# Fundamental Mathematics/Arithmetic/Arithmetic Number/Complex Number

## Complex Number

Complex Number

${\displaystyle Z=A+iB=|Z|\angle \theta ={\sqrt {B^{2}+A^{2}}}\angle Tan^{-1}{\frac {B}{A}}=|Z|e^{i\theta }\,}$

Complex conjugate Number

${\displaystyle Z^{*}=A-iB=|Z|\angle -\theta ={\sqrt {B^{2}+A^{2}}}\angle -Tan^{-1}{\frac {B}{A}}=-|Z|e^{i\theta }\,}$

## Mathematical Operations

### Operation on 2 different complex numbers

 Addition ${\displaystyle (A+iB)+(C+iD)=(A+C)+i(B+D)}$ Subtraction ${\displaystyle (A+iB)-(C+iD)=(A-C)+i(B-D)}$ Multilication ${\displaystyle (A+iB)+(C+iD)=(AC+BD)+i(AD+BC)}$ Division ${\displaystyle {\frac {(A+iB)}{(C+iD)}}={\frac {(A+iB)}{(C+iD)}}{\frac {(C+iD)}{(C+iD)}}={\frac {(AC+BD)+i(AD+BC)}{(C+iD)^{2}}}}$

### Operation on complex numbers and its conjugate

 Addition ${\displaystyle (A+iB)+(A-iB)=2A}$ Subtraction ${\displaystyle (A+iB)-(A+iB)=i2B}$ Multilication ${\displaystyle (A+iB)+(A-iB)=A^{2}-B^{2}}$ Division ${\displaystyle {\frac {(A+iB)}{(A-iB)}}={\frac {(A+iB)}{(A-iB)}}{\frac {(A-iB)}{(A-iB)}}={\frac {A^{2}-B^{2}}{(A-iB)^{2}}}}$

In Polar form

${\displaystyle Z\times Z^{*}=|Z|\angle \theta \times |Z|\angle -\theta =|Z|^{2}\angle (\theta -\theta )=|Z|^{2}}$
${\displaystyle {\frac {Z}{Z^{*}}}={\frac {|Z|\angle \theta }{|Z|\angle -\theta }}=1\angle 2\theta }$

## Power of Z

Since

${\displaystyle Z\times Z=Z^{2}=(|Z|\angle \theta )(|Z|\angle \theta )=|Z|^{2}\angle (\theta +\theta )=|Z|^{2}\angle 2\theta }$

Hence

${\displaystyle Z^{n}=Z\times Z\times Z...=|Z|^{n}\angle n\theta }$

Since

${\displaystyle Z^{*}\times Z^{*}=(Z^{*})^{2}=(|Z|\angle -\theta )(|Z|\angle -\theta )=|Z|^{2}\angle (-\theta -\theta )=|Z|^{2}\angle -2\theta }$

Hence

${\displaystyle (Z^{*})^{n}=Z^{*}\times Z^{*}\times Z^{*}...=|Z^{*}|^{n}\angle -n\theta }$

## Euler's formula

### Euler formula

${\displaystyle e^{i\theta }=\cos \theta +i\sin \theta \,}$


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

${\displaystyle e^{i\pi }=-1\,}$

More generally,

${\displaystyle z=x+yi=r(\cos \theta +i\sin \theta )=re^{i\theta }\,}$


Eucleur's power can be expressed as complex number

${\displaystyle e^{i\theta }=\cos \theta +i\sin \theta \,}$

Hence, conjugate of the complex number

${\displaystyle e^{-i\theta }=\cos \theta -i\sin \theta \,}$

Adding complex number and its conjugate

${\displaystyle e^{i\theta }+e^{-i\theta }=2Cos\theta }$
${\displaystyle Cos\theta ={\frac {1}{2}}(e^{i\theta }+e^{-i\theta })}$

Minus complex number and its conjugate

${\displaystyle e^{i\theta }-e^{-i\theta }=2iSin\theta }$
${\displaystyle Sin\theta ={\frac {1}{2i}}(e^{i\theta }-e^{-i\theta })}$

### de Moivre's formula

${\displaystyle z^{n}=(\cos(x)+i\sin(x))^{n}=\cos(nx)+i\sin(nx)=re^{in\theta }\,}$


for any real ${\displaystyle x}$ and integer ${\displaystyle n}$. This result is known as