Complex Numbers

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Complex Number[edit]

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


Complexplane.JPG Euler's formula.svg
  in Cartesian form,
  in trigonometric polar form,
  in polar exponential form.

Complex conjugate Number[edit]

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 Operations[edit]

Operation on 2 different complex numbers[edit]


Operation on complex numbers and its conjugate[edit]


In Polar form[edit]

Operation on complex number and its conjugate

Operation on 2 different complex numbers

Complex power[edit]

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

Euler formula[edit]

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

More generally,

de Moivre's formula[edit]

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

Transcendental functions[edit]

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:


Complexplane.JPG Euler's formula.svg

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 Also[edit]

"Complex Numbers".