Analysis and calculus · Maths EE idea · Ambitious
Proving numbers are irrational
A research question to start from
How can the irrationality of √2, e and π be proved, and what do the methods have in common?
A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.
Why it works as a maths EE
Proofs of increasing difficulty with a comparison of techniques as the line of argument.
Mathematics you would need
- Proof by contradiction
- Series bounds
- Integrals and polynomials (Niven-style proof)
- Inequalities
Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.
One possible line of attack
- Prove √2 and √n (non-square) irrational.
- Prove e irrational from its series.
- Work through a proof that π is irrational, explaining each step.
- Compare the strategies.
Scope and difficulty
Ambitious. Ambitious; the π proof needs careful explanation.
Pitfalls
- Copying the π proof.
- No comparison.
Where to start reading
Search a library catalogue or a university's open lecture notes for: proof e irrational series; Niven proof pi irrational. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).