e = 1/0! + 1/1! + 1/2! + ...
let e = p/q, S(n) = 1/0! + 1/1! + ... + 1/n!
e - S(n) = 1/(n+1)! + 1/(n+2)! + .... < 1/(n+1)! (1 + 1/(n+1) + 1/(n+1)^2 + ...) = 1/((n)(n!))
e - S(n) > 0
0< n n! e - n n! S(n) < 1
Therefore n n! e - n n! S(n) is not an integer
Since e = p/q, and take n > q, then n! e is an integer
So n n! e - n n! S(n) is an integer
So n n! e - n n! S(n) is an integer and is not an integer
Contradiction!
Therefore, e is irrational
(I don't take credit for this proof, its the classic one)