Why Euler's identity is not magic

People call $e^{i\pi} + 1 = 0$ beautiful because it connects five constants that seem to have nothing to do with each other. That framing makes it sound like a coincidence. It is closer to a tautology, and the real question is why the definition that makes it a tautology is the right one.

The exponential as a definition

Define $\exp$ by its power series:

$$\exp(z) = \sum_{n=0}^{\infty} \frac{z^n}{n!}$$

This converges for every complex $z$, and it is the unique function with $\exp' = \exp$ and $\exp(0) = 1$. Plug in $z = i\theta$ and separate the even and odd terms:

$$\exp(i\theta) = \sum_{k} \frac{(-1)^k \theta^{2k}}{(2k)!} + i \sum_{k} \frac{(-1)^k \theta^{2k+1}}{(2k+1)!} = \cos\theta + i\sin\theta$$

So $\exp(i\pi) = \cos\pi + i\sin\pi = -1$. There is no step where anything surprising happens.

Where the content actually lives

The content is in the claim that the series definition of $\exp$ deserves the name “exponential” at all. It agrees with $e^x$ on the reals, it satisfies $\exp(a+b) = \exp(a)\exp(b)$, and it turns the additive group of the imaginary axis into the multiplicative group of the unit circle. That last fact is what physics uses constantly, and it is why the identity feels deep. The depth is in the isomorphism, not in the digits.