Annotated geometric proof that Euler's number is irrational

fermatslibrary.com

29 pointsluisb6 comments

This reminds me the shortest proof of irrationality of e I know.

http://i.imgur.com/uEXJkF7.png

This actually seems too complicated. If e = a/b, then if we multiply by b! then the right side is an integer, while the left side is an integer plus a fractional tail 0 < 1/(b+1)+1/(b+1)(b+2)+... < sum{1/(b+1)^n} = 1/b.

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.

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!)

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.

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

Reply on news.ycombinator.com