No magic required: complex numbers as pairs

Last lesson, you accepted i = √−1 because a mathematician declared it. It works, but it can feel like cheating — as if a new number appeared just because someone wished it into existence.

Maths can do better: build complex numbers from nothing but ordered pairs of real numbers — two numbers written together, like (3, 4). Add a sum rule and a product rule, and i turns out not to be magic at all: it is simply the pair (0, 1).