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 (turn

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.