Definition

$$z = a + bi$$
where a is the real part and b is the imaginary part, i² = -1

Modulus & Argument

$$|z| = \sqrt{a^2 + b^2}$$
$$\arg(z) = \arctan\!\left(\frac{b}{a}\right)$$

Polar Form

$$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$$

Euler's Formula

$$e^{i\theta} = \cos\theta + i\sin\theta$$

Multiplication Rule

$$z_1 \cdot z_2 = r_1 r_2 \, e^{i(\theta_1 + \theta_2)}$$
Multiply moduli, add arguments

Conjugate Properties

  • $$\overline{a+bi} = a - bi$$
  • $$z \cdot \bar{z} = |z|^2$$
  • $$\overline{z_1 + z_2} = \bar{z}_1 + \bar{z}_2$$
  • $$|\bar{z}| = |z|$$

Values

z₁ (blue)
z₂ (red)
Result (green)
Orbit Prep