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

*e* = 2.718281828459…

The e constant is defined as the limit:

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*+c

^{x}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.

The e constant can be approximately calculated by using the infinite series:

= 1 + 1 + 0.5 + 0.166667 + …

= 2.71828182…

≈ 2.71828183

2.71828183 is an approximated value of the e constant. The e constant has an infinite number of digits.