On the lexicographic representation of numbers

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.Comment: 15 page

Non-power positional number representation systems, bijective numeration, and the Mesoamerican discovery of zero

Pre-Columbian Mesoamerica was a fertile crescent for the development of number systems. A form of vigesimal system seems to have been present from the first Olmec civilization on wards, to which succeeding peoples made contributions. We discuss the Maya use of the representational redundancy present in their Long Count calendar, a non-power positional number representation system with multipliers 1, 20, 18 x 20, ..., 18 x 20(n). We demonstrate that the Mesoamericans did not need to invent positional notation and discover zero at the same time because they were not afraid of using a number system in which the same number can be written indifferent ways. A Long Count number system with digits from 0 to 20 is seen later to pass to one using digits 0 to 19, which leads us to propose that even earlier there may have been an initial zeroless bijective numeration system whose digits ran from 1 to 20. Mesoamerica was able to make this conceptual leap to the concept of a cardinal zero to perform arithmetic owing to a familiarity with multiple and redundant number representation systems