Fundamental Mathematics/Arithmetic/Arithmetic Number/Complex Number

Complex Number Complex Number

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

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

Operation on 2 different complex numbers

 Addition $(A+iB)+(C+iD)=(A+C)+i(B+D)$ Subtraction $(A+iB)-(C+iD)=(A-C)+i(B-D)$ Multilication $(A+iB)+(C+iD)=(AC+BD)+i(AD+BC)$ Division ${\frac {(A+iB)}{(C+iD)}}={\frac {(A+iB)}{(C+iD)}}{\frac {(C+iD)}{(C+iD)}}={\frac {(AC+BD)+i(AD+BC)}{(C+iD)^{2}}}$ Operation on complex numbers and its conjugate

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

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

Since

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

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

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

$(Z^{*})^{n}=Z^{*}\times Z^{*}\times Z^{*}...=|Z^{*}|^{n}\angle -n\theta$ Euler's formula

Euler formula

$e^{i\theta }=\cos \theta +i\sin \theta \,$ of which there is the famous case (for θ = π):

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

$z=x+yi=r(\cos \theta +i\sin \theta )=re^{i\theta }\,$ Eucleur's power can be expressed as complex number

$e^{i\theta }=\cos \theta +i\sin \theta \,$ Hence, conjugate of the complex number

$e^{-i\theta }=\cos \theta -i\sin \theta \,$ Adding complex number and its conjugate

$e^{i\theta }+e^{-i\theta }=2Cos\theta$ $Cos\theta ={\frac {1}{2}}(e^{i\theta }+e^{-i\theta })$ Minus complex number and its conjugate

$e^{i\theta }-e^{-i\theta }=2iSin\theta$ $Sin\theta ={\frac {1}{2i}}(e^{i\theta }-e^{-i\theta })$ de Moivre's formula

$z^{n}=(\cos(x)+i\sin(x))^{n}=\cos(nx)+i\sin(nx)=re^{in\theta }\,$ for any real $x$ and integer $n$ . This result is known as