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Comment by mci | Hacker News Reader

This discussion would really go better if you followed the links. See lines 92–96 of cf.py:

      By the
    # Gauss-Kuzmin theorem [5, 6], a partial quotient in the
    # continued fraction expansion of almost every number exceeds
    # 1e+31 with probability log2(1+1e-31) =~= 1e-31/ln(2) =~=
    # 1.5e-31.

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I'm somewhat familiar with Gauss-Kuzmin theorem. If I understand it well, it means that large partial quotients are rare in most numbers. This is fairly intuitive, and conforming to the experiences of anybody who has done practical computations of continued fractions, as you certainly have.

Notice that this result does not say that most numbers have bounded partial quotients. In fact, a classical result in diophantine approximation theory, if I recall correctly, says that the set of numbers whose partial fractions are bounded (called badly aproximable numbers) is of Lebesgue measure zero. Thus, if your system represents real numbers by sequences of bounded partial quotients, then it can only represent a negligible (but uncountable) set of real numbers. A "random" number in [0,1] will certainly have an unbounded sequence of partial quotients (i.e., with probability 1).

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