OCR OCR 2018-style practice: Find the three cube roots of 8i, giving your answers in the form re^iθ, and state what they look like on an Argand diagram.

Write 8i in modulus-argument form (M1): 8i = 8\,e^iπ/2, and include all coterminal arguments π2 + 2π k. Take cube roots: modulus 8^1/3 = 2, arguments 13(π2 + 2π k) = π6 + 2π k3 for k = 0, 1, 2 (M1, A1). The roots are 2e^iπ/6, 2e^i 5π/6 and 2e^i 3π/2 (equivalently 2e^-iπ/2) (A1, A1). On an Argand diagram they lie on a circle of radius 2, equally spaced 2π3 apart, forming an equilateral triangle (A1). Markers reward the modulus-argument form with the 2π k terms, dividing modulus and argument, the three roots, and the geometric description.

EnglandFurther MathsSyllabus dot point

How do you find the nth roots of a complex number, and how do conditions on z describe loci on the Argand diagram?

The nth roots of unity and of a general complex number, their geometric arrangement as a regular polygon, and loci on the Argand diagram defined by modulus and argument conditions (circles, perpendicular bisectors, half-lines and regions).

A focused answer to the OCR A-Level Further Mathematics A content on roots of unity and loci, covering the nth roots of unity and of a general complex number and their arrangement as a regular polygon on a circle, and loci on the Argand diagram from modulus and argument conditions, including circles, perpendicular bisectors, half-lines and shaded regions.

Generated by Claude Opus 4.812 min answerUpdated 2026-06-02

Reviewed by: AI editorial process; not yet individually human-reviewed

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Quick answer

To find the $n$ th roots of a complex number $w = R\,e^{i\phi}$ , take the real $n$ th root of the modulus and divide each coterminal argument by $n$ : the roots are $R^{1/n}\,e^{i(\phi + 2\pi k)/n}$ for $k = 0, 1, \ldots, n-1$ . They share modulus $R^{1/n}$ , are spaced $\dfrac{2\pi}{n}$ apart, and form a regular $n$ -gon. The $n$ th roots of unity are the powers of $\omega = e^{2\pi i/n}$ and sum to zero. For loci, $|z - a| = r$ is a circle centre $a$ radius $r$ ; $|z - a| = |z - b|$ is the perpendicular bisector of $a$ and $b$ ; $\arg(z - a) = \theta$ is a half-line from $a$ at angle $\theta$ (start point excluded); inequalities shade regions.

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What this dot point is asking
The nth roots of a complex number
Roots of unity as a regular polygon
Loci: circles and perpendicular bisectors
Loci: half-lines and regions
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What this dot point is asking

OCR wants you to find all $n$ th roots of unity and of a general complex number using de Moivre's theorem, recognise that they sit equally spaced on a circle as the vertices of a regular polygon, and sketch and describe loci on the Argand diagram defined by modulus and argument conditions: a circle $|z - a| = r$ , a perpendicular bisector $|z - a| = |z - b|$ , a half-line $\arg(z - a) = \theta$ , and regions defined by inequalities.

The nth roots of a complex number

The key idea is that a complex number has infinitely many arguments, differing by $2\pi$ , and dividing each by $n$ produces $n$ distinct roots before they repeat. Take the positive real $n$ th root of the modulus and spread the arguments evenly.

Roots of unity as a regular polygon

The $n$ th roots of unity are $1, \omega, \omega^2, \ldots, \omega^{n-1}$ where $\omega = e^{2\pi i/n}$ . They are equally spaced around the unit circle, forming a regular $n$ -gon with one vertex at $1$ . Because they are the roots of $z^n - 1 = 0$ , whose $z^{n-1}$ coefficient is zero, they sum to zero, a fact OCR often asks you to use or to confirm from the diagram.

Loci: circles and perpendicular bisectors

A locus is the set of points $z$ satisfying a condition. Modulus conditions measure distance, so they give circles and perpendicular bisectors.

Loci: half-lines and regions

Argument conditions measure direction, so $\arg(z - a) = \theta$ gives a half-line. Inequalities give regions, which you shade.

An inequality such as $|z - a| \le r$ shades the closed disc inside the circle, and combining two conditions picks out the points or arcs satisfying both.

Roots of unity and loci pull together the whole complex strand: de Moivre supplies the roots, and the Argand diagram turns algebraic conditions into geometry.

Try this

Q1. State the centre and radius of the locus $|z + 1 - 2i| = 3$ . [2 marks]

Cue. Rewrite as $|z - (-1 + 2i)| = 3$ : centre $(-1, 2)$ , radius $3$ .

Q2. How many distinct fifth roots does a non-zero complex number have, and how are they arranged? [2 marks]

Cue. Five roots, equally spaced $\tfrac{2\pi}{5}$ apart on a circle, forming a regular pentagon.

Exam-style practice questions

Practice questions written in the style of OCR exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

OCR 20186 marksFind the three cube roots of

8i

, giving your answers in the form

re^{i\theta}

, and state what they look like on an Argand diagram.

Show worked answer →

Write $8i$ in modulus-argument form (M1): $8i = 8\,e^{i\pi/2}$ , and include all coterminal arguments $\dfrac{\pi}{2} + 2\pi k$ .

Take cube roots: modulus $8^{1/3} = 2$ , arguments $\dfrac{1}{3}\left(\dfrac{\pi}{2} + 2\pi k\right) = \dfrac{\pi}{6} + \dfrac{2\pi k}{3}$ for $k = 0, 1, 2$ (M1, A1).

The roots are $2e^{i\pi/6}$ , $2e^{i 5\pi/6}$ and $2e^{i 3\pi/2}$ (equivalently $2e^{-i\pi/2}$ ) (A1, A1).

On an Argand diagram they lie on a circle of radius $2$ , equally spaced $\dfrac{2\pi}{3}$ apart, forming an equilateral triangle (A1).

Markers reward the modulus-argument form with the $2\pi k$ terms, dividing modulus and argument, the three roots, and the geometric description.

OCR 20225 marksSketch on a single Argand diagram the locus of points satisfying

|z - 3 - 4i| = 5

and the locus satisfying

\arg(z - 1) = \dfrac{\pi}{4}

, and describe each.

Show worked answer →

The first locus is $|z - (3 + 4i)| = 5$ , the set of points at distance $5$ from the point $3 + 4i$ (M1): a circle, centre $(3, 4)$ , radius $5$ (A1). Note this circle passes through the origin since $|3 + 4i| = 5$ .

The second locus is $\arg(z - 1) = \dfrac{\pi}{4}$ (M1): a half-line (ray) starting at the point $1$ (on the real axis), making an angle $\dfrac{\pi}{4}$ with the positive real direction, with the start point $z = 1$ excluded (A1).

A correct sketch shows the circle and the ray on the same axes (A1).

Markers reward identifying the circle's centre and radius, the half-line's start point and angle, the exclusion of the start point, and a labelled sketch.

Related dot points

Sources & how we know this

OCR A Level Further Mathematics A (H245) specification — OCR (2017)