Continued Fraction Expansion
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Quadratic irrationals (numbers of the form, where a, b and c are integers), and in particular, square roots of integers, have periodic continued fractions. Sometimes what is desired is finding not the numerical value of a square root, but rather its continued fraction expansion. The following iterative algorithm can be used for this purpose (S is any natural number that is not a perfect square):
Notice that mn, dn, and an are always integers. The algorithm terminates when this triplet is the same as one encountered before. The expansion will repeat from then on. The sequence is the continued fraction expansion:
Read more about this topic: Methods Of Computing Square Roots
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