# 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

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

## Complex Conjugate Number

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

## Rules

If there are two Complex Numbers

${\displaystyle Z_{1}=A+jB}$
${\displaystyle Z_{2}=C+jD}$
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. ${\displaystyle {\frac {(A+jB)}{(C+jD)}}}$ = ${\displaystyle {\frac {(A+jB)(C-jD)}{(C+jD)(C-jD)}}}$ = ${\displaystyle {\frac {(AC+BD)+j(BC-AD)}{C^{2}+D^{2}}}}$
1. ${\displaystyle Z_{1}\times Z_{2}=|Z_{1}||Z_{2}|\angle (\theta _{1}+\theta _{2})}$
2. ${\displaystyle Z_{1}/Z_{2}={\frac {|Z_{1}|}{|Z_{2}|}}\angle (\theta _{1}-\theta _{2})}$