Kids are better at algebra than you think

Word problems take place in a graded ring, from (the recently relocated) Mathematics under the Microscope (Alexandre Borovik), via The Unapologetic Mathematician (John Armstrong).

In short, Borovik claims that elementary school word problems take place over , where the x_i represent different things that could be added. In this formalism, it makes sense to add, say, apples and oranges, going against the usual rule that you're only allowed to add quantities with the same "dimension". (Indeed, Borovik illustrates the idea with an example of this nature.)

I'm reminded of "dimensional analysis" as taught in, say, introductory physics classes, where we only allow monomials to have meaning, namely that the monomial x m^a kg^b sec^c measures some physical quantity with dimensions L^aM^bT^c where L, M, T stand for length, mass, and time. (For example, in the case of speed, a = 1, b = 0, c = -1.) I can't think of situations in physics where one deals with a quantity of the form, say, a kg + b m. Is this because they don't exist, or because I don't know as much physics as some people?

Bringing Down The House

From the MIT News Office: Film loosely based on MIT blackjack team opens Friday. (That's Friday, March 28.) The film is based on the book Bringing Down The House. I haven't seen it, and I probably won't, because it hasn't gotten the best reviews. Also, I generally tend to dislike movies that take place in fictionalized versions of places I know; they look all wrong. (A Beautiful Mind and Good Will Hunting come to mind. Also, 10th and Wolf, which has nothing to do with math, but is set around the corner from where I was born in Philadelphia, and was filmed in Pittsburgh.)

Anyway, the press release says:

Most notably, the character played by Kevin Spacey, portrayed as an MIT professor, is entirely fictional. While his irresponsible acts may enliven the Hollywood script, they are entirely unrepresentative of the Institute.

I wonder if their legal department told them to say that. They also point out that real MIT students are good at math too! This is true.

It's almost baseball season again...

A Journey To Baseball's Alternate Universe, in today's New York Times, by Samuel Arbesman and Steven Strogatz.

Arbesman and Strogatz ask the question: how likely is it that some major league baseball player, at some point, would have had a 56-game hitting streak, as Joe DiMaggio did in 1941?

I've seen attempts to determine this before, but they're usually handwaving things that start out by saying "assume everybody bats .266 and gets 3.83 at-bats a game" (actual averages for the 2007 National League), and then let's compute the probability that such a player has a 56-game hitting streak in any given sequence of 56 games. In this case that's easy; the average player gets has a probability (1-.266)^3.83 = 0.306 of not getting a hit in any given game, thus a probability 0.694 of getting a hit in any given game; raising this to the 56th power tells you that the average player has a probability of 1.31 in a billion of getting hits in, say, the 56 games starting tomorrow and ending sometime in early June.

So what's the expected number of 56-game hitting streaks this season, according to this model? There are 107 ways any given player could get a 56-game streak -- starting in game 1, 2, ..., 107. So the expected number of 56-game streaks for Joe Qankee (yes, I'm reviving the Qankees) is this probability times 107, or 1.41 × 10^-7. Now, assume there are eight Qankees that play every day. (The Qankees are an extraordinarily healthy National League team. The fact that their name rhymes with that of the American League team that DiMaggio actually had his streak with is purely coincidence.) The expected number of 56-game hitting streaks by Qankees this season is thus 1.13 × 10^-6. (Note that this is not the probability that one of them has such a streak. A 57-game streak would get counted twice here, a 58-game streak three times, and so on. However, it is an upper bound for the probability of a Qankee having a streak of at least 56 games.)

Now, there have been something less than three thousand "team seasons" in Major League Baseball (one team playing for one season). So the expected number of 56-games streaks is bounded above by (1.13 × 10^-6) × 3000 = 0.0338, or about one in 300.

But we've had one. That seems like a lot.

What's the problem here? Well, the average player isn't the one that's going to have that streak. A .280 hitter will put together a streak in 7.40 56-game frames out of every billion. A .300 hitter, in 69 out of every billion. A .320 hitter, in 498 out of every billion. (And I'm still assuming such a player only gets 3.83 at-bats a game; that's probably not true, because the player who hits well will lead his team as a whole to have more at bats.) But an equally bad hitter doesn't drag down the expectation nearly as much. I've ignored batting order (which Arbesman and Strogatz did take into account, implicitly; their inputs for each player are the total number of hits, number of games played, and number of plate appearances, and number of plate appearances varies with position in the batting order).

