### Number *e*

The **number e** is a transcendental number with the value

It is an important number because it forms the connection between powers, logarithms and trigonometric functions.

##### Power series

The power series for the number *e* can be written as

or if you prefer

##### Exponential functions

The number *e* is the basis of the exponential function

where *x* is a variable.

##### Base of the natural logarithm

You can write every number as an exponential function

Because *b ^{x}* conforms to the rules for logarithms as exponents, it applies

for every real number *x*.

##### Own derivative

The derivative of the exponential function is

and therefor the exponential function is its own derivative!

##### Eulers formula

The formula of Euler shows the relation between complex exponential functions and trigonometric functions, as in

##### Hyperbolic functions

All trigonometric and hyperbolic functions are defined by exponential functions, as for the hyperbolic sine

## HistoryThe letter |