Article ID: CBB191900919

### Early Development of the Sexagesimal Place Value Notation in Mesopotamia: Analysis of Some New Evidence (2015)

The use of the sexagesimal place value notation( hereafter SPVN) is one of the most striking features of cuneiform mathematics. The earliest attestations of a systematic use of the SPVN are found in a small set of Ur III mathematical texts,but traces of positional notations also appear in about two dozen Ur III administrative texts. These notations are located outside the main text,and hence we refer to such notations as "marginal numbers. "This article studies the marginal numbers in two Ur III documents. The numbers testify to diverse practices with positional notations and sexagesimal factors in administrative contexts during this period. A close analysis of the evidence reveals a diversity of graphical systems for what has been considered until now as a uniform notion of the SPVN. Thus,in addition to the SPVN fully developed during the Old Babylonian period,we identify other systems that we label as"partially sexagesimal place value notation"( hereafter partial-SPVN). The partial-SPVNs do not appear to be floating,while the SPVN does. The partial-SPVNs convert fractions of a measurement unit into an integer number of a sexagesimal sub-unit. This transformation would facilitate subtractions of measurement values with fractions.This diversity of place value notations reflects computational methods that vary according to the operation( multiplication,reciprocal,addition,subtraction) or the context( administrative or mathematical).The observations above may lead to a reconstruction of the process of calculation behind the Ur III administrative records. Some steps in the calculation process seem to have taken place external to the tablet,as no trace of their details occurs on the tablet. The marginal numbers thus provide evidence of an intermediate step,which converts quantities noted in the main text into the input of an external device where calculations are executed,or conversely,converts the output of the device into quantities noted in the main text.

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Authors & Contributors
Burnett, Charles
Michaela Wiesinger
Giardino, Valeria
Rabii, Maryam
Zou, Dahai
Yap, Audrey
Journals
Archive for History of Exact Sciences
Tarikh-e Elm (The Iranian Journal for the History of Science)
Ziran Kexueshi Yanjiu (Studies in the History of Natural Sciences)
VIET: Voprosy Istorii Estestvoznaniia i Tekhniki
Studies in History and Philosophy of Science
Science in Context
Publishers
MIT Press
Franz Steiner Verlag
Ashgate, Variorum
Concepts
Mathematics
Numerals
Number notation; mathematical notation
Number systems
Sexagesimal system
Arab/Islamic world, civilization and culture
People
Śivayog ī, Nijaguṇa
Dedò, Modesto
Gauss, Carl Friedrich
Leonardo da Pisa
Artin, Emil
Time Periods
Medieval
Ancient
15th century
14th century
12th century
Qing dynasty (China, 1644-1912)
Places
India
Russia
Europe
Mesopotamia
Moscow (Russia)
Italy
Institutions
British Museum
Imperial University of Peking