Comments on Euler's Formula
Vishnu Gupta said:
It unifies algebra, trigonometry, complex numbers, and calculus.
I vividly remember my maths teacher impressing us upon what you stated above. 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.
Cuspy Code said:
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.
User23 said:
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.
Lumost said:
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?
Rnhmjoj said:
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.