  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 onwards, 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$\times$ 20, $\ldots$,
18$\times$ 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 in
different 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.
