IA idea · Numerical methods & error analysis

When the computer gets it wrong: rounding error and cancellation

AA HLAA SLAI HL Ambitious Also in: Pure maths, Calculus

Research question

Why does a spreadsheet give badly wrong values of (1 − cos x)/x² for very small x, at what x does the error take over, and how can rewriting the expression fix it?

Adapt it: change the place, the data or the comparison until the question is yours.

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

Students trust technology. Finding exactly where it fails, and why, is a rigorous and surprising exploration of limits, series and floating-point arithmetic.

The mathematics you'll need

Limits (the true value tends to 1/2)
Maclaurin series for cos x
Relative error and catastrophic cancellation
Rewriting with trigonometric identities (2 sin²(x/2))
Log-log error plots

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

No data needed; experiment in a spreadsheet and on a GDC and compare.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Find the limit with series.
Tabulate the spreadsheet values as x shrinks.
Explain the failure with cancellation.
Fix it with an identity and compare.
Reflect on trusting technology in your own IA.

Pitfalls that cost marks

Blaming 'the computer' without explaining the mechanism.
Not quantifying where the error takes over.
Using only one device.

Showing personal engagement

Find another expression that breaks the same way.
Compare your calculator, spreadsheet and Desmos.
Predict the x at which errors appear before testing.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Fits — ambitious at SL	Limits (the true value tends to 1/2); Maclaurin series for cos x
AA HL	Good fit	Limits (the true value tends to 1/2); Maclaurin series for cos x
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Good fit	Limits (the true value tends to 1/2); Maclaurin series for cos x

Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Choose the function or equation yourself and predict how each method will behave before you run it. Hunting for the cases that break a method is engagement an examiner can see.

Reflection (D)

Reflect on error: how it changes with step size or iterations, why a method converges slowly or not at all, and how you know your 'exact' comparison value is correct. For this idea, start with: blaming 'the computer' without explaining the mechanism — say how it affects your answer.

Use of mathematics (E)

SL: The trapezoidal rule or a simple iteration applied correctly, errors tabulated against step size and explained, with any method outside the syllabus explained step by step.

HL: Convergence analysed rather than observed: an error bound derived with calculus or a series, an order of convergence measured and justified, or Euler's method studied against an exact solution.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Investigate summing a series forwards and backwards and explain why the results differ.

Extending it for HL

Derive an error bound with a Maclaurin series or calculus, then show your numerical results follow it.

See a complete IA, marked

Our annotated exemplar Is a hanging chain a parabola? Comparing catenary and quadratic models (AA SL) asks a different question, but shows how a complete numerical methods exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).