Arithmetic/Types of Numbers/Complex Number

Complex Number

Complex Number is a number that can be expressed mathematically as a sum of a Real Number and an Imaginary Number

$Z=A+jB$

$Z=|Z|\angle \theta$

$|Z|={\sqrt {A^{2}+B^{2}}}$

$\theta =Tan^{-}1{\frac {B}{A}}$

Complex Conjugate Number

$Z=A-jB$

$Z=|Z|\angle -\theta$

$|Z|={\sqrt {A^{2}+B^{2}}}$

$\theta =-Tan^{-}1{\frac {B}{A}}$

Rules

If there are two Complex Numbers

$Z_{1}=A+jB$

$Z_{2}=C+jD$

1. (A + jB) + (C + jD) = (A + C) + j (B + D)
2. (A + jB) - (C + jD) = (A - C) + j (B - D)
3. (A + jB) x (C + jD) = (AC + BD) + j (AD + BC)
4. ${\frac {(A+jB)}{(C+jD)}}$ = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikibooks.org/v1/":): {\displaystyle \frac{(A + jB) (C - jD)}{(C+ jD) (C - jD)}} = ${\frac {(AC+BD)+j(BC-AD)}{C^{2}+D^{2}}}$

1. $Z_{1}\times Z_{2}=|Z_{1}||Z_{2}|\angle (\theta _{1}+\theta _{2})$
2. $Z_{1}/Z_{2}={\frac {|Z_{1}|}{|Z_{2}|}}\angle (\theta _{1}-\theta _{2})$