The square root of 3 and mock theta functions. (No, they're not connected.)

Mark Dominus writes about why it really isn't so mysterious that Archimedes had the approximation √3 = 265/153 (which is correct to four decimal places). Apparently historians of mathematics have been mystified by this. Dominus points out that tabulating n² and 3n² for the first few hundred integers would be enough. And it might even be enough to go up to 100 or so, observing where n² and 3m² are close to each other (which gives an approximation √3 ~ n/m), guess the pattern (it comes from the continued fraction of √3, but Archimedes didn't need to know that), and extrapolate. He suggests that's because the historians themselves weren't so good at arithmetic. Many of these historians date from the late 19th and early 20th century, after when mathematics generally turned more abstract and before computers existed, so it's plausible. If I were a historian I'd have something serious and insightful to say about this.

A more general question: if you're trying to work out the history of mathematics by examining the original sources, how important is it to be a good mathematician? I saw a lecture by George Andrews last week on Ramanujan's lost notebook; he and Bruce Berndt are working on an edited version of it (first volume, second volume, review of first volume in the October 2006 Bulletin of the AMS). Andrews happened to be looking through some papers at the library of Trinity College, Cambridge, when he came across these papers. The manuscript conventionally called "Ramanujan's lost notebook" consists of many pages of formulas and almost no words and is concerned with mock theta function; Andrews claims that he would not have recognized the significance of what he was looking at had he not wrote a PhD thesis on mock theta functions.

Some partition identities due to Euler

I wrote a couple weeks ago about my love of the identity

which is an analytic version of the fact that every integer is uniquely a sum of distinct powers of 2. What I didn't realize at the time, but learned from George Andrews article Euler's "De Partitio Numerorum", in the current issue (Vol. 44, No. 4) of the Bulletin of the American Mathematical Society, is that this is due to Euler. This is hardly surprising, though, as Euler was as far as I know the first person to apply generating functions to partitions. (The result was known before that; it's the generating-function statement that's due to Euler.)

But here's one I didn't know:

This is an analytic version of the fact that every integer is uniquely a sum of distinct powers of 3 and their negatives, and is Theorem 4 of the Andrews paper. For example, 1 = 1, 2 = 3-1, 3 = 3, 4 = 3+1, 5 = 9-3+1, 6 = 9-3, 7 = 9-3+1, and so on.

To see this fact combinatorially, note that we can make 3^k sums out of the numbers 3⁰, 3¹, ..., 3^k-1; for each of these we include the number, its negative, or neither. If we can show that these are the distinct integers from -l to l, where l = (3^k-1)/2, then that's enough. (For example, when k = 2, we have l = 4, and the nine possible sums of 1, 3, and their negatives are
-3-1, -3, -3+1, -1, 0, 1, 3-1, 3, 3+1
where "0" denotes the empty sum. These are just the integers from -4 to 4.)

But this can be shown by induction on k. Clearly it's true for k = 0. Then, for the induction, consider the 3^k sums of the powers of three up to 3^k-1 and their negatives; by the inductive hypothesis these are just the integers in the interval . Now consider the 3^k+1 sums of the powers of three up to 3^k and their negatives. There are 3^k of these which are just the ones which don't include 3^k. Those that include +3^k add up to , and those which include -3^k add up to . These three intervals put together are , as desired. (For example, when k=2, these three intervals are [-4, 4], [5, 13], and [-13, -5], which together make up [-13, 13].) By induction, we can make each integer in uniquely as a sum of distinct powers of 3 which are less than 3^k and their negatives, which is essentially the desired result.

The last section of the article talks about partitions with some negative parts, or "signed partitions". For example, 9+8-6-5 is a signed partition of 6. These are a bit more awkward to work with in terms of generating functions -- the generating function above about the powers of 3 doesn't converge, ever! So it's basically a curiosity, but one can actually prove things about these signed partitions using generating functions; see the article for examples.