Vaughn Climenhaga

Having played around with continued fractions for quite some time, I've come to appreciate these fascinating mathematical objects, which usually receive short shrift in the standard undergraduate curriculum.  This is a brief introduction to these wonderful creatures which I wrote in order to put my thoughts in order for a talk during the Penn State REU in summer 2007.  Since giving that lecture, I have updated the notes slightly to include my belated realisation that decimal and continued fraction expansions alike are simply specialisations of the Cauchy sequence construction, so that what is offered here is not a different construction of the real numbers, but a different way of representing the reals, which has many benefits in the realm of rational approximations.

Real numbers, continued fractions, and rational approximations