Complex Numbers

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Complex NumberEdit

The set of complex numbers is denoted  . A complex number   can be written in Cartesian coordinates as

 

where  .   is called the 'real part' of   and   is called the 'imaginary part' of  . These can also be written in a trigonometric polar form, as

 

where   is the 'magnitude' of   and   is called the 'argument' of  . These two forms are related by the equations

 

The trigonometric polar form can also be written as

 

by using Euler's Identity

 

Coordination

   
   in Cartesian form,
   in trigonometric polar form,
   in polar exponential form.

Complex conjugate NumberEdit

A complex number   is a complex conjugate of a number   if and only if

 

If a complex number   is written as  , then the conjugate is

 

Equivalently in polar form if   then

 

Mathematical OperationsEdit

Operation on 2 different complex numbersEdit

Addition  
Subtraction  
Multiplication  
Division  

Operation on complex numbers and its conjugateEdit

Addition  
Subtraction  
Multiplication  
Division  

In Polar formEdit

Operation on complex number and its conjugate

 
 

Operation on 2 different complex numbers

 
 

Complex powerEdit

A careful analysis of the power series for the exponential, sine, and cosine functions reveals the marvelous

Euler formulaEdit

 

of which there is the famous case (for θ = π):

 

More generally,

 


de Moivre's formulaEdit

 

for any real   and integer  . This result is known as .

Transcendental functionsEdit

The higher mathematical functions (often called "transcendental functions"), like exponential, log, sine, cosine, etc., can be defined in terms of power series (Taylor series). They can be extended to handle complex arguments in a completely natural way, so these functions are defined over the complex plane. They are in fact "complex analytic functions". Many standard functions can be extended to the complex numbers, and may well be analytic (the most notable exception is the logarithm). Since the power series coefficients of the common functions are real, they work naturally with conjugates. For example:

 
 

SummaryEdit

   

Complex number

   . In Rectangular plane 
   . In Polar plane
   . In trigonometry
   . In Complex plane

Complex conjugate number

   . In Rectangular plane 
   . In Polar plane 
   . In trigonometry angle
   . In Complex plane

ReferencesEdit

See AlsoEdit

"Complex Numbers".