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Complex Numbers
A-Level H2 Math
Basics
$z = a + bi$
$|z| = r = \sqrt{a^2 + b^2}$
Modulus
$\arg(z) = \theta \in (-\pi,\, \pi]$
Principal argument
Argument by Quadrant
1st quadrant2nd quadrant3rd quadrant4th quadrant
$\tan^{-1}\!\left|\dfrac{b}{a}\right|$$\pi - \tan^{-1}\!\left|\dfrac{b}{a}\right|$$-\!\left[\pi - \tan^{-1}\!\left|\dfrac{b}{a}\right|\right]$$-\tan^{-1}\!\left|\dfrac{b}{a}\right|$
Purely real: $k\pi,\; k \in \mathbb{Z}$Purely imaginary: $\dfrac{(2k+1)\pi}{2},\; k \in \mathbb{Z}$
Properties of Conjugates
$(z^*)^* = z$
$(z \pm w)^* = z^* \pm w^*$
$z + z^* = 2\operatorname{Re}(z)$
$(zw)^* = z^* w^*$
$z - z^* = 2i\operatorname{Im}(z)$
$\left(\dfrac{z}{w}\right)^* = \dfrac{z^*}{w^*}$
$zz^* = |z|^2$
$(z^n)^* = (z^*)^n$
$z = z^* \iff z \text{ is real}$
$|z| = 1 \implies z^* = \dfrac{1}{z}$
$z^n + (z^n)^* = 2\cos n\theta \quad \text{(for } |z|=1\text{)}$
$z^n - (z^n)^* = 2i\sin n\theta \quad \text{(for } |z|=1\text{)}$
$z + \dfrac{1}{z} = 2\cos\theta \qquad z - \dfrac{1}{z} = 2i\sin\theta \quad \text{(for } |z|=1\text{)}$
Geometrical Representation
$iz$Rotation of $\dfrac{\pi}{2}$ anticlockwise about origin
$-iz$Rotation of $\dfrac{\pi}{2}$ clockwise about origin
$-z$Rotation of $\pi$ about origin
$z^*$Reflection in the real axis
Fundamental Theorem of Algebra
$\text{A polynomial of degree } n \text{ has exactly } n \text{ roots (real or non-real)}$
Conjugate Root Theorem
$\text{If } p(z) = 0 \text{ and } p \text{ has real coefficients, then } p(z^*) = 0$
Non-real roots occur in conjugate pairs $z$ and $z^*$.

Formulas for the Singapore A-Level H2 Mathematics syllabus