IA idea · Numerical methods & error analysis

Why does the ancient square-root method double the correct digits?

AA SLAA HL Solid Also in: Pure maths, Calculus

Research question

Why is the 'average of x and a/x' method for √a the same as Newton–Raphson, why does the number of correct digits roughly double each step, and how does it compare with bisection?

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

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Why it makes a good exploration

An ancient method, a modern one and a striking pattern in the errors all connect. Showing the error is squared each step is a proof within reach of a strong SL student.

The mathematics you'll need

Iteration and recurrence relations
Newton–Raphson from the tangent line (new: derive it)
Error eₙ₊₁ ≈ eₙ²/(2√a): quadratic convergence
Bisection and its linear convergence
Logarithms to count correct digits

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; iterate in a spreadsheet with enough decimal places.

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

Run the averaging method for several a.
Derive Newton–Raphson and show it gives the same formula.
Tabulate errors and show the squaring pattern.
Prove the error relation algebraically.
Compare with bisection and reflect on starting values.

Pitfalls that cost marks

Spreadsheet rounding hiding the error pattern; say how many digits you trust.
No proof of the convergence rate.
Forgetting what happens for a bad starting value.

Showing personal engagement

Find the method's history yourself and test it by hand.
Predict the number of steps for 15 correct digits.
Extend to cube roots and compare.

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Iteration and recurrence relations; Newton–Raphson from the tangent line (new: derive it)
AA HL	Good fit	Iteration and recurrence relations; Newton–Raphson from the tangent line (new: derive it)
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: Solid. Needs some independent work beyond class examples. 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: spreadsheet rounding hiding the error pattern; say how many digits you trust — 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

Derive and test the method for cube roots, or prove the iteration always converges for any positive start.

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).