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



   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


Operation on complex numbers and its conjugateEdit


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:




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


See AlsoEdit

"Complex Numbers".