Abstract: Dedekind and Cantor gave independent constructions of the real numbers from the rational numbers during the mid to late 1800s. Cantor considered a natural equivalence relation on Cauchy sequences of rational numbers with the resulting cosets representing abstract real numbers. It turns out that Cantor's ideas can be modified and applied to sequences of real numbers - via ultrafilters - in order to give an explicit construction of non standard or hyperreal numbers. The purpose of this talk will be to develop both constructions (reals and hyperreals) and to explain some non standard principles which arise in the process.