Number theory · Maths EE idea · Solid
Best rational approximations to π
A research question to start from
Why are the convergents of a continued fraction the best rational approximations to a real number, and what does this say about approximations such as 22/7 and 355/113?
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
A precise notion of 'best' that you can define, prove and test numerically.
Mathematics you would need
- Continued fraction algorithm
- Convergents and their recurrences
- Error bounds |x − p/q| < 1/q²
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
- Compute the continued fraction of π and its convergents.
- Prove the error bound for convergents.
- Prove (or carefully present) the best-approximation property.
- Compare with approximations to e and √2 and explain the differences.
Scope and difficulty
Solid. Solid; narrow to one or two constants.
Pitfalls
- Treating numerical agreement as proof.
- Confusing best approximations of the first and second kind.
Where to start reading
Search a library catalogue or a university's open lecture notes for: best rational approximation continued fraction convergents; Hurwitz theorem. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).