Finite precision lexicographic continued fraction number systems
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Lexicographic continued fraction binary (LCF) representation provides an order preserving bitstring representation of the non negative real numbers where every rational number has a finite length bitstring representation. We investigate the precision of k-bit LCF approximation. The maximum gap size over [0,1] for (k+1)-bit LCF representation is shown to be less than 2<sup>−.81k</sup>, comparable to binary coded decimal in worst case representation efficiency. The distribution of gap sizes for (k+1)-bit LCF representation over [0,1] is shown on a logarithmic scale to be bell shaped between 2<sup>−.81k</sup> and 2<sup>−1.39k</sup>, becoming more peaked near the value corresponding to uniform spacing, 2<sup>−k</sup>, with increasing k.