On the lexicographic representation of numbers

Abstract: It is proven that, contrarily to the common belief, the notion of zero is not necessary for having positional representations of numbers. Namely, for any positive integer $k$, a positional representation with the symbols for $1, 2, \ldots, k$ is given that retains all the essential properties of the usual positional representation of base $k$ (over symbols for $0, 1, 2 \ldots, k-1$). Moreover, in this zero-free representation, a sequence of symbols identifies the number that corresponds to the order number that the sequence has in the ordering where shorter sequences precede the longer ones, and among sequences of the same length the usual lexicographic ordering of dictionaries is considered. The main properties of this lexicographic representation are proven and conversion algorithms between lexicographic and classical positional representations are given. Zero-free positional representations are relevantt in the perspective of the history of mathematics, as well as, in the perspective of emergent computation models, and of unconventional representations of genomes.