IA idea · Numerical methods & error analysis

How many terms do you need? Fast and slow series for π

AA HL Ambitious Also in: Pure maths

Research question

How many terms do the Leibniz series and a Machin-type arctangent formula need to give π to ten decimal places, and can you predict that number from an error bound before you calculate?

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

Two infinite series for the same number behave very differently. Bounding the error in advance, then checking it, is HL mathematics doing real work.

The mathematics you'll need

Maclaurin series for arctan x
Alternating-series error bound
Machin's formula checked with compound-angle identities
Comparing rates of convergence
Logarithms to predict the number of terms

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; use a spreadsheet with enough precision and say how many digits you trust.

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

Derive the arctan series.
Use the Leibniz series and its error bound.
Verify a Machin-type formula with tan identities.
Predict and check the number of terms each needs.
Reflect on spreadsheet precision.

Pitfalls that cost marks

Quoting Machin's formula without verifying it.
Exceeding spreadsheet precision without noticing.
History instead of mathematics.

Showing personal engagement

Find your own arctan identity for π/4.
Predict before you compute.
Compare with a method you already know (continued fractions).

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Not a natural fit	The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics.
AA HL	Good fit	Maclaurin series for arctan x; Alternating-series error bound
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	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

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: quoting machin's formula without verifying it — 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

Find other identities of Machin type and rank them by speed.

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 How far can a stack of books lean over the edge of a table? (AA HL) 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).