You probably haven't heard of the p-adic numbers, but they are full-fledged number systems on par with the real numbers—and given there's a p-adic number system for every prime p, they outnumber ℝ infinity to one! They're also weird and wild landscapes for which Alice's Adventures in Wonderland provides a better guide than common sense does: big becomes small, short becomes long, and geometry can be described but not easily drawn. In this talk we'll meet the p-adics and see how p-adic calculus makes short work of testing for convergence of infinite series in a way calculus students only dream about. We'll also see how the nice properties of p-adic numbers led to them conquering number theory in the 20th century (spoiler: they played a part in Wiles' proof of Fermat's Last Theorem).