Mathematics · Ch 4 — Complex Numbers and Quadratic Equations

Summary

A complex number is of the form $z = a + ib$ , where $a, b \in \mathbb{R}$ and $i = \sqrt{-1}$ (so $i^2 = -1$ ). $a$ is the real part, $b$ the imaginary part.
Equality: $a + ib = c + id$ iff $a = c$ and $b = d$ .
Algebra: Addition, subtraction, multiplication, and division (by rationalising the denominator) follow the usual algebraic rules, treating $i$ as a symbol with $i^2 = -1$ .
Conjugate: $\overline{z} = a - ib$ . Key properties: $z \overline{z} = a^2 + b^2$ (a real number), $\overline{z_1 \pm z_2} = \overline{z_1} \pm \overline{z_2}$ , $\overline{z_1 z_2} = \overline{z_1} \cdot \overline{z_2}$ .
Modulus: $|z| = \sqrt{a^2 + b^2}$ . Properties: $|z_1 z_2| = |z_1||z_2|$ , $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$ , $|z_1 + z_2| \leq |z_1| + |z_2|$ (triangle inequality).
Argand plane: $z = a + ib$ is plotted as $(a, b)$ . The $x$ -axis is the real axis, $y$ -axis the imaginary axis.
Polar form: $z = r(\cos\theta + i\sin\theta)$ , where $r = |z|$ and $\theta = \arg(z)$ (principal argument, usually $-\pi < \theta \leq \pi$ ). To convert: $a = r\cos\theta$ , $b = r\sin\theta$ .
De Moivre’s Theorem (for integer $n$ ): $(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$ . Used to find powers and roots.
Square roots of a complex number: For $z = a + ib$ , solve $(x+iy)^2 = a+ib$ by equating real and imaginary parts, using $x^2 + y^2 = |z|$ . …