Definition:Euler's Number

Definition

As the Limit of a Sequence

The sequence $\sequence {x_n}$ defined as $x_n = \paren {1 + \dfrac 1 n}^n$ 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 $\ds \sum_{n \mathop = 0}^\infty \frac 1 {n!}$ 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:

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

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.

Some sources give it as the Euler number, but this is not endorsed on $\mathsf{Pr} \infty \mathsf{fWiki}$ in order not to confuse this with the Euler numbers.

Also see

• Results about Euler's number can be found here.

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 $\ds \lim_{n \mathop \to \infty} \paren {1 + \dfrac 1 n}^n$