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

$Z=A+jB$

$Z=|Z|\mathrm{\angle }\theta$

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

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

Complex Conjugate Number

$Z=A-jB$

$Z=|Z|\mathrm{\angle }-\theta$

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

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

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. $\frac{(A+jB)}{(C+jD)}$ = $\frac{(A+jB)(C-jD)}{(C+jD)(C-jD)}$ = $\frac{(AC+BD)+j(BC-AD)}{C^2+D^2}$

1. $Z_1\times Z_2=|Z_1||Z_2|\mathrm{\angle }({\theta }_1+{\theta }_2)$
2. $Z_1/Z_2=\frac{|Z_1|}{|Z_2|}\mathrm{\angle }({\theta }_1-{\theta }_2)$