Now that we’ve got continued fractions under our belt, from chapter 10, we can go on and start looking at “Approximation of Irrationals by Rationals”, chapter 11. One of the (many) cool things about continued fractions, is that they provide “best” rational approximations. We decided, yet again, to split the chapter into two weeks.
In our meeting today, discussing the content of 11.1-11.9, we spent most of our time trying to sort out some typos and see how a few of the inequalities came about. In particular, a typo on page 211, in the theorem that at least one in three consecutive convergents is particularly close to a starting irrational, took us quite a while to sort out.
Eric brought up a comment from the chapter notes that is quite fascinating. The first several sections talk about “the order of an approximation”. Given an irrational , is there a constant (depending on ) so that there are infinitely many approximations with ? This would be an order approximation. In theorem 191, they show that an algebraic number of degree (solution to polynomial of that degree) is not approximable to any order greater than (which seems to be a slightly weaker (by 1) statement than Lioville’s Approximation Theorem). The note Eric pointed out was about Roth’s theorem which states that, in fact, no algebraic number can be approximated to order greater than 2. According to the Mathworld page, this earned Roth a Fields medal.
This reminded me about some things I had seen about the irrationality measure of a number. Roth’s theorem, reworded, says something like: every algebraic number has irrationality 1 (in which case it is rational) or 2. So if a number has irrationality measure larger than 2, you know it is transcendental. Apparently, finding the irrationality measure of a particular value is quite a trick. According to the Mathworld page, has irrationality measure 2, so you can’t use that to decide about it being transcendental.
The whole thing is interesting, as pointed out in H&W, because you think of algebraic numbers as sort of nice (it doesn’t get much nicer than polynomials), but, in terms of rational approximations, they are the worst.