How is the number e defined?

There are several useful representations of the constant e.

1. e = the unique number a such that if f(x) = a^x, then f'(x) = a^x.

2. e = lim(x->infinity)(1 + 1/x)^x

3. e = the infinite sum 1/0! + 1/1! + 1/2! + 1/3! + ...

All three of these representations can be shown to be equal.

In base 10, e is approximately 2.718281828.