Why does an imaginary exponent correspond to the argument of a complex number? More importantly, why does an exponential function with an imaginary input output trigonometric curves?

Question

Why does an imaginary exponent correspond to the argument of a complex number?  More importantly, why does an exponential function with an imaginary input output trigonometric curves?

anonymous · Answer

I know $$e^{i\theta}=i\sin \theta + \cos \theta$$and it's multiple derivations.  But is there a intuitive way to think about this relationship?

anonymous · Answer

could be proved by Euler's Formula

anonymous · Answer

I got to know about it through taylor series about zero (the derivation)

anonymous · Answer

Again, I know Euler's formula, and I know how it's derived and verified, but is there an intuitive way to think about this relationship?

anonymous · Answer

I would say not, although someone is supposed to have said that anyone who cannot see it will never make a mathematician.

Personally, I dislike the way this whole area is approached in mathematics, to me it is simply a number system (an algebra) to which has been appended the element i along with some rules for operating with it.

There is another algebra which does away with the somewhat artificial distinctions between real and complex numbers, vectors and points, directions and positions and that algebra includes various "i" elements that square to -1, all quite "naturally".

anonymous · Answer

What is this alternative algebra that you speak of?

unklerhaukus · Answer

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