---
tag: 0MVCFMW7S4
kind: object
name: e
aliases: [euler-number, napier-constant]
msc: [11Y60]
uses: [0RMVS7RXZX, 0G4D3ZKTP8]
sources:
  - url: https://en.wikipedia.org/wiki/E_(mathematical_constant)
    locator: Definitions
same_as:
  wikidata: Q82435
  wikipedia: E (mathematical constant)
---

## Description

The number $e$ is the value of the [[exponential function|0RMVS7RXZX]] at $1$:
$$e = \exp(1) = \sum_{n=0}^{\infty} \frac{1}{n!} \approx 2.718281828.$$
It is also the limit $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$.
It is the base of the natural logarithm, so $\ln e = 1$.

## Remarks

The series is the [[sum of the reciprocals of the factorials|0G4D3ZKTP8]].
The number $e$ is [[irrational|0W4GQSG7B6]]. Charles Hermite proved in 1873 that it is transcendental.
Jacob Bernoulli introduced the constant in 1683, in a problem of continuous compound interest. Leonhard Euler began to use the letter $e$ for it.

## Explanations

### L1 human

The number $e$ is about $2.718$. It is the natural base for exponential growth.
An amount that grows by $100\%$ per year, compounded continuously, becomes $e$ times larger in one year.