Definition:Euler's Number

Definition

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 as $n$ increases without bound.

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

As the Limit of a Series

The series $\displaystyle \sum_{n \mathop = 0}^\infty \frac 1 {n!}$ converges to a limit.

This limit is Euler's number $e$.

As the Base of the Natural Logarithm

As the base of the Natural Logarithm:

The number $e$ can be defined as the number satisfied by:

$\ln e = 1$.

where $\ln e$ denotes the natural logarithm of $e$.

That $e$ is unique follows from Logarithm is Strictly Increasing.

In Terms of the Exponential Function

In terms of the exponential function:

The number $e$ can be defined as the number satisfied by:

$e := \exp 1 = e^1$

where $\exp 1$ denotes the exponential function of $1$.

As the Base of the Exponential with Derivative One at Zero

There is a number $x \in \R$ such that:

$\displaystyle \lim_{h \to 0} \frac{ x^{h} - 1 }{ h } = 1$

This number is called Euler's Number and is denoted $e$.

Decimal Expansion

The decimal expansion of Euler's number $e$ starts:

$2 \cdotp 71828 \, 18284 \, 59045 \, 23536 \, 02874 \, 71352 \, 66249 \, 77572 \, 47093 \, 69995 \ldots$

Also known as

Euler's Number is also known as Napier's Constant for John Napier.

Source of Name

This entry was named for Leonhard Paul Euler.

Historical Note

Euler's number was named $e$ by Euler himself.

He originally defined it as the limit of the sequence $\displaystyle \lim_{n \mathop \to \infty} \left({1 + \dfrac 1 n}\right)^n$