Definition:Euler's Number

As the Limit of a Sequence
The sequence $$\left \langle {x_n} \right \rangle$$ defined as $$x_n = \left({1 + \frac 1 n}\right)^n$$ converges to a limit.

That limit is called Euler's Number and is denoted $$e$$.

Its value is approximately $$2.718281828 \ldots$$

As the Limit of a Series
The series $$\sum_{k \ge 0} \frac 1 {k!}$$ also converges to the same limit.

As the Base of the Natural Logarithm
The number $$e$$ can also be defined as the number satisfied by $$\ln e = 1$$.

Equivalence of Definitions
See Exponential as the Limit of a Sequence for how $$\lim_{n \to \infty} \left({1 + \frac 1 n}\right)^n = e$$ follows from the definition of $$e$$ as the number satisfied by $$\ln e = 1$$.

Comment
This is the most famous irrational constant in mathematics after $$\pi$$, and equally far-reaching in scope and usefulness.

The proof that it is irrational is straightforward.