You've heard about real numbers– but what about hyperreal numbers? In the 1960s, logician Abraham Robinson discovered that, by an unexpected application of model theory, one could construct a field which satisfies all the same first-order statements as the real numbers, but additionally contains elements both infinitely large and infinitely small. Using these "hyperreal numbers", Robinson developed the field of nonstandard analysis, in which these exotic elements allow quick, intuitive proofs of familiar results in real analysis. In this talk, we'll follow the construction of the hyperreals as an ultrapower of $\mathbb R$ modulo a nonprincipal ultrafilter on the natural numbers, and showcase a few of the slicker proofs of a nonstandard flavor. The question of whether all calculus textbooks should be rewritten is left as an exercise.