Article ID: CBB634644866

Three Thousand Years of Sexagesimal Numbers in Mesopotamian Mathematical Texts (2019)

unapi

The Mesopotamian system of sexagesimal counting numbers was based on the progressive series of units 1, 10, 1·60, 10·60, …. It may have been in use already before the invention of writing, with the mentioned units represented by various kinds of small clay tokens. After the invention of proto-cuneiform writing, c. 3300 BC, it continued to be used, with the successive units of the system represented by distinctive impressed cup- and disk-shaped number signs. Other kinds of “metrological” number systems in the proto-cuneiform script, with similar number signs, were used to denote area numbers, capacity numbers, etc. In a handful of known mathematical cuneiform texts from the latter half of the third millennium BC, the ancient systems of sexagesimal counting numbers and area numbers were still in use, alongside new kinds of systems of capacity numbers and weight numbers. Large area numbers, capacity numbers, and weight numbers were counted sexagesimally, while each metrological number system had its own kind of fractional units. In the system of counting numbers itself, fractions could be expressed as sixtieths, sixtieths of sixtieths, and so on, but also in terms of small units borrowed from the system of weight numbers. Among them were the “basic fractions” which we would understand as 1/3, 1/2, and 2/3. In a very early series of metro-mathematical division exercises and an equally early metro-mathematical table of squares (Early Dynastic III, c. 2400 BC), “quasi-integers” of the form “integer plus basic fraction” play a prominent role. Quasi-integers play an essential role also in a recently found atypical cuneiform table of reciprocals. The invention of sexagesimal numbers in place-value notation, in the Neo-Sumerian period c. 2000 BC, was based on a series of innovations. The range of the system of sexagesimal counting numbers was extended indefinitely both upward and downward, and the use of quasi-integers was abolished. Sexagesimal place-value numbers were used for all kinds of calculations in Old Babylonian mathematical cuneiform texts, c. 1700 BC, while traditional metrological numbers were retained in both questions and answers of the exercises. Examples of impressive computations of reciprocals of many-place regular sexagesimal place-value numbers, with no practical applications whatsoever, are known from the Old Babylonian period. In the Late Babylonian period (the latter half of the first millennium BC), such computations were still popular, performed by the same persons who constructed the many-place sexagesimal tables that make up the corpus of Late Babylonian mathematical astronomy.

...More
Citation URI
https://data.isiscb.org/isis/citation/CBB634644866/

Similar Citations

Article Sarma, Sreeramula Rajeswara; (2012)
The Kaṭapayādi System of Numerical Notation and Its Spread Outside Kerala (/isis/citation/CBB001210557/)

Article Ouyang, Xiaoli; Christine Proust; (2015)
Early Development of the Sexagesimal Place Value Notation in Mesopotamia: Analysis of Some New Evidence (/isis/citation/CBB191900919/)

Book Dietmar Herrmann; (2019)
Mathematik im Vorderen Orient: Geschichte der Mathematik in Altägypten und Mesopotamien (/isis/citation/CBB894339081/)

Article Dejić, Mirko; (2014)
How the Old Slavs (Serbs) Wrote Numbers (/isis/citation/CBB001420032/)

Article Yap, Audrey; (2011)
Gauss' Quadratic Reciprocity Theorem and Mathematical Fruitfulness (/isis/citation/CBB001024184/)

Article Eric Vandendriessche; (2022)
The concrete numbers of “primitive” societies: A historiographical approach (/isis/citation/CBB201980517/)

Book Crossley, John N.; Leigh-Lancaster, David; (2007)
Growing Ideas of Number (/isis/citation/CBB000830869/)

Article Comes, Rosa; (2006)
Notación alfanumérica griega y notaciones derivadas: uso científico-técnico (/isis/citation/CBB001023867/)

Book Chemla, Karine; (2012)
The History of Mathematical Proof in Ancient Traditions (/isis/citation/CBB001320137/)

Article Yuste, Piedad; (2005)
Estudio geométrico de AO 17264 (/isis/citation/CBB000933617/)

Article Robson, Eleanor; (2001)
Mathematical cuneiform tablets in Philadelphia. Part 1: Problems and Calculations (/isis/citation/CBB000102368/)

Book Montelle, Clemency; (2010)
Chasing Shadows: Mathematics, Astronomy, and the Early History of Eclipse Reckoning (/isis/citation/CBB001031330/)

Book Hunger, Hermann; Pingree, David; (1999)
Astral Sciences in Mesopotamia. (Handbook of Oriental Studies; The Near and Middle East) (/isis/citation/CBB000111268/)

Chapter Høyrup, Jens; (2002)
Seleucid Innovations in the Babylonian; “Algebraic” Tradition and their Kin Abroad (/isis/citation/CBB000203452/)

Article Muroi, Kazuo; (2002)
Expressions of Multiplication in Babylonian Mathematics (/isis/citation/CBB000330344/)

Book Proust, Christine; (2007)
Tablettes mathématiques de Nippur (/isis/citation/CBB000931495/)

Article Clark, Kathleen; Robson, Eleanor; (2007)
Ancient Accounting in the Modern Mathematics Classroom (/isis/citation/CBB000831568/)

Article Burna, Bob; (2013)
Root 2: The Early Evidence and Later Conjectures (/isis/citation/CBB001213289/)

Article John M. Steele; (2015)
Late Babylonian Metrological Tables in the British Museum (/isis/citation/CBB416694210/)

Authors & Contributors
Robson, Eleanor
Proust, Christine
Ouyang, Xiaoli
Herrmann, Dietmar
Vandendriessche, Eric
Yusta, Piedad
Concepts
Mathematics
Number notation; mathematical notation
Tablets; papyri
Number theory; number concept
Number systems
Sexagesimal system
Time Periods
Ancient
19th century
Medieval
Renaissance
9th century
20th century
Places
Mesopotamia
Middle and Near East
Greece
India
Sumer
Serbia
Institutions
British Museum
Comments

Be the first to comment!

{{ comment.created_by.username }} on {{ comment.created_on | date:'medium' }}

Log in or register to comment