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Funding for this project generously provided by Overdeck Family Foundation

around 1900–1600 BCE

Sumerian 7 Quotient and 10 Product Tablet

Reciprocals and multiplication by 7 and 10 in cuneiform

Many mathematical Babylonian tablets contain tables of divisions or multiplications. Tablet N3914, held by the Penn Museum in Philadelphia, tabulates 7/n and a number of products involving those quotients for integers n from 1 to 10. All are written in the base 60, or sexagesimal, number representation used by the Babylonians.

Sumerian 7 Quotient and 10 Product Tablet

This old Babylonian tablet contains division and multiplication calculations laid out in a tabular format. The first column of the table is a simple count of the integers 1 to 10, though the numbers 5–10 are partially or completely missing. The second column gives 7 divided by the numbers in the first column, while the third column gives the numbers in the second column multiplied by 10. The fourth and final column gives the products of the first and third columns which, because n × (7/n × 10), is always the number 70, written in base-60 numbers as 1 10, i.e. 1 × 60 + 10 × 1. The number at the top of the third column is the sum of the numbers in that column. The significance of an additional number written at the end of the first row is not known.

Artifact dimensions

7 in. × 7 in. × 2 in.

Original artifact location

Nippur (historical name), Afak, Al-Qadisiyah, Iraq (current name)

Current artifact location

Penn Museum, Philadelphia

Catalog number



Counting timeline Sumerian Earliest Known Math Table Sumerian 7 Quotient and 10 Product Tablet Babylonian School Multiplication Tablet Babylonian Reciprocals Tablet Rhind Papyrus Roman Stylus Tablet Bamboo-Strip Multiplication Table Venerable Bede's De temporum ratione South American Quipu Aztec Dates from Codex Mendoza Midewiwin Birchbark Scroll

Interactive Content

Computational Explanation

Other Resources

Additional Reading

  • Cajori, F. A History of Mathematical Notations: Two Volumes Bound as One, Vol. 1. New York: Dover, pp. 2–4, 1928.
  • Robson, E. Fig. 10 in "Mathematical Cuneiform Tablets in Philadelphia, Part 1: Problems and Calculations." SCIAMVS, Vol. 1, pp. 11–48, 2000.

Image Credits

Penn Museum