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Numerical Notation: A Comparative History

Stephen Chrisomalis

Comprehensive, encyclopaedic and scholarly are the first three words that spring to mind when reading Chrisomalis’ mammoth work on numerical notation. Not for the faint-hearted, this thoroughly researched academic tome, with an extensive bibliography stretching to some 30 pages, covers over 100 different numerical notation systems spanning over 5,000 years.

The ordering of the numerical systems is not immediately obvious, and is not chronological, but thematic. The different systems are categorised into eight major phylogenies (a term borrowed from biology meaning the study of evolutionary relatedness among groups of organisms): Hieroglyphic, Levantine, Italic, Alphabetic, South Asian, Mesopotamian, East Asian and Mesoamerican. The major advantage in such an ordering of the text is that most systems discussed were not created completely independently of each other, and the origins of and interrelations between these numerals are set in their anthropological and historical context, rather than merely as a mathematical or linguistic technical endeavour. Moreover, such context explains why most of the systems that have been used over past millennia have fallen into decline, for reasons such as the administrative convenience of calculating with positional numbers, leaving little over a dozen systems left in active usage today. All systems of notation discussed in the book are illustrated clearly in tabular form.

A separate chapter is dedicated to the so-called miscellaneous systems, with examples from a Zuni irrigation stick of a sequential-ordinal tally to the weird and wonderful Bamum numerals from the southwest of Cameroon. The book concludes with essays on Cognitive and Structural Analysis, and Social and Historical Analysis of numerals.

Our ten Western Arabic numerals, from their origin as a ‘foreign and suspicious novelty during the medieval period’, have become so familiar that it is easy to forget that there are other numerical notation systems. The first example of Western numerals is generally held to be the Codex Vigilanus, written in 976AD in the monastery of Albelda in northern Spain, although it only contained the digits one through nine and not zero (whose history as a numerical concept is a separate issue, and one which is particularly involved). We are warned that ‘it is common practice to end studies of numerical notation with the Western numerals’, however in placing them in the middle of his study Chrisomalis has chosen ‘to emphasize that their present triumph is neither inevitable nor eternal’.

I approached the text from the point of view of a mathematician with an interest in the history of mathematics, however this work will equally be of interest to the anthropologist, historian of science, or linguist.

George Matthews AMIMA
Mathematics Today April 2012

Numerical Notation: A Comparative History can be purchased at