What does the complex numbers strand in OCR Further Maths A cover?

It covers the arithmetic of complex numbers (addition, multiplication and division by the conjugate), the Argand diagram, solving polynomial equations with conjugate-pair roots, the modulus and argument and the modulus-argument and exponential forms, the multiplication and division rules in which moduli multiply or divide and arguments add or subtract, de Moivre's theorem with its use for powers and for deriving multiple-angle identities, expressing powers of sine and cosine in terms of multiple angles, the nth roots of unity and of a general complex number arranged as a regular polygon, and loci on the Argand diagram. It is part of the mandatory Pure Core.

Which complex numbers topics carry the most marks in OCR Further Maths A?

De Moivre's theorem and roots are heavy because they support multiple-angle identities, powers, and the geometry of roots of unity, all of which appear as structured questions. Loci are a reliable source of marks through sketch-and-describe questions, and conjugate-pair factorisation of cubics and quartics recurs. Fluency with the modulus-argument and exponential forms underpins everything, since powers and roots act simply on the modulus and argument.

What techniques recur across the OCR complex numbers content?

Dividing by multiplying by the conjugate, finding a modulus and argument with the correct quadrant, converting between Cartesian, modulus-argument and exponential form, multiplying and dividing by combining moduli and arguments, expanding with the binomial theorem after de Moivre to get a multiple-angle identity, linearising a power of cosine or sine using z plus or minus its reciprocal, finding all n roots by dividing the argument by n, and turning a modulus or argument condition into a labelled sketch. These are automatic by exam day.

What is the most common complex numbers mistake in OCR Further Maths A?

Reading an argument straight from a calculator without checking the quadrant, and drawing a full line instead of a half-line for an argument locus. Other frequent errors are forgetting that a real polynomial's complex roots come in conjugate pairs, taking the wrong real or imaginary part when deriving an identity, mishandling powers of i in a binomial expansion, and reading the centre of a circle locus with the wrong sign. Always sketch z first and track i^2 = -1 carefully.

How should I revise the OCR complex numbers content?

Work topic by topic against the specification, drilling each technique until it is automatic, then practise mixed problems under timed conditions. Keep a sheet of standard results: Euler's relation, de Moivre's theorem, the z plus and minus reciprocal identities, the conjugate-pair fact, and the four loci shapes. Always sketch a complex number before reading its argument, and check the number of roots and their equal spacing when finding nth roots.

EnglandFurther Maths

OCR A-Level Further Maths A Complex numbers: arithmetic, modulus-argument form, de Moivre and loci

A deep-dive OCR A-Level Further Mathematics A guide to the complex numbers strand of the Pure Core: arithmetic and the Argand diagram, modulus-argument and exponential form, de Moivre's theorem and multiple-angle identities, and the nth roots of unity and loci, with the techniques OCR repeats in the Pure Core papers.

Generated by Claude Opus 4.818 min readH245Updated 2026-06-02

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

Jump to a section

What the complex numbers strand demands
Arithmetic and the forms
De Moivre and roots
How the complex numbers content is examined
Check your knowledge

What the complex numbers strand demands

The complex numbers strand of OCR A-Level Further Mathematics A (H245) is part of the mandatory Pure Core, assessed across both Pure Core papers (Y540 and Y541). It extends the real numbers with a new object, $i$ with $i^2 = -1$ , and develops both its algebra and its geometry on the Argand diagram. The examiners reward fluent arithmetic, careful handling of the argument's quadrant, clear use of de Moivre's theorem, and labelled loci sketches.

This guide walks through the four complex numbers topics in a logical order, then sets out the exam patterns OCR repeats. Each topic has a matching dot-point page with worked exam questions; this overview ties them together.

Arithmetic and the forms

Complex arithmetic and the Argand diagram establishes addition, multiplication and division (by the conjugate), plotting on the Argand diagram, and the conjugate-pair fact that lets you solve real cubics and quartics once one complex root is known. Modulus-argument and exponential form introduces $r = |z|$ and $\theta = \arg z$ , the forms $r(\cos\theta + i\sin\theta)$ and $re^{i\theta}$ , and the rules that moduli multiply or divide while arguments add or subtract, which turn multiplication into rotation and enlargement.

De Moivre and roots

De Moivre and trigonometric identities uses $(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$ to compute powers, to derive multiple-angle identities by binomial expansion, and to linearise powers of sine and cosine with $z \pm z^{-1}$ . Roots of unity and complex loci finds all $n$ th roots of a complex number, arranged as a regular polygon, and turns modulus and argument conditions into loci: circles, perpendicular bisectors, half-lines and shaded regions.

How the complex numbers content is examined

A typical OCR profile for this strand:

Arithmetic questions. Divide two complex numbers, or solve a real cubic given one complex root.
Form conversion. Write a number in modulus-argument or exponential form, or multiply and divide in those forms.
De Moivre questions. Derive a multiple-angle identity, compute a power, or linearise $\cos^n\theta$ before integration.
Roots and loci. Find the $n$ th roots of a complex number and describe their geometry, or sketch and describe one or more loci.

Check your knowledge

A mix of recall and technique questions covering the complex numbers content. Attempt them under timed conditions, then check against the solutions.

State the value of $i^2$ . (1 mark)
Find the conjugate of $3 - 5i$ . (1 mark)
State de Moivre's theorem. (1 mark)
What shape is the locus $|z - 2i| = 4$ ? (1 mark)
How many distinct cube roots does a non-zero complex number have? (1 mark)
State Euler's relation linking $e^{i\theta}$ to trigonometric functions. (1 mark)

Check your knowledge: complex numbers

Recall questions covering the core definitions and results from the complex numbers strand.

Step 1: State the value of $i^2$

The imaginary unit $i$ is defined so that $i^2 = -1$ ; this single rule drives all complex arithmetic.

Answer: $i^2 = -1$ .

Step 2: Find the conjugate of $3 - 5i$

The complex conjugate is formed by negating the imaginary part while keeping the real part unchanged.

Answer: $3 + 5i$ .

Step 3: State de Moivre's theorem

De Moivre's theorem extends the modulus-argument form to integer powers: raising a complex number to the $n$ th power multiplies the argument by $n$ and leaves the modulus unchanged.

Answer: $(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$ for integer $n$ .

Step 4: Identify the shape of the locus $|z - 2i| = 4$

The condition $|z - a| = r$ means the distance from $z$ to the fixed point $a$ is constant, which is the definition of a circle. Reading the equation, the fixed point is $2i$ , i.e. $(0, 2)$ in the Argand diagram, and the radius is $4$ .

Answer: A circle, centre $(0, 2)$ , radius $4$ .

Step 5: State how many distinct cube roots a non-zero complex number has

A non-zero complex number always has exactly $n$ distinct $n$ th roots, equally spaced around a circle in the Argand diagram. For $n = 3$ the roots are $\tfrac{2\pi}{3}$ apart.

Answer: Three, equally spaced $\tfrac{2\pi}{3}$ apart on a circle.

Step 6: State Euler's relation linking $e^{i\theta}$ to trigonometric functions

Euler's relation connects the exponential and trigonometric forms of a complex number, and is the foundation for the exponential form $re^{i\theta}$ .

Answer: $e^{i\theta} = \cos\theta + i\sin\theta$ .

Sources & how we know this

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