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 $$2.71828 \, 18284 \, 59045 \, 23536 \, 0287 \ldots$$

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

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.