IA idea · Pure maths, number & proof
Why is 355/113 such a good approximation to π?
Research question
How do continued fractions produce the best rational approximations to irrational numbers such as π, √2 and e, and why is 355/113 so unusually accurate?
Make it yours, then agree it with your teacher.
Next step: plan this IA free — the planner opens with this question filled in. For feedback on your own draft, the IA package (€39 once) gives you 5 AI feedback credits and every annotated exemplar in full.
Why it makes a good exploration
22/7 and 355/113 are not lucky guesses: they are convergents of π's continued fraction. Exploring why some convergents are exceptionally good (a large next term) reveals structure in numbers that looked random.
The mathematics you'll need
- Continued fractions and convergents (explain)
- Recurrence relations for numerators and denominators
- Error bounds |x − p/q| < 1/q²
- Periodic continued fractions of square roots
Where the data comes from
No data needed; check expansions against OEIS.
- OEIS (On-Line Encyclopedia of Integer Sequences) — Check a sequence you have found and read its known formulas and references.
A possible outline
- Compute continued fractions for several numbers.
- Derive the convergent recurrences.
- Measure errors and compare with 1/q².
- Explain 355/113 via the large term 292.
- Reflect on √2's periodic expansion and on e's pattern.
Pitfalls that cost marks
- Rounding errors in calculations — keep exact fractions.
- Quoting error bounds without justification.
- Too many numbers without depth.
Showing personal engagement
- Use the method to design a gear ratio or calendar approximation (e.g., leap years).
- Compare with decimal truncations.
- Find the continued fraction of your own favourite number.
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 | Continued fractions and convergents (explain); Recurrence relations for numerators and denominators |
| AA HL | Good fit | Continued fractions and convergents (explain); Recurrence relations for numerators and denominators |
| 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
Reflection (D)
For this idea, start with the weak points: rounding errors in calculations — keep exact fractions; and quoting error bounds without justification. Say how each one affects your answer.
Use of mathematics (E)
SL: continued fractions and convergents (explain); recurrence relations for numerators and denominators; error bounds |x − p/q| < 1/q²; periodic continued fractions of square roots, each explained in your own words and carried out correctly.
How pure maths explorations reach the top bands in general: Pure maths, number & proof IA ideas. Guides: A · B · C · D · E · self-grade sheet.
Taking it further
Prove that a periodic continued fraction represents a quadratic irrational.
Extending it for HL
Add an HL technique the question genuinely needs: what an HL pure maths exploration adds.
See a complete IA, marked
How far can a stack of books lean over the edge of a table? (AA HL) is a complete pure maths exploration, marked on every criterion.
Frequently asked questions
Which IB Maths course is this IA idea for?
It works for AA HL and AA SL. We rate it Solid: it needs some independent work beyond class examples.
What mathematics does this IA use?
Continued fractions and convergents (explain), recurrence relations for numerators and denominators, error bounds |x − p/q| < 1/q² and periodic continued fractions of square roots.
Where can I get data for this IA?
No data needed; check expansions against OEIS. Checked sources and tools: OEIS (On-Line Encyclopedia of Integer Sequences).
Turn this idea into your IA
The planner opens with “Why is 355/113 such a good approximation to π?” filled in: the question, the maths and the data sources.
Your plan is kept in this browser. A free account keeps it with your drafts on any device: Join free · Sign in
Also: free IA self-check · A–E checklist (PDF) · have a draft? AI feedback or a review with Pete (€100)
Writing it up: how to write a Maths IA, step by step · structure and format · mistakes that cost marks · annotated exemplars, marked on every criterion
Similar ideas
- How long is a coastline? Measuring fractal dimensionAA SLAA HLAI SLAI HLSolid
- The Collatz conjecture: what can we say about stopping times?AA SLAA HLAI SLAI HLSolid
- How many primes are there below n? Testing the prime number theoremAA SLAA HLAI SLSolid
- Beyond Fibonacci: what ratio do other recurrence sequences approach?AA HLAA SLSolid
All pure maths ideas · AA HL ideas · AA SL ideas · All 239 IA ideas