I know that if this expansion is finite, then I can go to the lowest denominator in the whole fraction and turn it into a fraction and keep doing so until I get a fraction which means the number is rational. But how do I prove the opposite? What I thought of, is that if it is rational, then I can use the process to make it a continued fraction which is finite, I believe. But it doesn't seem like a proof to me. I would really appreciate your assessment.
2026-03-28 08:51:11.1774687871
A real number is rational $\iff$ its continued fraction expansion is finite.
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Now conversely suppose that continued fraction is finite then we'll show that number is rational.