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!)