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Number e

The number e is a transcendental number with the value

e = 2.7182…

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 bx 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

 


Additional information

The letter e stands for exponent, but everyone calls it the e from Euler, after the famous mathematician who explained its great significance.

 


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