Complex Numbers

Basics, conjugate and modulus

Form
$z = x + i y,$ $i^{2} = - 1,$ $\overset{z}{ˉ} = x - i y,$ $∣ z ∣ = x^{2} + y^{2}$
Conjugate
$z \overset{z}{ˉ} = ∣ z ∣^{2},$ $\overline{z_{1} \pm z_{2}} = \overset{z}{ˉ}_{1} \pm \overset{z}{ˉ}_{2},$ $\overline{z_{1} z_{2}} = \overset{z}{ˉ}_{1} \overset{z}{ˉ}_{2},$ $Re (z) = \frac{z + z ˉ}{2}$
Modulus
$∣ z_{1} z_{2} ∣ = ∣ z_{1} ∣∣ z_{2} ∣,$ $\frac{z _{1}}{z _{2}} = \frac{∣ z _{1} ∣}{∣ z _{2} ∣},$ $∣ z_{1} + z_{2} ∣^{2} = ∣ z_{1} ∣^{2} + ∣ z_{2} ∣^{2} + 2 Re (z_{1} \overset{z}{ˉ}_{2})$
Powers of i
$i^{4 k} = 1, i^{4 k + 1} = i, i^{4 k + 2} = - 1, i^{4 k + 3} = - i$

Polar / Euler form
$z = r (cos θ + i sin θ) = r e^{i θ},$ $r = ∣ z ∣, θ = ar g z \in (- π, π]$
Argument rules
$ar g (z_{1} z_{2}) = ar g z_{1} + ar g z_{2},$ $ar g \frac{z _{1}}{z _{2}} = ar g z_{1} - ar g z_{2} (mod 2 π),$ $ar g \overset{z}{ˉ} = - ar g z$

De Moivre
$(cos θ + i sin θ)^{n} = cos n θ + i sin n θ$
Cube roots of unity
$1, ω, ω^{2};$ $ω = \frac{- 1 + i 3}{2},$ $1 + ω + ω^{2} = 0,$ $ω^{3} = 1$
nth roots of unity
$z_{k} = e^{\frac{2 π ik}{n}}, k = 0, 1, \dots, n - 1;$ $\sum z_{k} = 0, \prod z_{k} = (- 1)^{n - 1}$

Distance and section
$∣ z_{1} - z_{2} ∣ =$ distance $;$ $z = \frac{m z _{2} + n z _{1}}{m + n}$
Rotation
$\frac{z _{3} - z _{1}}{z _{2} - z _{1}} = \frac{∣ z _{3} - z _{1} ∣}{∣ z _{2} - z _{1} ∣} e^{i θ}$
$θ$ is the angle from $z_{2} - z_{1}$ to $z_{3} - z_{1}$ (anticlockwise).
Circle
$∣ z - z_{0} ∣ = r;$ $\frac{z - z _{1}}{z - z _{2}} = k (k \neq = 1)$ is a circle $, k = 1$ a line
Collinear points
$\frac{z _{3} - z _{1}}{z _{2} - z _{1}} \in R$
Triangle inequality
$∣ z_{1} ∣ - ∣ z_{2} ∣ \leq ∣ z_{1} + z_{2} ∣ \leq ∣ z_{1} ∣ + ∣ z_{2} ∣$