## Complex number – Algebra Formulas

### Complex number

A Complex Number is a combination of a Real Number and an Imaginary Number

### 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:

$$[r_1(cos \theta_1 + i \cdot sin \theta_1)] \cdot [r_2(cos \theta_2 + i \cdot sin \theta_2)] = r_1 \cdot r_2[cos( \theta_1 + \theta_2)+i \cdot sin( \theta_1 + \theta_2)]$$
$${r_1(cos \theta_1 + i \cdot sin \theta_1) \over r_2(cos \theta_2 + i \cdot sin \theta_2)} = \frac{r_1}{r_2}[cos( \theta_1 – \theta_2)+i \cdot sin( \theta_1 – \theta_2)]$$