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.

Stable Marriages and Designing Markets


A few days ago Jeremy Kun with the Math ∩ Programming blog wrote about the problem of stable marriages, by which here is meant pairing off people so that everyone is happy with their pairing. Put like that it almost sounds like the sort of thing people used to complain about in letters to Ann Landers about mathematicians doing foolish things — don’t mathematicians know that feelings matter in this, and, how does this help them teach kids to do arithmetic.

But the problem is just put that way because it’s one convenient representation of a difficult problem. Given a number of agents that can be paired up, and some way of measuring the collection of pairings, how can you select the best pairing? And what do you mean by best? Do you mean the one that maximizes whatever it is you’re measuring? The one that minimizes it (if you’re measuring, say, unhappiness, or cost, or something else you’d want as little of)? Jeremy Kun describes the search for a pairing that’s stable, which requires, in part, coming up with a definition of just what “stable” means.

The work can be put to describe any two-party interaction, which can be marriages, or can be the choice of people where to work and employers who to hire, or can be people deciding what to buy or where to live, all sorts of things where people have preferences and good fits. Once the model’s developed it has more applications than what it was originally meant for, which is part of what makes this a good question. Kun also write a bit bout how to expand the problem so as to handle some more complicated cases, and shows how the problem can be put onto a computer.

Math ∩ Programming

Here is a fun puzzle. Suppose we have a group of 10 men and 10 women, and each of the men has sorted the women in order of their preference for marriage (that is, a man prefers to marry a woman earlier in his list over a woman later in the list). Likewise, each of the women has sorted the men in order of marriageability. We might ask if there is any way that we, the omniscient cupids of love, can decide who should marry to make everyone happy.

Of course, the word happy is entirely imprecise. The mathematician balks at the prospect of leaving such terms undefined! In this case, it’s quite obvious that not everyone will get their first pick. Indeed, if even two women prefer the same man someone will have to settle for less than their top choice. So if we define happiness in this naive way…

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How I Make Myself Look Foolish


It seems to me that I need to factor numbers more often than most people do. I can’t even attribute this to my being a mathematician, since I don’t think along the lines of anything like mathematical work; I just find that I need to know, say, that 272,250 is what you get by multiplying 2 and 3 to the second power and 5 to the third power and 11 to the second power. And I reliably go to places I know will do calculations quickly, like the desktop Calculator application or what you get from typing mathematical expressions into Google, and find that since the last time I looked they still haven’t added a factorization tool. I have tools I can use, particularly Matlab or its open-source work-just-enough-alike-to-make-swapping-code-difficult replica Octave, which takes a long time to start up for one lousy number.

So I got to thinking: I’ve wanted to learn a bit about writing apps, and surely, writing a factorization app is both easy and quick and would prove I could write something. The routine is easy, too: take a number (272,250) as input; then divide by two as many times as you can (just one, giving 136,125), then divide by three as many times as you can (twice, giving 15,125), then by five as many times as you can (three times, reaching 121), then by seven (you can’t), then eleven (twice, reaching 1), until you’ve run the whole number down. You just need to divide repeatedly by the prime numbers, starting at two, and going up only to the square root of whatever your input number is.

Without bothering to program, then, I thought about how I could make this a more efficient routine. Figuring out more efficient ways to code is good practice, because if you think long enough about how to code efficiently, you can feel satisfied that you would have written a very good program and never bother to actually do it, which would only spoil the beauty of the code anyway. Here’s where the possible inefficiency sets in: how do you know what all the prime numbers up to the square root of whatever you’re interested in is?

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