In particle physics, we study how elementary particles, such as electrons, interact with each other. The force that one particle exerts on another depends crucially on the distance between particles. To measure distances, we use rulers. An implicit assumption in using rulers to measure distances is that space and time coordinates must be real numbers. What would happen if we lived in a world where space and time were *not* real numbers? I will provide an answer to this question by describing a world where space and time coordinates are valued in an exotic number system, rather than real numbers. The space-time in this world has an intriguing structure, leading to a notion of "distance" completely different from the one familiar to us from our everyday lives. Surprisingly, despite these significant differences, I'll show that these worlds are intimately tied together, making us wonder: What is the "correct" notion of space-time, if any?