The title of this post is misleading, because you might think there's a substantial connection between its two halves. The connection is only that I happened to come across both of these things this morning and it seemed silly to make separate posts about them.
Todd Trimble gives two proofs of the continued fraction expansion for e, namely e = [2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, ...]. This is in the usual notation for continued fractions, so it actually means
e = 2 + 1/(1 + 1/(2 + 1/(1 + 1/(1 + 1/(4 + ...)))))
but all the extra 1s obscure the pattern. This is one of those things that it's much easier to state than it is to prove. And I must admit it's always seemed a bit strange to me that e has such a nice continued fraction expansion, while π doesn't -- e has a special place in continued-fraction land.
From the arXiv, via the physics arXiv blog: A Monte Carlo Approach to Joe DiMaggio and Streaks in Baseball, by Samuel Arbesman and Steven Strogatz, which is what it sounds like. This expands upon this piece in the New York Times on the same subject, which I wrote about back in March. And no, I'm not sure why it's in a physics category at the arXiv.
04 August 2008
Continued fractions and baseball hitting streaks
Posted by Michael Lugo at 7:34 AM
Labels: baseball, links, number theory, probability
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It's a little silly, but I also gave a rather tenuous connection between continued fractions and baseball hitting *averages* in the blog post POW-8 and its solution. I'll also mention, since I have continued fractions on my mind, that An Idelic Life has a new post on musical tuning systems, where continued fractions are used to explain 12-tone and 53-tone scales (the latter being used in some forms of Turkish, Arabic, and Indian music).
As for why e has a nice continued fraction whereas pi doesn't: putting aside the fact that there are some cute *non-regular* (or non-simple) continued fractions for pi, I don't have any especially great answer, but my friend James Dolan has a kind of interesting take on related matters. The rough slogan is something like, "there is a bijective correspondence between methods for solving differential equations and ways of defining e," in the sense that if you take a method for solving an ODE (power series, fixed-point methods, separation of variables, etc.) and apply it to y' = y, you get a corresponding definition of e. According to that philosophy, the continued fraction definition of e should correspond to a continued fraction method for solving (at least certain classes of) ODE, like perhaps the Riccati equation. That would be more an answer to the question "who ordered e?" rather than "why didn't anyone order pi?", but I think it's an interesting point of view.
The 'physics' arxiv is only for people who are considered crackpots by the establishments. Mathematicians cannot really appreciate the social machinations here.
Strogatz is most definitely not a crackpot.
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