Non-Power Positional Number Representation Systems, Bijective Numeration, and the Mesoamerican Discovery of Zero

TL;DR

The paper explores Mesoamerican number systems, showing they used non-power positional representations with redundancy, which facilitated the conceptual leap to zero and arithmetic without needing to invent zero simultaneously.

Contribution

It reveals that Mesoamerican numeration evolved from non-zero-based systems with redundancy, challenging the idea that zero was independently invented at the same time as positional notation.

Findings

Mesoamerican systems used non-power positional representations with redundancy.

They transitioned from digits 0-20 to 0-19, implying earlier zeroless systems.

Redundant representations helped conceptualize zero and perform arithmetic.

Abstract

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.