For p a prime, the p-adics are a number system that lets us apply geometric concepts like distance and curvature to the study of congruences modulo powers of p. Formally, the p-adics behave in many ways like the real numbers — for example, they have no “holes” — but, as one moves deeper into p-adic geometry, the fractal objects that appear rapidly divorce from our usual physical intuition. Despite this intuition gap, in the past 60 years, p-adic geometry has played an outsized role in the resolution of many fundamental questions in number theory about symmetries and primes. In this talk, I will give an example-based history of calculus and differential geometry over the p-adics, highlight some of the applications of p-adic geometry to number theory, and then explain some of my work that reintroduces calculus and geometric intuition into the study of perfectoid spaces and related exotic objects in modern p-adic geometry that, until recently, were inaccessible by differential methods.