Complex number
A Complex Number is a combination of a Real Number and an Imaginary Number
Complex number – Algebra Formulas
1. Equality of Complex Numbers Formula:
$$ a+bi = c+di ⇔ a=c and b=d $$
2. Addition of Complex Numbers:
$$ (a+bi)+(c+di) = (a+c)+(b+d)i $$
3. Subtraction of Complex Numbers:
$$ (a+bi)−(c+di) = (a−c)+(b−d)i $$
4. Multiplication of Complex Numbers:
$$ (a+bi)×(c+di) = (ac−bd)+(ad+bc)i $$
5. Multiplication Conjugates:
$$ (a+bi)(a+bi) = a^2+b^2 $$
6. Division of Complex Numbers:
$$ {(a+bi) \over (c+di)} = {a+bi \over c+di} \times {c-di \over c-di} $$
$$ {(a+bi) \over (c+di)} = {ac+bd \over c^2+d^2} \times {bc-ad \over c^2-d^2}i $$
7. Euler’s Formula:
$$ z = re^{i \theta} = r(cos \theta + i sin \theta) $$
8. Modulus of a Complex Number:
$$ \vert{a+bi}\vert = \sqrt{a^2+b^2} $$
9. De Moivre’s Theorem:
$$ z^n = (re^{i \theta})^n = r^n[cos (n\theta) + i sin (n\theta)] $$
10. Multiplication and division of complex numbers in polar form:
$$ [r_1(cos \theta_1 + i \cdot sin \theta_1)] \cdot [r_2(cos \theta_2 + i \cdot sin \theta_2)] = r_1 \cdot r_2[cos( \theta_1 + \theta_2)+i \cdot sin( \theta_1 + \theta_2)] $$
$$ {r_1(cos \theta_1 + i \cdot sin \theta_1) \over r_2(cos \theta_2 + i \cdot sin \theta_2)} = \frac{r_1}{r_2}[cos( \theta_1 – \theta_2)+i \cdot sin( \theta_1 – \theta_2)] $$