Pure maths · Solid · Dissertation

Proving numbers are irrational: a maths EPQ idea

Why it works as an EPQ

Proof by contradiction is on the A Level course, and the proofs for √2, √n, log₂3 and e get steadily harder. You can judge where the method stops being enough.

Scope and difficulty

Solid. Solid. The proofs for √n and log₂3 are achievable; e needs a series argument; π is beyond the scope and should be referenced, not reproduced.

The maths

Builds on these A Level topics: Proof · Exponentials and logarithms · Sequences and series.

You would learn:

• The fundamental theorem of arithmetic
• The series for e
• Algebraic and transcendental numbers (for the conclusion)

Learning maths beyond the course is expected in a maths EPQ, but you must understand and explain everything you use.

One possible plan

1. Prove √2 irrational in two different ways and compare them.
2. Generalise to √n for non-square n and to log₂3.
3. Prove e is irrational from its series.
4. Discuss why π and e + π are much harder, citing the results rather than proving them.

Pitfalls

• Copying a proof for π that you cannot explain line by line.
• Forgetting to justify 'in lowest terms' at the start of each contradiction.

Where to start reading

• What Is Mathematics? (Courant and Robbins)
• Search for: proof e is irrational series

Prefer books, lecture notes, journal articles and official data to a single website, and record every source as you read (how to reference).

Similar ideas

• Fermat's Last Theorem for n = 4Why can Fermat's Last Theorem be proved for n = 4 with school mathematics when the general case took over 350 years?PureAmbitiousDissertation
• Are some infinities bigger than others?Are some infinities bigger than others, and how convincing is Cantor's argument?PureSolidDissertation
• Why mathematicians accepted imaginary numbersWhy did mathematicians come to accept imaginary numbers, and what can they do that real numbers cannot?PureSolidDissertation
• Is the golden ratio really everywhere?Is the golden ratio really found throughout nature and art, or is it a mathematical myth?PureAccessibleDissertation or artefact

All pure maths ideas · all 93 ideas

Your EPQ must be your own work. These pages coach: ideas, structure, checklists and planning. Submitting text, proofs, code or analysis written by someone else or by an AI tool as your own is malpractice. How to use help and AI honestly.