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

Reference

• Complex Number