Complex Numbers: The Map

A2 · P3 · 2 min

Complex numbers extend the real numbers with a single new symbol, $i$ , whose square is $-1$ . With it every polynomial equation has solutions, and a number becomes a point in a plane rather than a point on a line. This unit is a sequence of skills that build on each other; use this page to see how they fit.

The notes in order

Introduction: $i^2 = -1$ , real and imaginary parts, adding, subtracting, multiplying, the conjugate, and equating parts.
Division: multiply by the conjugate; solving equations with complex coefficients.
Square roots: $\sqrt{a + bi}$ by equating parts.
Complex roots of polynomials: conjugate pairs and factorising cubics and quartics.
The complex plane: Argand diagram, modulus $|z|$ and argument $\arg z$ .
Geometry of operations: what conjugation, addition and multiplication do to a point.
Polar form: $r(\cos\theta + i\sin\theta)$ and multiplying or dividing by adding or subtracting arguments.
Exponential form: $re^{i\theta}$ .
Loci: circles, perpendicular bisectors and half-lines from $|z - a| = k$ , $|z - a| = |z - b|$ , $\arg(z - a) = \alpha$ .

The notation at a glance

Key result

For $z = x + iy$ with $x, y$ real:

Symbol	Meaning	Value
$\operatorname{Re} z$	real part	$x$
$\operatorname{Im} z$	imaginary part	$y$ (not $iy$ )
$z^*$	conjugate	$x - iy$
$\lvert z \rvert$	modulus	$\sqrt{x^2 + y^2}$
$\arg z$	argument	angle from the positive real axis, usually $-\pi < \arg z \leq \pi$

Two complex numbers are equal if and only if both their real parts and their imaginary parts are equal.

The three forms

z = x + iy = r(\cos\theta + i\sin\theta) = re^{i\theta}, \qquad r = |z|,\ \theta = \arg z.

Cartesian form is for adding and subtracting; polar and exponential forms are for multiplying, dividing and reading off geometry.

A tour through the forms

Let $z = 1 + i\sqrt{3}$ . Find $z^*$ , $|z|$ , $\arg z$ , and write $z$ in polar and exponential form. Then find $z^2$ two ways.

Solution

$z^* = 1 - i\sqrt{3}$ . $|z| = \sqrt{1 + 3} = 2$ . $\arg z = \tan^{-1}\sqrt{3} = \tfrac{\pi}{3}$ (first quadrant).

$z = 2\left(\cos\tfrac{\pi}{3} + i\sin\tfrac{\pi}{3}\right) = 2e^{i\pi/3}$ .

Cartesian: $z^2 = 1 + 2i\sqrt{3} - 3 = -2 + 2i\sqrt{3}$ . Polar: $z^2 = 4e^{2i\pi/3} = 4\left(-\tfrac{1}{2} + i\tfrac{\sqrt{3}}{2}\right) = -2 + 2i\sqrt{3}$ ✓.

Exam tip

A typical P3 question walks through several of these skills in order: solve a quadratic with complex roots, plot them, find modulus and argument, then sketch a locus. Each part is short; the marks come from doing the standard step cleanly and showing the working the syllabus insists on (multiplication, division and square roots in full).

The notes in order

The notation at a glance

The three forms

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