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

e constant or Euler's number is a mathematical constant. The e constant is real and irrational number.

e = 2.718281828459...

Definition of e

The e constant is defined as the limit:

$e=\lim_{x\rightarrow \infty }\left ( 1+\frac{1}{x} \right )^x = 2.718281828459...$

Alternative definitions

The e constant is defined as the limit:

$e=\lim_{x\rightarrow 0 }\left ( 1+ \right x)^\frac{1}{x}$

The e constant is defined as the infinite series:

$e=\sum_{n=0}^{\infty }\frac{1}{n!}=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+...$

Properties of e

Reciprocal of e

The reciprocal of e is the limit:

$\lim_{x\rightarrow \infty }\left ( 1-\frac{1}{x} \right )^x=\frac{1}{e}$

Derivatives of e

The derivative of the exponential function is the exponential function:

(e^x)' = e^x

The derivative of the natural logarithm function is the reciprocal function:

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

Integrals of e

The indefinite integral of the exponential function e^x is the exponential function e^x.

∫ e^xdx = e^x+c

The indefinite integral of the natural logarithm function log_ex is:

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

The definite integral from 1 to e of the reciprocal function 1/x is 1:

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

Base e logarithm

The natural logarithm of a number x is defined as the base e logarithm of x:

ln x = log_ex

Exponential function

The exponential function is defined as:

f (x) = exp(x) = e^x

Euler's formula

The complex number e^iθ has the identity:

e^iθ = cos(θ) + isin(θ)

i is the imaginary unit (the square root of -1).

θ is any real number.

e constant