A Level · Cambridge (CIE) · 9709

Describe modulus circles disks and annular regions

Interpret |z − a| as distance from a fixed point, draw circular boundaries and distinguish strict and inclusive radial inequalities.

Section 3.9 · Complex numbers

Pure Mathematics 3 · Complex numbers · 3.9

Exam scope and assessment notes

Pure Mathematics 3 (Paper 3, A Level) · Mathematics 9709, 2026–2027, section 3.9. Paper 1 algebra and radian trigonometry are assumed. Use Cartesian working for products, quotients and the two square roots of a non-zero complex number. Arguments usually use −π < θ ≤ π; an alternative interval 0 ≤ θ < 2π is acceptable unless specified. The argument of zero is undefined. General nth-root methods are beyond this mapped sequence. Written practice is self-checked; human subject-teacher review remains pending.

Before you begin

Interpret complex differences as displacements.
Use the equation of a circle and inequality boundaries.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

What does |z − a| measure on an Argand diagram?

What you will learn

Convert a fixed-distance locus to a circle and Cartesian equation.
Describe disks and annuli with the correct included or excluded boundaries.

The shaded open disk is |z − (1 + i)| < 2, centred at (1,1) with radius two. The dashed circumference is excluded. The filled centre is included, while the open point at 3 + i is excluded because its distance from the centre is exactly two. The horizontal and vertical scales are equal. Equality would select only the circumference; replacing < with ≤ would also include that boundary.

A shifted modulus measures distance from its centre

For a = u + iv and z = x + iy, |z − a| = √[(x − u)² + (y − v)²]. Thus |z − a| = k, for k > 0, is the circumference of a circle centred at (u,v) with radius k. The equation |z + 2 − i| = 3 uses z − (−2 + i), so its centre is (−2,1) and Cartesian equation is (x + 2)² + (y − 1)² = 9. Read the whole shifted value before naming the centre.

The locus |z − (1 + i)| < 2 is the open disk centred at (1,1), not just its circumference. Its circular boundary is excluded, while its centre is included because distance zero is less than two. Use a dashed or otherwise clearly labelled excluded boundary. The inclusive condition ≤ 2 adds the circumference; a condition > 2 selects the exterior and excludes the circumference.

Two radial bounds define an annulus

The conditions 1 ≤ |z − (2 − i)| ≤ 3 give an annulus centred at (2,−1), with both radius-one and radius-three boundaries included. Its centre is excluded by the lower bound. A strict lower bound with inclusive upper bound excludes the inner circle but includes the outer one. Describe each boundary separately rather than calling the whole region open or closed without checking both inequalities.

Test membership using squared distance where radii are non-negative: distance < √5 is equivalent to squared distance < 5. At a = −1 + 2i, point 1 + 3i has squared displacement 4 + 1 = 5, so it lies on the boundary and fails a strict inequality. The centre has distance zero and satisfies the open-disk condition. Negative distance bounds need a direct check: |z − a| < −1 has no solutions because modulus cannot be negative.

Pause and explain

For |z − (1 + i)| < 2, which boundary statement is correct?

Put the idea to work

Worked example

Describe and sketch |z − (1 + i)| < 2, including its boundary.

Show the worked solution

The shifted modulus is distance from (1,1).
The condition selects every point less than two units from that centre.
Shade the interior and exclude the radius-two circumference; the centre itself is included.

Answer Open disk centred at (1,1), radius 2; circumference excluded, centre included.

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

Describe |z + 2 − i| = 3 and write its Cartesian equation for z = x + iy.

Give me a hint

Rewrite the shifted term as z minus the centre’s complex value.

Compare my reasoning

Use z + 2 − i = z − (−2 + i).
The centre is (−2,1) and the radius is 3.
Squaring the distance gives (x + 2)² + (y − 1)² = 9, describing the circumference.

Circle centred at (−2,1), radius 3; (x + 2)² + (y − 1)² = 9.

Look for these in your work

Read the centre with the correct shifted signs.
Distinguish the circumference from its interior.

2 · Independent

Describe 1 ≤ |z − (2 − i)| ≤ 3, specifying the centre and both boundaries.

Give me a hint

Apply each distance bound separately.

Compare my reasoning

The centre is (2,−1).
Permitted distances are from 1 to 3 inclusive, so the region is an annulus.
Both circular boundaries are included, while the central disk of radius less than 1 is excluded.

Annulus centred at (2,−1), inner radius 1 and outer radius 3; both boundaries included.

Look for these in your work

Identify both radii and the signed centre.
Explain membership at each boundary and at the centre.

3 · Transfer

For |z − (−1 + 2i)| < √5, decide whether z = 1 + 3i, z = −1 + 2i and z = 2 + 2i belong. Explain each decision.

Give me a hint

Compare each squared displacement with 5; equality fails a strict inequality.

Compare my reasoning

At 1 + 3i the displacement is 2 + i with squared length 5, so this boundary point is excluded.
At −1 + 2i the displacement is zero, so the centre is included.
At 2 + 2i the displacement is 3 with squared length 9 > 5, so it is outside and excluded.

1 + 3i excluded (boundary); −1 + 2i included (centre); 2 + 2i excluded (outside).

Look for these in your work

Calculate distances from the supplied centre.
Respect the strict boundary rather than accepting equality.

Common mistakes

Read z − a carefully to avoid reversing the centre’s signs.
An inequality describes a region, while equality describes its boundary.
Test both annular boundaries and do not infer a direction from modulus alone.

Recall without your notes

What can you explain now?

Does |z − a| < −1 have any solutions?

Compare with the explanation

No.

A distance is non-negative, so it cannot be less than a negative bound.

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Make it stick

Translate a shifted modulus equality or inequality into a circle, disk or annulus, stating each boundary’s membership.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics

Check the official syllabus 2026–2027

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.