e

Description

The number $e$ is the value of the exponential function 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. The number $e$ is irrational. 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.