numbers that’s a little weird, but, change the plus sign to a minus: I.

Now for some trouble.

Sometimes we want the positive numbers and zero, or in the lingo, the “nonnegative numbers”. Good luck with that. Mathematicians haven’t quite settled on what this should be called, or abbreviated. The “Natural numbers” is a common name for the numbers 0, 1, 2, 3, 4, and so on, and this makes perfect sense and gets abbreviated N. You can double-brace the left vertical stroke, or the diagonal stroke, as you like and that will be understood by everybody.

That is, everybody except the people who figure “natural numbers” should be 1, 2, 3, 4, and so on, and that zero has no place in this set. After all, every human culture counts with 1 and 2 and 3, and for that matter crows and raccoons understand the concept of “four”. Yet it took thousands of years for anyone to think of “zero”, so how natural could that be?

So we might resort to speaking of the “whole numbers” instead. More good luck with that. Besides leaving open the question of whether zero should be considered “whole” there’s the linguistic problem. “Whole” number carries, for many, the implication of a number that is an integer with no fractional part. We already have the word “integer” for that, yes. But the fact people will talk about rounding off to a whole number suggests the phrase “whole number” serves some role that the word “integer” doesn’t. Still, W is sitting around not doing anything useful.

Then there’s “counting numbers”. I would be willing to endorse this as a term for the integers 0, 1, 2, 3, 4, and so on, except. Have you ever met anybody who starts counting from zero? Yes, programmers for some — not all! — computer languages. You know which computer languages. They’re the languages which baffle new students because why on earth would we start counting things from zero all of a sudden? And the obvious single-letter abbreviation C is no good because we need that for complex numbers, a set that people actually use for domains a lot.

There is a good side to this, if you aren’t willing to sit out the 150 years or so mathematicians are going to need to sort this all out. You can set out a symbol that makes sense to you, early on in your writing, and stick with it. If you find you don’t like it, you can switch to something else in your next paper and nobody will protest. If you figure out a good one, people may imitate you. If you figure out a really good one, people will change it just a tiny bit so that their usage drives you crazy. Life is like that.

Eric Weisstein’s Mathworld recommends using Z* for the nonnegative integers. I don’t happen to care for that. I usually associate superscript * symbols with some operations involving complex-valued numbers and with the duals of sets, neither of which is in play here. But it’s not like he’s wrong and I’m right. If I were forced to pick a symbol right now I’d probably give Z0+. And for the nonpositive itself — the negative integers and zero — Z0- presents itself. I fully understand there are people who would be driven stark raving mad by this. Maybe you have a better one. I’d believe that.

Let me close with something non-controversial.

These are some sets that are too important to go unmentioned. But they don’t get used much in the domain-and-range role I’ve been using as basis for these essays. They are, in the terrain of these essays, some rubble.

You know the rational numbers? They’re the things you can write as fractions: 1/2, 5/13, 32/7, -6/7, 0 (think about it). This is a quite useful set, although it doesn’t get used much for the domain or range of functions, at least not in the fields of mathematics I see. It gets abbreviated as Q, though. There’s an extra vertical stroke on the left side of the loop, just as a vertical stroke gets added to the C for complex-valued numbers. Why Q? Well, “R” is already spoken for, as we need it for the real numbers. The key here is that every rational number can be written as the quotient of one integer divided by another. So, this is the set of Quotients. This abbreviation we get thanks to Bourbaki, the same folks who gave us Z for integers. If it strikes you that the imaginary French mathematician Bourbaki used a lot of German words, all I can say is I think that might have been part of the fun of the Bourbaki project. (Well, and German mathematicians gave us many breakthroughs in the understanding of sets in the late 19th and early 20th centuries. We speak with their language because they spoke so well.)

If you’re comfortable with real numbers and with rational numbers, you know of irrational numbers. These are (most) square roots, and pi and e, and the golden ratio and a lot of cosines of angles. Strangely, there really isn’t any common shorthand name or common notation for the irrational numbers. If we need to talk about them, we have the shorthand “R \ Q”. This means “the real numbers except for the rational numbers”. Or we have the shorthand “Qc”. This means “everything except the rational numbers”. That “everything” carries the implication “everything in the real numbers”. The “c” in the superscript stands for “complement”, everything outside the set we’re talking about. These are ungainly, yes. And it’s a bit odd considering that most real numbers are irrational numbers. The rational numbers are a most ineffable cloud of dust the atmosphere of the real numbers.

But, mostly, we don’t need to talk about functions that have an irrational-number domain. We can do our work with a real-number domain instead. So we leave that set with a clumsy symbol. If there’s ever a gold rush of fruitful mathematics to be done with functions on irrational domains then we’ll put in some better notation. Until then, there are better jobs for our letters to do.

A Venn Diagram of the Real Number System


I’m aware that it isn’t properly exactly a Venn diagram, now, but the mathematics-artist Robert Austin has a nice picture of the real numbers, and the most popular subsets of the real numbers, and how they relate. The bubbles aren’t to scale — there’s just as many counting numbers (1, 2, 3, 4, et cetera) as there are rational numbers, and there are far more irrational numbers than there are rational numbers — but if you don’t mind that, then, this is at least a nice little illustration.

RobertLovesPi.net

real number system

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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.