Continued Fractions and Their Generalizations: A Short History of f-expansions

The subject of representations of real numbers is an old one. Decimal expansions, in particular, have some rather deep problems associated with them: since the decimal expansion of an irrational number does not repeat we are left wondering if we really “know” this number at all. An overly simplistic description says that the sequence of decimal digits in such a number appears “random” or “unpredictable”. The second description is false (we can, after all, produce deterministic algorithms that produce each digit in turn for numbers such as \(\sqrt{2}\)) and the jury is out on the first. About the best we can typically hope for is that the expansion is normal, i.e. that each digit occurs with the same frequency as the others. Even here the list of known normal numbers is rather small compared with the set of irrationals. So is there a better way of representing irrationals? Indeed there is — continued fraction expansions of the form \[a_0+\cfrac{b_1 }{a_1+\cfrac{b_2}{a_2+\dotsb}}.\] In fact, an early (Babylonian?) result is that \[ \sqrt{2}=\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\dotsb}}},\] an entirely predictable representation connected with rational approximations to \(\sqrt{2}\) as well as solutions to Pell’s equation \(x^2-2y^2=\pm 1\). Even more amazing is the fact that the first rigorous proof (J. H. Lambert, 1761) of the irrationality of \(\pi\) used the following beautiful expansion of \(\tan(x)\): \[ \tan\left(\frac{m}{n}\right) = \cfrac{m}{n-\cfrac{m^2}{3n-\cfrac{m^2}{5n-\cfrac{m^2}{7n-\dotsb}}}}.\]

The book under review is a somewhat terse precis of all past research into c.f. expansions. Since iterates of the transformation \(T(x)=\frac{1}{x}-\left[\frac{1}{x}\right]\) (for \(0<x<1\) play a large role in determining the sequence of partial convergents, one might expect right away that ergodic theory and dynamical systems would play a large role in this research. The story begins with work of Lambert, Legendre and Gauss and continues to more modern work of the Russian school (Kuzmin and many others). Invariant measures and entropy also make appearances in the latter chapters. In all, 164 pages bring us up to current research into the subject of continued fractions and fibered systems. There is little mention of applications beyond analysis and, as mentioned before, the delivery is terse but thorough. Those looking for an easy road into the subject might well prefer to start with C. D. Olds’ classic book (which is number 9 in the MAA New Mathematical Library). Those looking for a brisk introduction in this delightful subject need look no further than the one under review. 

Jeff Ibbotson holds the Smith Teaching Chair in Mathematics at Phillips Exeter Academy.