
Annotated geometric proof that Euler's number is irrational - luisb
http://fermatslibrary.com/s/a-geometric-proof-that-e-is-irrational
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m1el
This reminds me the shortest proof of irrationality of e I know.

[http://i.imgur.com/uEXJkF7.png](http://i.imgur.com/uEXJkF7.png)

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conceit
Is that the sum of _( -1 )^n / ( n! )_ or _( ( -1 )^n / n )!_ and maybe the
tranformation is trivial, but why does this equal e^(-1)?

Edit: I guess the factorial of a fraction doesn't make much sense.

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baghira
You can definitely extend the definition of factorial to the complex numbers
using Euler's Gamma function, but the notation is different (indeed ! is
reserved for integers).

As far as the second question goes, that's taken to be the definition e^x :=
\sum_{n=0}^{\infty} (x^n)/(n!)

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bsder
These are cute, but I'm pretty sure that the ancient Greeks had the ability to
prove quite a few of these numbers were irrational.

Transcendental proofs, on the other hand, took a while.

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Grue3
Ancient Greeks did not know _e_. However the person who first came up with it
(Jacob Bernoulli?) most certainly could prove it's irrational.

