Agreed, the full formula is much more intersting and deep: it shows there is a mapping between the reals and the circle group that preserves the structure (turns addition into multiplication). This concept leads to the very general exponential map of a Lie algebra to a Lie group and Pontryagin duality, which is the essence of Fourier transform.

It unifies algebra, trigonometry, complex numbers, and calculus.

I vividly remember my maths teacher impressing us upon what you stated. By far the best math teacher I had. He didn't work out the proof on the board but made us work it out, took us on a journey, him as a guide, so to speak.

In a way it's analogous to how Einstein's \( E = mc^2 \) is a special case of how the norm of the four-momentum is defined in special relativity, which is \( mc = \sqrt{(E/c)^2 - p^2 \}. For the special case of a stationary object we have \( p = 0 \), so \( E = mc^2 \) follows automatically. But the special case is somehow more memorable and more famous, and I believe something similar has happened with Euler's identity.

Visual Complex Analysis is the text that introduced me to this concept. The functions \( \cos \) and \( \sin \) suddenly look a lot like accessor functions for a two member object.

It's a unification of geometry and algebra in 2 dimensions, but generalizations beyond that are scarce. Effectively we're left with a solution where we can represent any space in dimensions modulo 2.

While quaternions and some higher dimensional complex numbers exist, is there a unified formula expressing Euler's formula for arbitrary numbers of dimensions? Is there one for an infinite dimension space?