Why is the reciprocal of the fractional part (vs the integer part) taken at each step? Perhaps this is obvious, if so I don't see it. Appreciate your guidance.
Example from Wiki showing reciprocal taken at each step:
2026-03-30 21:49:57.1774907397
Continued fractions, why the reciprocal?
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It's just the Euclidean algorithm.
$$\cfrac{649}{200}=3+\cfrac{49}{200}=3+\cfrac{1}{\cfrac{200}{49}}=3+\cfrac{1}{4+\cfrac{4}{49}}=3+\cfrac{1}{4+\cfrac{1}{\cfrac{49}{4}}}=3+\cfrac{1}{4+\cfrac{1}{12+\cfrac{1}{4}}}$$