Questions tagged [eulers-number-e]

Euler’s Number, written as $~e~$, is probably the second most famous mathematical constant after $~\pi~$. It is often defined either through an integral, viz. $e$ is the unique positive number such that $\int_1^e t^{-1}dt=1$, or by a series, viz. $e:= \sum_{n=0}^{\infty} \frac{1}{n!}$. Euler’s number has a value of $~2.718281828459045…~$; like $~\pi~$, it is transcendental (and thus irrational), which was first shown by Hermite.

It has many interesting mathematical properties, such as:

• Among exponential functions of the form $f_b(x)=b^x$, $b>0$, $e$ is the only base satisfying $f_b'(0)=1$.
• It has a central role in Euler's Identity, $e^{\pi i}+1=0$.
• The probability that a permutation of $[n]=\{1,2,\ldots,n\}$ is a derangement (has no fixed point) approaches $1/e$ for large $n$
• $e$ is present in Stirling's formula; one phrasing of this is $e=\lim\limits_{n\to\infty} \frac{n}{(n!)^{1/n}}$
• Another important property of $e$ is $\lim\limits_{n\to \infty}\left(1+\frac{x}{n}\right)^n = e^x$, a formula useful in calculus and for calculating interest

$~e~$ is a fascinating and useful number for scientists, engineers, and mathematicians alike.

Note this is not the same as Euler's constant $\gamma$, defined by $\gamma:=\lim_{n\to\infty}-\log(n) + \sum_{k=1}^{n}\frac{1}{k}$. Questions about this number should use the tag euler-mascheroni-constant.

**For more details, **

https://en.wikipedia.org/wiki/E_(mathematical_constant)

https://www.mathscareers.org.uk/article/calculating-eulers-constant-e/