This is the first of three chapters which originated in presentations given by Alf van der Poorten a few years before his death. As such they should be read like informal lectures and mined for their nuggets of gold.

Continued fractions of algebraic numbers

As we might expect by now, there is much more we can say about the continued fractions of quadratic irrationalities. First we look further in a more general way at algebraic numbers.

In spite of the expected unbounded behaviour of the continued fraction expansion of an algebraic non-quadratic irrational, there is a simple algorithm to compute its expansion. Indeed, it is quite straightforward [42, 95, 141] to find the beginning of the expansion of a real root of a polynomial equation.

Example 4.1 We illustrate this for the polynomial f(X) = X3 − X2 − X − 1. Then f has one real zero, say α, where 1 < α < 2. So a0 = 1 and α1 = 1/(α − a0) is a zero of the polynomial f1(X) = −X3f(X−1 + a0) = 2X3 − 2X − 1.