Definition:Euler's Number

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

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

Its value is approximately:
 * $2.71828 \ 18284 \ 59045 \ 23536 \ 0287 \ldots$

As the Limit of a Series
The series $\displaystyle \sum_{n=0}^\infty \frac 1 {n!}$ 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 $\ln e = 1$.

In Terms of the Exponential Function
We can also define $e$ as:
 * $e := \exp 1 = e^1$

Equivalence of Definitions
It is shown in Equivalence of Definitions of Euler's Number that all the methods given here for specifying $e$ are logically equivalent.

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.