Is zero even?

Did you know that there are actually things to say about whether zero is even or odd (from Wikipedia)? Obviously it is, but the math-ed folks have seriously looked at this.

I found this via a comment by John Thacker at The Volokh Conspiracy. There's a poll there; right now 2% of people have said 0 is odd, 51% even, 43% both, 4% neither. I can kind of understand what's going on with people saying "neither" (perhaps they're getting this from some elementary-school notions), but how is 0 odd?

My answer: yes, zero is even, because it's twice an integer.

(Or because the identity permutation on n letters is an element of the alternating group A_n -- I've been thinking about permutations a lot lately. But if you understand that, you probably are like me, think zero is even, and didn't even think there was anything to discuss.)

Incidentally, sometime recently -- I forget the context -- I saw something that referred to the Gaussian integer a+bi as "uneven" if and only if a and b had different parity.

Economic impact of mathematics?

Tim Gowers wrote in The Importance of Mathematics:

If you were to work out what mathematical research has cost the world in the last 100 years, and then work out what the world has gained, in crude economic terms, then you would discover that the world has received an extraordinary return on a very small investment.

I don't doubt this. But has anyone actually tried to do this? (And would the numbers even be meaningful?)

Eponyms in mathematics

Let S be the standard Smith class of normalized univalent Matcuzinski functions on the unit disc, and let B be the subclass of normalized Walquist functions. We establish a simple criterion for the non-Walquistness of a Matcuzinski function. With this technique it is easy to exhibit, using standard Hughes-Williams methods, a class of non-Walquist polynomials. This answers the Kopfschmerzhaus-type problem, posed by R. J. W. (“Wally”) Jones, concerning the smallest degree of a non-Walquist polynomial.

This fake abstract of a paper is from Merv Henwood and Ivan Rival, Eponymy in Mathematical Nomenclature: What's in a Name, and What Should Be? (PDF), from the Mathematical Intelligencer in 1980. It sounds to me like slight caricature -- but only slight. Henwood and Rival point out that such names are lazy. Names have at least two important functions -- to describe and to label -- and eponyms only label.

Perhaps such abstracts would be more common in areas which are small enough that all the major players talk to each other. I imagine that Smith, Matcuzinski, Walquist, etc. know each other.

Also of interest is David Rusin's list of eponyms occurring in the MSC classification. These names in general seem a bit less obscure than the names one would find in the abstract of a random paper, which isn't surprising as they're names of concepts big enough to get areas named after them.

(And can someone confirm or refute the story that Banach, in the paper in which he introduced Banach spaces, called them "spaces of type B" in an effort to get them named after himself? I've heard this one a few times but always unsourced.)

Steen on mathematics and biology

Here's a fascinating article on what math is good for in biology: The "Gift" Of Mathematics in the Era of Biology, by Lynn Arthur Steen. Steen gives lots of examples about what math is good for in biology. Somewhat surprisingly to me, he doesn't really mention one of the first things that came to mind, namely the use of combinatorial techniques to study the genome, which is nothing but a word on a four-letter alphabet. It's possible that he subsumes this in "statistics", though; to take a simple example, one might want to know how many times a certain sequence of bases would appear in a "random" genome in order to determine whether the fact that such a pattern appears often is signal or noise. Still, he makes the point that while the traditional mathematics curriculum (with lots of calculus and differential equations) takes its scientific inspiration from physics, biology is ascending.

A shorter version of this article is available at The Chronicle of Higher Education.

(How did I find this? Steen was one of the authors of Counterexamples in Topology, which I mentioned yesterday, so I went over to his web site.)

Perfection "squared" on standardized tests

I came across an article about a student who got a perfect score on both the ACT and the SAT. (These are the two standardized tests used for university admissions in the US; generally schools on the coasts use the SAT and schools in the interior of the country use the ACT, although this is a vast generalization. The geographical separation seems to be a function of where the tests originated, in Iowa and New Jersey respectively.

This article (which I'm not linking to because I found it by googling a student, and the student is probably already not happy that this is all over the Internet) points out that less than 1 percent of students get a perfect score on each of these tests. (As you'll see below, this is quite an understatement.) I think we're supposed to come to the conclusion that less than 1 in 10000 students would get a perfect score on both.

But of course scores on these tests are positively correlated! So the probability of getting a perfect score on both tests is much higher than the product of the probability of getting a perfect score on each. (I don't think knowing that would help you on the SAT. But it's been a while. In my day they were out of 1600.)

This article indicates that 294 of the high school seniors graduating in 2008 got a perfect score on the SAT, and 514 out of 1.4 million got a perfect score on the ACT. Wikipedia puts the number of SAT takers at 1.5 million per year; let's knock this down to 1 million since some people take the test more than once and we're talking about the total number of students. So the probability that a random student who takes both tests gets a perfect score on both is something like (294/1000000) (514/1400000), which is about one in 1.3 million. The number of students taking both tests is less than this (many people only take one of the two), so assuming independence there should be less than one student per year who gets a perfect score on both tests.

But a quick glance at the Google results will convince you that there are a few students per year who pull this off.

Counterexamples in X

Counterexamples in Probability And Statistics (Joseph P. Romano and A. F. Siegel) and Counterexamples in Probability and Real Analysis (Gary L. Wise and Eric B. Hall) both seem to be books in the tradition of Counterexamples in Analysis (Bernard Gelbaum and John Olmsted) and Counterexamples in Topology (Lynn Arthur Steen and J. Arthur Seebach. These are books that collect the examples just "outside" the boundaries of the various standard theorems, the point being to explain why one needs the seemingly strange collections of hypotheses that seem to begin every analytic theorem. (Hence the tags "education" and "teaching"; I've often seen these counterexample books described as "anti-textbooks", and as being complementary to standard textbooks which often spend most of their time telling you what's true.)

It seems that these books are concentrated on the analytic end of mathematics; I couldn't find, for example, books of counterexamples in algebra, combinatorics, or number theory. There is, however, Theorems and Counterexamples in Mathematics. My sense is that the nonexistence of these books is connected to the fact that those fields don't seem quite as rife with theorems where all the work is hidden in the definitions.

The hidden mathematics of bathrooms

From the xkcd blog: urinal protocol vulnerability.

The basic premise here is the following: there's a long row of urinals (n of them), and a line of men who want to use them. The first man picks a urinal at the end. Each man after that picks one of the urinals which is the furthest from any of the occupied urinals. Nobody ever leaves. How many men have to show up before one of them will be forced into using a urinal adjacent to one that's already occupied? Call this number f(n).

You might think that f(n)/n approaches some limit, but it doesn't; it oscillates between 1/3 and 1/2 based on the fractional part of log₂ n. If n = 2^k + 1 then this "greedy" algorithm for filling the urinals works and every other urinal gets filled: f(2^k + 1) = 2^k-1 + 1. If n = 3 x 2^k-1 + 1 then the worst possible thing happens and only every third urinal gets filled, and f(3 x 2^k-1 + 1) = 2^k-1 + 1. (Yes, that's the same number, and the function's constant in between.) f(5) = f(6) = f(7) = 3, f(9) = ... = f(13) = 5, and so on.) Oscillations like this -- periodic in the logarithm of the problem size -- happen a lot in problems involving binary trees counted by the number of nodes. Still, it was a bit surprising to see this, because I'd never thought about the problem in the case of "unphysically" large n.

Exercise for the reader: invent a mathematically equivalent version of this problem that doesn't involve urinals.