A Level · Cambridge (CIE) · 9709

Verify numerical accuracy without mistaking stable digits for proof

Combine a given iteration, original-equation evidence and rounding bands, recognising that small residuals and close successive terms need interpretation.

Pure Mathematics 2 & 3 · Numerical solution of equations · 2.6 / 3.6

Exam scope and assessment notes

Pure Mathematics 2 (Paper 2, AS only) and Pure Mathematics 3 (Paper 3, A Level) · Mathematics 9709, 2026–2027. Paper 1 graph and algebra knowledge is assumed. The AS route using Papers 1 + 2 cannot be carried forward to A Level. Shared sections 2.6 and 3.6 include locating roots graphically or by sign changes, notation for convergent approximation sequences, and using a given iteration or rearrangement to a prescribed accuracy. An iteration may fail to converge. Knowledge of the general condition for convergence is not required here. Check the original equation, domains and requested rounding. Written practice is self-checked.

Before you begin

Run given iterations and recognise failure or domain problems.
Check original continuous sign brackets and decimal rounding bands.

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

Quick readiness check

Must a correctly rounded approximate root always give exactly zero in the original equation?

What you will learn

Report a numerical root with its starting value, given recurrence and justified requested precision.
Explain why close iterates or a small equation residual alone do not guarantee root accuracy.

A root report should show what the numbers establish

A useful numerical solution records the original equation, given recurrence and seed, enough terms to show the calculation, and the requested final precision. A convergent approximation is not normally an exact root after rounding. For the supplied rule xₙ₊₁ = (xₙ + 2/xₙ)/2, x₀ = 1, the terms 1.5, 1.416666667 and 1.414215686 approach the positive root of x² − 2 = 0. Its fixed-point equation recovers x² = 2 for x ≠ 0, and the positive iterates retain the requested branch.

The original sign bracket (1.4142, 1.4143) justifies 1.414 to three decimal places. Do not replace the given recurrence with a memorised unrelated method, and do not require the rounded value to produce exactly zero. For a cosine fixed-point equation, the standard numerical arguments use radians; a degree-mode calculation would generate a different sequence and solve a different equation.

Close values and small residuals need an independent accuracy check

Consider xₙ₊₁ = 0.9999xₙ from x₀ = 1, whose only fixed point is 0. The first values 1, 0.9999 and 0.99980001 all round to 1.000 at three decimal places, although they are far from the root 0. Agreement of displayed digits alone is therefore not a universal stopping proof. A small residual f(x) also depends on the equation's scale: f(x) = 10^(−9)(x − 100) has a tiny value at x = 0 despite that input being 100 units from its root.

For x³ + x − 1, a valid bracket (0.6823, 0.6824) establishes the three-decimal answer 0.682, but crosses the four-decimal rounding threshold 0.68235. Further original sign checks at 0.68232 and 0.68233 narrow the bracket to one four-decimal band, establishing 0.6823. Report only the precision supported by the evidence and retain working digits. A sketch, residual or pair of iterates can guide checks without automatically certifying every requested digit.

Pause and explain

Which bracket justifies the cubic root to four decimal places as 0.6823?

Put the idea to work

Worked example

Use the supplied rule xₙ₊₁ = (xₙ + 2/xₙ)/2, x₀ = 1, for the positive root of x² − 2 = 0. Report the first three new terms and justify a three-decimal answer.

Show the worked solution

Generate x₁ = 1.5, x₂ = 17/12 ≈ 1.416666667 and x₃ = 577/408 ≈ 1.414215686, retaining full precision.
The positive inputs avoid the rule's zero denominator; its fixed point gives x² = 2 on the non-zero branch.
Original signs at 1.4142 and 1.4143 bracket a root inside one three-decimal rounding band, giving 1.414.

Answer x₁ = 1.5, x₂ = 17/12, x₃ = 577/408; positive root approximately 1.414 to three decimal places.

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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

For xₙ₊₁ = 0.9999xₙ, x₀ = 1, calculate x₁ and x₂, round them to three decimals and decide whether that proves the fixed point is near 1.

Give me a hint

Solve r = 0.9999r separately from examining two displayed terms.

Compare my reasoning

The new terms are x₁ = 0.9999 and x₂ = 0.99980001.
Both round to 1.000 at three decimal places.
The fixed-point equation gives 0.0001r = 0, so r = 0. The displayed agreement does not establish closeness to the root.

x₁ = 0.9999, x₂ = 0.99980001; both display 1.000, but the fixed point is 0, so the accuracy claim fails.

Look for these in your work

Generate and round the actual sequence values.
Use the original fixed-point equation to expose the false stopping claim.

2 · Independent

Use the given rule xₙ₊₁ = cos xₙ in radians, x₀ = 0.5, to approximate a solution of cos x = x to three decimal places. Check the original bracket (0.7390, 0.7391).

Give me a hint

Retain the full radian calculator values and evaluate f(x) = cos x − x at both bracket ends.

Compare my reasoning

The sequence begins approximately 0.877582562, 0.639012494, 0.802685101, … and approaches 0.739085133.
Original endpoint values are approximately 0.000142477295 and −0.000024881312, giving a continuous sign-change bracket.
Every value in the bracket rounds to 0.739 to three decimal places; using degree arguments would solve a different fixed-point rule.

Root approximately 0.739 to three decimal places, using radian iteration and the original bracket.

Look for these in your work

Use the given radian recurrence and seed with retained precision.
Support the final rounding by original-equation signs.

3 · Transfer

For x³ + x − 1 = 0, the verified root bracket is (0.6823, 0.6824). State what it proves at three and four decimal places, then check (0.68232, 0.68233) to improve the report.

Give me a hint

Compare each interval with the relevant rounding threshold before increasing the claimed precision.

Compare my reasoning

The first bracket lies wholly in the three-decimal band for 0.682, but crosses four-decimal boundary 0.68235.
Original values at 0.68232 and 0.68233 are approximately −0.000018703417 and 0.000005263605, so continuity gives the finer root bracket.
The finer bracket lies wholly in the four-decimal band for 0.6823; the supported report can now be improved.

First bracket proves 0.682 to three decimals but not a unique four-decimal value; finer signs justify 0.6823 to four decimals.

Look for these in your work

Match each accuracy claim to its full rounding band.
Use additional original-function evidence before reporting an extra digit.

Common mistakes

Matching displayed digits can occur long before a sequence is close to its root.
A small residual is not a universal root-error bound without information about the equation.
Use the requested decimal places or significant figures and verify the entire supporting bracket.

Recall without your notes

What can you explain now?

Can two successive values both rounding to 1.000 alone prove a root rounds to 1.000?

Compare with the explanation

No.

A slowly changing sequence can display stable rounded digits while remaining far from its actual fixed point.

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Make it stick

Explain how a complete numerical-root report links the recurrence, domain, original equation and requested precision. Give a counterexample to relying only on matching rounded iterates or a tiny residual.

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.