Suppose we can write a real number x as a continued fraction like this(adsbygoogle = window.adsbygoogle || []).push({});

x=a0+1/(a1+1/(a2+1/(a2+...=[a0; a1, a2, a3, a4, ... an...].

Is there a binary operation f(i,x) so that f(i,x)=ai? I was wondering if there was a formula which gives the ith item in the sequence of integers which is connected to every x in the context of this expansion.

Every rational number has a unique continued fraction expansion so I think this is a valid question. Moreover, every irrational number has a unique, infinite continued fraction expansion.

My first guess was to combine the inputs a, b in the Euclidean algorithm from which the continued fraction expansion arises but I don't know how to extract the ith item in the sequence of quotients. Any thoughts?

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# Continued Fractions

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