Rather than making some assumptions on how batting averages are distributed (which would probably be wrong, and even if they were right in the peak of the distribution would still be wrong because what really matters is the tails), I'll defer to Arbesman and Strogatz. Their method is to simulate the entire history of baseball 10,000 times, which is enough to get a nice basically-smooth curve for the distribution of the length of the longest streak. The median length of the longest streak, in their simulations, is 53 games.

Simulation might not be entirely necessary, though. It's routine to calculate the distrbution of the length of the longest streaks in sequences of biased coin flips; aggregating that information together is a little harder. But I don't care enough to do it, so I'll stop here.

Furstenberg's topological proof of the infinitude of primes

I just learned about Furstenberg's proof of the infinitude of primes (link goes to Wikipedia article); it was published in 1955 in the American Mathematical Monthly. (Link goes to JSTOR. Full citation: The American Mathematical Monthly, Vol. 62, No. 5. (May, 1955), p. 353. If you have to do any work to get the article from JSTOR -- for example, logging in through a proxy server because you're at home instead of on campus -- or you don't have JSTOR access -- don't worry, you're not missing much. The article's a third of a page, and the Wikipedia article is probably longer than Furstenberg's original.) Surprisingly, I hadn't seen this before.

Anyway, here's my version of the proof: topologize the integers by taking the (doubly infinite) arithmetic sequences as a basis. (The only thing that needs checking here, to show these form a basis, is that the nonempty intersection of two basis elements contains a basis element; this is true since the intersection of two arithmetic sequences is another arithmetic sequence.) Consider the set which consists of all the integers except -1 and 1; this is the union of the sequences pZ over all primes. Now, there are no nonempty finite open sets in this topology, so there are no cofinite closed sets except Z itself. Thus the union of the pZ isn't closed. So it can't be an union of finitely many closed sets, since finite unions of closed sets are closed. So there are infinitely many sets pZ, and thus infinitely many primes!

(Thanks to an anonymous commentator for pointing out some errors.)

Open and closed?

At Language Log there's a post about how English-speakers use "open" and "closed", which are not grammatically the same sort of thing, in opposition to each other -- "open"/"close" or "opened"/"closed" would, on the surface, make more sense. (Compare French ouvert and fermé, which are both past participles.)

I won't try to summarize the linguistic content of the post; I'm not a linguist, although I did go through a phase where that seemed interesting.

But in mathematics-land, open and closed aren't even opposites, in the sense that open means not-closed and closed means not-open. Of course the complement of an open set is closed, and vice versa, but that's a more complicated relationship, because now we're talking about two sets, not one. This is one of about a zillion examples of how we take perfectly good natural-language words and give them specific meanings (group, ring, field, set, class, ...), which may or may not be preferable to making up entirely new words as some other fields (biology comes to mind) prefer.

The Mandelbrot monk

The Mandelbrot Monk, Udo of Aachen (1200-1270), began developing probability theory, computed an approximation of the Mandelbrot set, and like some other people got a suspiciously good approximation for π by stick-throwing.

Stuff mathematicians like?

You probably by now know about the blog Stuff White People Like, which talks about stuff white people like.

By "white people", the blog doesn't mean all white people, but rather white urban twenty- and thirty-somethings with money to burn.

Stuff white people like (according to the blog) that I'd guess a lot of mathematicians also like include coffee (if you don't know why, you're probably not a mathematician), "Gifted" children (lots of mathematicians probably were), Apple products (mostly judging from what sort of computer colloquium and seminar speakers are running, although this could be skewed by the fact that Apple users, being in the minority, are probably more likely to bring their own machine), not arts degrees (arts degrees apparently make people more interesting to talk to at parties, a concept entirely foreign to us), graduate school (although it seems to be humanities PhD programs that they're referring to), and bad memories of high school.

There are also a lot of less-good imitations: Stuff Educated Black People Like, Stuff Lesbians Like, Stuff Gay Guys Like, Stuff College People Like, Stuff Asian People Like, Stuff White Trash People Like, Stuff Unimaginative Bloggers Like, and perhaps others.

But what about "Stuff Mathematicians Like"? I know that I wouldn't do a good job with it -- but someone should. I'll get you started -- the set of stuff mathematicians like includes coffee, Rubik's Cubes, saying things are trivial, being proud of the uselessness of one's work, and the prefix "co-". (So what is "ffee"? Okay, there's another thing mathematicians like -- bad jokes.) Feel free to name other elements of this set.