Numerical Notation

Numerical Notation: A Comparative HistoryNumerical Notation: A Comparative History (Cambridge, 2010)
Stephen Chrisomalis, Wayne State University

Numerical Notation is a cross-cultural anthropological examination of systems of written numerals used over the past 5000 years. I argue that numerical notations (like Roman numerals or Egyptian hieroglyphic numerals) are mainly social and representational tools, rarely if ever used directly for arithmetic. Understanding how they develop, change, and decline requires us to analyze how their users actually perceived and used them. Computational efficiency is only rarely relevant to their transformation and replacement. There are regularities and patterns among the world’s written numerals, but they derive from more general principles of representation and cognition. It is a thoroughly interdisciplinary study incorporating theories and principles from anthropology, archaeology, linguistics, epigraphy/paleography, cognitive science, developmental psychology, semiotics, and history of science.

Numerical Notation is now available for purchase through the Cambridge University Press website (click the image or link above), and is frequently available at a 20% or greater discount from, Barnes and Noble, and other vendors.


The following scholarly reviews of Numerical Notation have appeared. I will list all reviews I know about, positive or negative.

  • A. Alexander, Comparative Studies in Society and History 54(2): 453-455.
  • H. Gropp, Anthropos 107(1): 237-8.
  • C. Heintz, Journal of the Royal Anthropological Institute 19(3): 664-6.
  • D.M. Hutton, Kybernetes 41(7-8): 1160.
  • J.-C. Martzloff, Zentralblatt MATH Zbl 1229.01002.
  • G. Matthews, Mathematics Today (04/2012)
  • J.V. Rauff, Mathematics and Computer Education 45(3): 280-1.
  • A. Aveni, Journal of Anthropological Research 66(4): 556-7.
  • S. Cuomo, Antiquity 85(328): 662-4.
  • P. Davis, SIAM News 43(5).
  • I. Grattan-Guinness, Annals of Science 70(2): 294-5.
  • J. Katz, Journal of the American Oriental Society 131(3): 497-8.
  • G.E.R. Lloyd, Isis 101(4): 864-5.
  • R.L. Pour, Choice 47(12): 2346.
  • G. Schuppener, Written Language & Literacy 15(2): 279-281.

Errata and Addenda

I sometimes tell my students that everything they ever do as scholars will be flawed, just as everything I’ve ever done or will ever do will be flawed. Meant to shock? Yeah, a little. But it’s worth remembering that this is the process of scholarship, error followed by correction. I’m sure that Numerical Notation has its share of boneheaded typos and brain-os, as well as corrections of fact or interpretation that can be handled in limited space. As readers contact me to note them, I’ll correct these below. Major re-evaluations will have to wait for an as-yet-extremely-hypothetical second edition.

On p. 237 and again on p. 408 I allude to an explanation for the diversity of proto-cuneiform numerals that I thought I had made explicitly, but apparently not. One possibility is that there are multiple systems, not because any one scribe was jumping between many different systems for different commodities, but because there were partly distinct scribal traditions, each one requiring its users to use only one numeral system. If future archaeologists excavated Detroit and found a document with a bunch of hexadecimal numbers, another with times in hours:minutes:seconds, and a third with Western numerals, they might conclude that we had no unified numerical system. It gets a little more complicated in the case of proto-cuneiform (because texts with different systems have been found in the same locations, suggesting they were related to one another in antiquity) but it is, I think, an interesting hypothesis.

Theodore Widom and Dirk Schlimm have proposed a revision to the typology I use in Numerical Notation. Its main improvement is that it disambiguates multiplicative-additive systems whose multipliers are expressed cumulatively from those expressed with ciphers (e.g. II M vs. 2K as two expressions for 2000). I consider this to be a very interesting approach – it creates fewer exceptions to some of the generalizations I have proposed in Numerical Notation, but (I think) is less closely aligned with how humans actually think about and use numerical notations. They presented their paper at the Society for Anthropological Sciences at my invitation in February 2011, and their article appeared in 2012 in Science and Context.

Widom, Theodore Reed, and Dirk Schlimm. 2012. “Methodological reflections on typologies for numerical notations.” Science in Context no. 25 (2):155-195. doi: 10.1017/S0269889712000038.

On page 127 I describe the Italian mathematician Luca Pacioli as an ‘accountant’. While it is certainly the case that that Pacioli’s Summa Arithmetica contains a great deal of information about accounting, including the first full description of double-entry bookkeeping, Pacioli was not a merchant or bookkeeper, but a mathematical scholar and Franciscan friar who was a collaborator of Leonardo da Vinci.

On page 64 I say, “The Linear A, Linear B, and Hittite hieroglyphic numerical notation systems are all decimal and cumulative-additive, and use a horizontal stroke for the units and a vertical stroke for the tens.” This is obviously reversed; all three of these systems use vertical strokes for the units and horizontal strokes for the tens.  The general point about the path of transmission remains correct.

p. 112: I make the assertion that “three vertical bars enclosing a numeral-phrase on the top and sides signified multiplication by 100,000.”  Obviously from the following text and table 4.11, this was not actually three vertical bars, but two vertical bars on either side of the numeral with a horizontal connector joining them on top.  The word ‘vertical’ should be omitted.

On pp. 120-121 I make two references to the system of Ocreatus as being a blend of the Roman numerals and Arabic ciphered-positional numerals.   This should indicate that the system blends Roman numerals and the Western ciphered-positional numerals, using the terminology I establish and use throughout the text to distinguish Arabic and Western numerals.

p. 122: The reference to Table 4.7 should read Table 4.13.

p. 459: Neugebauer 1979, title should be ‘Ethiopic Astronomy and Computus”.

2 thoughts on “Numerical Notation”

  1. I am Neelam Subedi from Nepal.Now I am doing research in mathematics and my research topic is” The development of Numeral System in Ancient period of Nepal”. One day I search the internet and see your book.I feel this book is important for my research and search ihe information about this book but I can not see inside the book . From India(Delhi),I bought a book around 7300 in Nepali currency.I read your bookand I feel about this book that the main prospective of analysis of numerical notation is linguistic and historical .In my research ,I will use both prospective and include mathematical prospective because I am mathematics lecturer in Nepal.

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