This paper is meant for the reader who knows something about continued fractions, and wishes to know more about the theory of chaotic dynamical systems; it is also useful for the person who knows something about chaotic dynamical systems but wishes to see clearly what the effects of numerical simulation of such a system are. This paper is not purely introductory, however: there are new dynamical systems results presented here and also in the companion paper [6], which contains some discussion of dynamical reconstruction techniques and dimension estimates.