Number theory · Maths EE idea · Accessible
How long is the repeating block of 1/n?
A research question to start from
What determines the length of the repeating block in the decimal expansion of 1/n, and for which primes is it as long as possible?
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
Starts from a calculator observation and leads to orders modulo n and primitive roots: real number theory from a simple question.
Mathematics you would need
- Long division and remainders
- Orders modulo n
- Fermat's little theorem
- Primitive roots
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 period lengths for n up to 50 and look at prime n first.
- Prove the period of 1/p equals the order of 10 modulo p.
- Explain why the period divides p − 1 and discuss full-period primes.
- Extend to composite n.
Scope and difficulty
Accessible. Accessible; composite n and the full-period question give depth.
Pitfalls
- Only computing, not proving.
- Ignoring factors of 2 and 5.
Where to start reading
Search a library catalogue or a university's open lecture notes for: period of decimal expansion order of 10 modulo p; full reptend primes. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).