The abacus representation of numeral systems

This paper introduces a novel theoretical framework for representing the internal structure of numeral systems. This framework is based on labels and reading conventions for the entries and columns of an abacus, which suffice to describe numeral systems in a systematic way (including ones that have sub-bases or are irregular). The abacus represents, for example, a decimal place-value numeral with columns of equal height (labelled from 0 (empty) to 9) by reading the label of the greatest filled entry in each column; and a Roman numeral with columns of different heights (1 and 4), repeating the label of each column as many times as it is filled. Owing to its uniform and abstract character, this abacus representation allows for the structural comparison of numeral systems across different representational formats (such as notational, verbal and body-based formats) and for the discussion of various irregularities of such systems.This article is part of the theme issue 'A solid base for scaling up: the structure of numeration systems'.