Advanced middle-school level video made for 3Blue1Brown's Summer of Math Exposition (SoME). It's about intuitively understanding why exponentiating an imaginary number should yield a periodic function: exp(ix)=cos(x)+i sin(x). It's because (-1)^n oscillates, but in discrete steps. To make the oscillation continuous, we have to take square roots of -1 (which is where the imaginary number i comes from), and take the continuous limit (which is where the natural exponential exp(x) comes from). The result is useful in practice, because it allows for straightforward factorization: exp(i(A+B))=exp(iA)exp(iB), but cos(A+B)=/=cos(A)cos(B).