IA idea · Pure maths, number & proof
The product of a regular polygon's diagonals
Research question
For a regular n-gon inscribed in a unit circle, why is the product of the distances from one vertex to all the others exactly n?
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Why it makes a good exploration
Measure the chords of a regular polygon and multiply them — you always get n. The proof uses roots of unity and a factorisation of zⁿ − 1, one of the most elegant uses of complex numbers in AA HL.
The mathematics you'll need
- Complex numbers in polar and Euler form
- Roots of unity and the factorisation of zⁿ − 1
- Modulus of products
- Limits and derivatives (the polynomial 1 + z + … + zⁿ⁻¹ at z = 1)
- Trigonometric product identities
Where the data comes from
Verify numerically and by measurement on drawn polygons in GeoGebra.
- GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.
A possible outline
- Measure for small n and conjecture.
- Represent vertices as roots of unity.
- Factorise and evaluate at z = 1.
- Derive a trigonometric identity as a corollary.
- Reflect on the power of changing representation.
Pitfalls that cost marks
- Numerical verification presented as proof.
- Skipping the factorisation step.
- Losing track of which roots are included.
Showing personal engagement
- Discover the result yourself with GeoGebra before reading about it.
- Find the corresponding result for sums of squared distances.
- Explain the proof to a classmate and note what confused them.
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 | Complex numbers in polar and Euler form; Roots of unity and the factorisation of zⁿ − 1 |
| 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
Reflection (D)
For this idea, start with the weak points: numerical verification presented as proof; and skipping the factorisation step. Say how each one affects your answer.
Use of mathematics (E)
HL: complex numbers in polar and Euler form; roots of unity and the factorisation of zⁿ − 1; modulus of products; limits and derivatives (the polynomial 1 + z + … + zⁿ⁻¹ at z = 1); trigonometric product identities, 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 the related identity for ∏ sin(kπ/n) and explore what happens for points not on a vertex.
Extending it for HL
Add an HL technique the question genuinely needs: what an HL pure maths exploration adds.
See a complete IA, marked
What is the quickest way to deliver a leaflet to every house on my estate? (AI 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. We rate it Ambitious: it suits confident students, who should expect to learn some mathematics on their own.
What mathematics does this IA use?
Complex numbers in polar and Euler form, roots of unity and the factorisation of zⁿ − 1, modulus of products, limits and derivatives (the polynomial 1 + z + … + zⁿ⁻¹ at z = 1) and trigonometric product identities.
Where can I get data for this IA?
Verify numerically and by measurement on drawn polygons in GeoGebra. Checked sources and tools: GeoGebra.
Turn this idea into your IA
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