Prove the transcendence of the number $e$ How to prove that the number $e=2.718281...$ is a transcendental number?   The truth is I have no idea how to do it. 
If I can recommend a book or reference on this topic thank you.
There are many tests on the transcendence of $ e $?   
I'd read several shows on the transcendence of $ e $

Your might be interested in the Lindemann-Weierstrass-theorem, which is useful for proving the transcendence of numbers, e.g., $\pi$ and $e$.  If you read further, you'll see that the transcendence of both $\pi$ and $e$ are direct "corollaries" of the Lindemann-Weierstrass theorem.
Indeed, $e^x$ is transcendent if $x$ is algebraic and $x \neq 0\,$ (by the Lindemann–Weierstrass theorem).
A sketch of a (much) more elementary proof is given here.
