Euler's number (idea)

e is generally defined as
lim(n -> inf) (1 + 1/n)^n
Alternately you could write
e=sum(1/n!)
Where the sum is clearly convergent for lim(n!/(n+1)!) = 0. The two definitions are of course equivalent.

Actually something a little more general is done . The WU below tries to explain how x^r is defined when r is real. Note that for rational and integral r we have no problem. When r is irrational the situation becomes a little problematic. So this is what we do.

We start by defining
e^x = sum(xⁿ/n!)
Where the power series has clearly has a radius of convergence equal to infinite.

We then show that this power series expansion satisfies common properties of exponentials such as exp(a+b)=exp(a)*exp(b) etc. After this we define an inverse to the exponential function and call this inverse log(x). Finally we define
x^r = e^r*log(x).
There. Thats it!