From: utzoo!decvax!ucbvax!G:shallit Newsgroups: net.math Title: Re: puzzle Article-I.D.: populi.190 Posted: Sat May 29 21:51:57 1982 Received: Sun May 30 03:00:33 1982 To answer your questions, it is not hard to prove that the continued fraction for x terminates if and only if x is rational. One way to see this is to associate CF's with the Euclidean algorithm for GCD; the two algorithms are sort of "mirrors" of each other, since in one you are interested in remainders; in the other, the quotients. Second, if the CF for x is eventually periodic, then x is the root of a quadratic equation. The converse is also true. Third, it is true that e = [ 2, 1, 2, 1, 1, 4, 1, 1, 6, ... ]. This representation is due to Euler and Hurwitz. There are similar expansions of interest for exp (1/m) where m is a positive integer and related transcendental quantities. Two books that have interesting things about CF's are "Continued Fractions by C. D. Olds, and Knuth, Art of Computer Programming, V. II. /Jeff Shallit, Department of Mathematics, University of California, Berkeley.