# e constant (e = 2.718)

E constant or Euler’s number – 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:

(ex) = ex

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

(logex) = (ln x) = 1/x

Integrals of e
The indefinite integral of the exponential function ex is the exponential function ex.

∫ ex dx = ex+c

The indefinite integral of the natural logarithm function logex is:

logex dx = ∫ lnx 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 = logex

### Exponential function

The exponential function is defined as:

f(x) = exp(x) = ex

### Euler’s formula

The complex number e has the identity:

e = cos(θ) + sin(θ)

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

θ – any real number.