using e^iw is just a notation.Suppose you have a signal
x(t)=Acos(fct).In frequency domain you represent it with a delta function at f=fc(X-axis) and height A.
Instead if x(t) is written as Real part of(e^iw),then in frequency domain you have two delta functions one representing a positive frequency and another a negative frequency both with amplitudes A/2.
The use of e^jw is just to simplify the integrations encountered and to get a reasonable mathematical frequency domain description.
exp(jx) is a point on unit circle. It is two dimentional vector having magnitude 1
When this vector is multiplied with any other it changes other vectors direction
(we call it phase shift) magnitude remain same. It is 1 at angle Θ.
In fourier transform we apply phase shift to each vector and then sum it up to have domain transform
Complex numbers are just vectors (not scalers) dont treat them as real number.
better understood as magnitude and phase ir r(exp(jΘ))
r = magnitude Θ = direction
Properties of calculus applies to natural no e and not 10, Evident from first equation