Fundamental Mathematics/Arithmetic/Arithmetic Number/Complex Number

Complex Number

Complex Number

$Z=A+iB=|Z|\mathrm{\angle }\theta =\sqrt{B^2+A^2}\mathrm{\angle }Ta{n}^{-1}\frac{B}{A}=|Z|e^{i\theta }$

Complex conjugate Number

$Z^{\ast }=A-iB=|Z|\mathrm{\angle }-\theta =\sqrt{B^2+A^2}\mathrm{\angle }-Ta{n}^{-1}\frac{B}{A}=-|Z|e^{i\theta }$

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

$Z\times Z^{\ast }=|Z|\mathrm{\angle }\theta \times |Z|\mathrm{\angle }-\theta =|Z{|}^2\mathrm{\angle }(\theta -\theta )=|Z{|}^2$

$\frac{Z}{Z^{\ast }}=\frac{|Z|\mathrm{\angle }\theta }{|Z|\mathrm{\angle }-\theta }=1\mathrm{\angle }2\theta$

Power of Z

Since

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

Hence

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

Since

$Z^{\ast }\times Z^{\ast }=(Z^{\ast }{)}^2=(|Z|\mathrm{\angle }-\theta )(|Z|\mathrm{\angle }-\theta )=|Z{|}^2\mathrm{\angle }(-\theta -\theta )=|Z{|}^2\mathrm{\angle }-2\theta$

Hence

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

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,

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

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

$z^n=(\cos (x)+i\sin (x){)}^n=\cos (nx)+i\sin (nx)=r{e}^{in\theta }$

for any real $x$ and integer $n$. This result is known as

Reference

• Complex Number