Maths IA · Algebra and functions · topic 8 of 40

Roots of unity: a pentagon by ruler and compass? · IB Maths IA idea: guide, sample graph and step-by-step plan

HL-level and elegant: go in prepared

A regular pentagon can be drawn with only a ruler and a compass. A regular heptagon cannot. The reason sits inside the complex numbers, and you're going to find it.

Difficulty●●● Stretch
KitPaper, compass and ruler, then graphing software
TimeAbout 13 to 15 hours
FitsAA HL
  • ✓ you like geometry and algebra together
  • ✓ you enjoy complex numbers and Euler form
  • ✓ you can follow a long chain of reasoning

The big idea

← geometry turned into algebra

The solutions of z^n = 1 are the points of a regular n-gon on the unit circle. Draw a pentagon, and you must build a length such as cos 72° from lengths you are given. Roots of unity let you turn that geometry into an algebra problem: find the polynomial that the length satisfies, and ask what ruler and compass can do to such polynomials. This is HL material and needs real HL-level maths for criterion E: de Moivre, polynomial factorisation, Euler form, with proofs and not only pictures.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

Swipe for more twists →

Pentagon by algebra

Find cos 72° exactly by factorising z⁵ − 1 and using symmetry of the roots.

→ A clean algebraic derivation, followed by a check with a calculator and a geometric construction.
Do the construction

Draw the pentagon with ruler and compass, and prove that your steps give the right angle or length.

→ A bridge between the algebra and the picture. The proof of the construction is the core of your criterion E.
Other polygons

Study the triangle, square, hexagon and the pentagon: which have simple constructions, and what do their roots have in common?

→ A comparison of cases that makes a pattern visible. Be careful to say what is shown and what is only observed.
Why the heptagon fails

Explore the polynomial for cos(2π/7) and what makes its roots hard to reach with ruler and compass.

→ A stretch into constructibility. You may need to state a result without proof; say so clearly.
Sum of the roots

Prove that the roots of unity add to zero and find other patterns in their products and powers.

→ Short proofs with geometric meaning: balance on the circle. Good for rigour in a smaller space.
Change the context

Polygons in architecture, tiles or logos with five-fold symmetry.

→ A real motivation. The maths stays the same, but the diagram and the introduction look different.

Your Maths IA research question: run the check

what will you PROVE?

Write your own question first. Then make sure it ticks every box:

  • Names the polygon(s) and the property you will study: roots, exact lengths, or constructibility.
  • Says what you will prove and what you will only state from a source.
  • Links the complex numbers to the geometry in a way that you explain, not just draw.
  • Needs HL-level maths to answer: de Moivre, Euler form, polynomial factorisation, proof.
  • Is narrow enough for 12 to 20 pages: one or two polygons, in depth.

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The picture to draw

label the exact values

The five roots of z⁵ = 1 on the unit circle. Draw yours on the Argand diagram, with the angles and the exact coordinates you find.

1ωω²ω³ω⁴ReImfive roots of z⁵ = 1each step turns by 2π/5which lengths can be built?

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

Introduction and aim~1.5 pages
Why a pentagon, what ruler and compass can and cannot do, and your question.
Roots of unity~2.5 pages
The solutions of z^n = 1 from de Moivre and Euler form, with the polygon picture and a proof that they are equally spaced.
The polynomial~3 pages
Factorising z⁵ − 1, the symmetric structure of the roots, and the exact value of a key length.
Checking the value~1 page
A numerical check and a graph, with accuracy stated.
Construction~3 pages
Ruler and compass steps for the pentagon, with a proof that each step is valid.
Another polygon~2 pages
The case where the method fails or changes, and why.
Reflection and evaluation~1.5 pages
What each method gave you, which proofs you can fully defend, and which results you took on trust.
Conclusion~1 page
Answer your aim using what you proved.
Referencesnot counted
Sources for the construction and any results stated without proof.

What your Maths IA graph could look like

Made-up illustration of the form of a strong graph, not the answer. Real roots of z^n = 1 against the complex ones. example only

-1.5-1.0-0.50.00.51.01.5-10-8-6-4-202468x (real axis only)f(x)x⁵ − 1x⁴ − 1only one real root here:where are the other four?ILLUSTRATIVE · MADE-UP DATA
  • Axis titles say what is plotted and that it is the real line only.
  • Both curves are drawn, so the reader sees the odd and even degree cases and the real roots.
  • The real graph shows only the real roots; the rest are in the complex plane, which is why the Argand diagram is needed.
  • Include the scale to the points where the curves cross the axis.
  • Another graph to consider: plot the points cos(2πk/n) on a number line for several n, and see how they pack.

Your game plan

prove the construction!

Tick steps off as you go. We'll remember on this device.

0 of 10 done

  1. Draw a pentagon, predictDraw a regular pentagon on the unit circle. Predict the coordinates of its vertices and what ruler and compass would need to produce.
  2. Trial run (1 lesson)Use de Moivre to find the fifth roots of unity in polar form and check them on an Argand diagram by hand.
  3. Show they are equally spacedProve that the roots are the vertices of a regular polygon, and say clearly what you assume.
  4. Factorise and use symmetryFactorise z⁵ − 1 and look for the way symmetry reduces the degree. Work out each step in your own notation.
  5. Find the exact valueSolve for the key length. Check numerically, and say how accurate your check is.
  6. ConstructPlan a ruler and compass construction that produces that length. Draw it carefully.
  7. Prove the constructionShow that each step gives the right length or angle. A drawn construction without a proof is only a picture.
  8. Test another polygonTry a case where the method behaves differently. What breaks, and why?
  9. ReflectWhich parts could you prove yourself? Which parts did you take from a source?
  10. Write, cut, checkWrite in the layout order. Check notation (z, ω, arguments in radians or degrees, kept consistent). Run the checklist before your teacher's draft.

Maths you'll need

Traps to avoid in this Maths IA

  • Drawing a pentagon by a recipe from a website, without a proof that it works.
  • Mixing degrees and radians in the same working.
  • Stating that a number is constructible without showing the steps, or the algebra that supports it.
  • At HL, using only SL trigonometry. Criterion E needs complex numbers and polynomials used with rigour.
  • Quoting a theorem on constructibility and applying it without saying what it actually says.
  • Decimal approximations used as exact values, so that the proof never closes.
  • A history of the Greek problems that does not connect to the maths in the rest.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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