Continued Fractions

The continued fraction representation of a real r is a stream $[s_0, s_1, s_2, \dots, s_i, \dots]$ of integers such that:

$$r = \lim_{i\rightarrow \infty} \; s_0 + \frac{\textstyle{1}}... ...\textstyle{\ddots + \frac{\textstyle{1}}{\textstyle{s_i}}}}}}}}$$

One of the benefits of the continued fraction is that some numbers which are complicated to represent using a digit notation have remarkably simple continued fraction representations. For example:

$$\begin{array} {ll} \phi = \frac{1+\sqrt{5}}{2} & = [1,1,1,1,... ...e & = [2,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12, \dots]\end{array}$$

The continued fraction representation is incremental. Jean Vuillemin [31] discusses the performance of an implementation using this representation.