A golden ratio notation for the real numbers

Several methods to perform exact computations on real numbers have been proposed in the literature. In some of these methods real numbers are represented by infinite (lazy) strings of digits. It is a well known fact that, when this approach is taken, the standard digit notation cannot be used. New forms of digit notations are necessary. The usual solution to this representation problem consists in adding new digits in the notation, quite often negative digits. In this article we present an alternative solution. It consists in using non natural numbers as ``base'', that is, in using a positional digit notation where the ratio between the weight of two consecutive digits is not necessarily a natural number, as in the standard case, but it can be a rational or even an irrational number. We discuss in full detail one particular example of this form of notation: namely the one having two digits ($0$ and $1$) and the golden ratio as base. This choice is motivated by the pleasing properties enjoyed by the golden ratio notation. In particular, the algorithms for the arithmetic operations are quite simple when this notation is used.