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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 $

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cs.toronto.edu/~yuvalf/… –  kush Nov 9 '12 at 17:25

2 Answers 2

up vote 6 down vote accepted

Try Michael Spivak's Calculus. I find it amusing that he would prove the transcendence of $ e $ in a calculus textbook.

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I thought the same exact thing! And he also devotes an entire chapter to the mere transcendence of $\pi$. –  Taylor Martin Nov 9 '12 at 18:41
Oh... I didn't realize that he also proved the transcendence of $ \pi $ in the same book. It is a beautiful book. –  Haskell Curry Nov 9 '12 at 19:33

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.

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$+1$ very nice Amy –  Babak S. Aug 10 '13 at 10:12

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