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Removing the mysteries of e

This is an explanation of why Imaginary powers of e (Complex exponentials) behave in the way that they do. It explains why e^ix identifies a point on a unit-radius circle at an angle of x radians, and why e^ix = cos x + i sin x.

It is an improved re-writing of an earlier explanation that I did in 2019, and it is written to be easy to understand. All the ideas, and the steps leading up to them, are also explained in my book about waves.

The explanation contains ideas that I haven't seen elsewhere. It also contains a reference to the number 4.30453032..., which I hadn't seen anywhere else when I wrote this. The number 4.30453032... is the solution to "x^xi = 1" and "x^x = e^2π".

Download the explanation here [1.8 MB PDF]
Written by me, Tim Warriner, and last edited in September 2021.