  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.
