I was posed this question on an exam (luckily a bonus) I had recently and I was wondering how you would go about doing it algebraically.
Prove that a number is rational if and only if its decimal expansion becomes periodic at some point.
I was posed this question on an exam (luckily a bonus) I had recently and I was wondering how you would go about doing it algebraically.
Prove that a number is rational if and only if its decimal expansion becomes periodic at some point.
I found an elegant prove of this, (way too hard as an exam question) with a stronger conclusion a/b has a periodical expansion AND the length of the period never exceeds the denominator!
The prove is by no means elementary because it uses infinite series.
LaTeX is currently offline, but here is the link try to make use of it. Heir