# Definition:Euler's Number

## Contents

## 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 $\displaystyle \sum_{n \mathop = 0}^\infty \frac 1 {n!}$ 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:

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

This sequence is A001113 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Also known as

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

## Also see

- Do not confuse with Euler's constant, also known as the Euler-Mascheroni constant

- $\pi$ (Pi), the other of the two most famous irrational constants in mathematics.

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

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3$: Appendix $\text A$: Euler - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.21$: Euler ($1707$ – $1783$) - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $5$: Eternal Triangles