# e constant

*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:

#### Alternative definitions

The e constant is defined as the limit:

The e constant is defined as the infinite series:

## Properties of e

#### Reciprocal of e

The reciprocal of e is the limit:

#### 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_{e }x)*' = *(ln* x*)'* = *1/*x*

#### Integrals of e

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

∫* e*^{x }dx = *e*^{x}+c

The indefinite integral of the natural logarithm function log_{e }x is:

∫ log_{e }x dx = ∫ ln*x dx* = *x *ln* x - x *+c

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

### Base e logarithm

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

ln *x* = log_{e }x

### 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*) + *i
*sin(*θ*)

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

θ is any real number.

## See also