Numerical Notation: A Comparative History

NUMERICAL NOTATION: A COMPARATIVE HISTORY by Stephen Chrisomalis Cambridge University Press, 2010, 486 pp. ISBN: 978-0-521-87818-0 Numerical Notation: A Comparative History is a new, comprehensive reference volume of all known numerical notation systems. Considered solely as a descriptive catalog of numerical systems, this work would be a "must-have" for any library. However, Chrisomalis has also combined this comprehensive catalog with an abundance of historical and cultural information and a new, well-considered classification system that make this work essential for all historians of mathematics and teachers of the history of mathematics. Chrisomalis classifies numerical systems along two axes that he calls "intraexponential" and "interexponential". The intraexponential axis looks at how the signs in a system are "combined within each power of the base" of the system. Here we find "cumulative" systems, in which many signs are added to achieve a total (e.g., Roman numerals); "ciphered" systems, in which a single sign represents the total (e.g., Greek alphabetic systems); and "multiplicative" systems, in which a unit sign is multiplied by a power sign to achieve the total (e.g., traditional Chinese). The interexponential axis categorizes the systems according to the way in which the values of the signs are combined to construct the entire numerical phrase. Here we have two types, "additive" and "positional". Roman numerals are additive, whereas Babylonian cuneiform is positional. Thus, we may classify numerical systems into one of five types: cumulative-additive (Roman numerals), cumulative-positional (Babylonian cuneiform), ciphered-additive (Greek alphabetic), ciphered-positional (Khmer), and multiplicative-additive (traditional Chinese). The sixth type, multiplicative-positional, is logically excluded. Having defined his categories and set down his criteria for historical relationships between systems, Chrisomalis presents detailed expositions of the world's numerical systems. The next eight chapters address, in turn, hieroglyphic systems (those descended from Egyptian hieroglyphs), Levantine systems (those descended from Phoenician and Aramaic), Italic systems (descended from Etruscan), alphabetic systems (descended from Greek alphabetic systems), South Asian systems (descended from Brahmi), Mesopotamian systems (descended from proto-cuneiform), East Asian systems (descended from Shang numerals), and Mesoamerican systems (descended from bar and dot systems). …