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


 * 1) $$Z_1 \times Z_2 = |Z_1| |Z_2| \angle (\theta_1 + \theta_2)$$
 * 2) $$Z_1 / Z_2 = \frac{|Z_1|}{|Z_2|} \angle (\theta_1 - \theta_2)$$