Polar Form: Multiplication, Division & Square Roots of Complex Numbers — Mathematics

Introduction

Polar form is the language in which the geometry of complex number arithmetic becomes transparent. When you write $z = r(\cos\theta + i\sin\theta)$ , the modulus $r = |z|$ captures the size of $z$ and the argument $\theta = \arg z$ captures its direction on the Argand diagram. The 9709 syllabus requires you to multiply and divide in this form — revealing elegant rules about how moduli and arguments combine — and to find both square roots of any complex number in exact Cartesian form. These skills appear regularly in Paper 3, both as standalone questions and as part of loci problems.

Core Concept

Polar (Modulus–Argument) Form

Every non-zero complex number $z = x + iy$ can be written as:

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

where $r = |z| = \sqrt{x^2 + y^2} \geq 0$ and $\theta = \arg z \in (-\pi, \pi]$ (the principal argument).

The exponential notation $re^{i\theta}$ follows from Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$ , and is accepted fully in 9709.

Multiplication in Polar Form

Let $z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$ . Then:

z_1 z_2 = r_1 e^{i\theta_1} \cdot r_2 e^{i\theta_2} = r_1 r_2\, e^{i(\theta_1 + \theta_2)}

This gives the two fundamental multiplication results:

|z_1 z_2| = |z_1||z_2|, \qquad \arg(z_1 z_2) = \arg z_1 + \arg z_2

Geometrically: multiply the moduli, add the arguments. If the sum of arguments falls outside $(-\pi,\pi]$ , adjust by adding or subtracting $2\pi$ to recover the principal argument.

Division in Polar Form

\frac{z_1}{z_2} = \frac{r_1 e^{i\theta_1}}{r_2 e^{i\theta_2}} = \frac{r_1}{r_2}\,e^{i(\theta_1 - \theta_2)}

The two division results:

\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}, \qquad \arg\!\left(\frac{z_1}{z_2}\right) = \arg z_1 - \arg z_2

Again, adjust the resulting argument into $(-\pi,\pi]$ if necessary.

Square Roots of a Complex Number

To find the square roots of $w = a + ib$ , let $\sqrt{w} = x + iy$ where $x, y \in \mathbb{R}$ .

Then $(x + iy)^2 = a + ib$ , which expands to:

x^2 - y^2 = a \quad \text{(equating real parts)}

2xy = b \quad \text{(equating imaginary parts)}

Combined with $|x+iy|^2 = |w|$ , i.e. $x^2 + y^2 = \sqrt{a^2 + b^2}$ , these three equations (only two are independent alongside $2xy=b$ ) determine $x$ and $y$ exactly. The syllabus demands full working shown — no guessing or short-cuts.

Key fact: A non-zero complex number always has exactly two square roots, and they are negatives of each other: if $z_0$ is one root, $-z_0$ is the other.

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