Why is 355/113 such a good approximation to π?

AA HLAA SL Solid

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?

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

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

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; check expansions against OEIS.

OEIS (On-Line Encyclopedia of Integer Sequences) — Check a sequence you have found and read its known formulas and references.

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

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.

Taking it further

Prove that a periodic continued fraction represents a quadratic irrational.

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).