Euler's number - e constant (e=2.71828183...)

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e constant
e constant or Euler's number is a mathematical constant. The e constant is real and irrational number.
e = 2.718281828459...
e definition
e is defined as the limit
$e=\lim_{x\rightarrow \infty }\left ( 1+\frac{1}{x} \right )^x = 2.718281828459...$
Alternative definitions
$e=\lim_{x\rightarrow 0 }\left ( 1+ \right x)^\frac{1}{x}$
or
$e=\sum_{n=0}^{\infty }\frac{1}{n!}=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+...$
Properties of e
Reciprocal of e
$\lim_{x\rightarrow \infty }\left ( 1-\frac{1}{x} \right )^x=\frac{1}{e}$
Derivatives of e
(e^x)' = e^x

(log_ex)' = (ln x)' = 1/x

Integrals of e
∫ e^xdx = e^x+c

∫ log_ex dx = ∫ lnx dx = x ln x - x +c

$\int_{1}^{e}\frac{1}{x}\: dx=1$

Euler's formula
The complex number e^iθ has the identity,
e^iθ = cos(θ) + i sin(θ)

See also
Natural logarithm
Normal distribution
Pi constant
e constant - wikipedia