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Funding for this project generously provided by Overdeck Family Foundation

around 1900–1600 BCE

Babylonian Reciprocals Tablet

Old Babylonian reciprocal table (with errors)

Tablet YPM BC 024321 from the Yale University Babylonian Collection has been dated to around 1900–1600 BCE. Both sides contain two columns of numbers written in the base-60 numeral system of the Babylonians. The obverse (front) of the tablet consists of the whole numbers from 56 through 71 with their reciprocals. The reverse side similarly contains the numbers 72 through 80 with their reciprocals. In both cases, the numbers sometimes wrap around onto the sides of the tablet.

Babylonian Reciprocals Tablet

Babylonian reciprocal tables are relatively common, clearly demonstrating the ability of Babylonian scribes to perform simple arithmetic calculations. This tablet is notable for giving approximate base-60 values for the reciprocals it contains that cannot be expressed finitely in base 60. It is also noteworthy that the tablet contains a number of computational (or transcription) errors. For example, in the eighth line of the front of the tablet, the approximate reciprocal 1/63 ≈ 0.015873 should be given as 57, 8, 34 (equal to 57/60² + 8/60³ + 34/60^4 ≈ 0.015873) but is instead written as 57, 8, 24 (equal to 57/60² + 8/60³ + 24/60^4 ≈ 0.0158722).

Artifact dimensions

4.7 cm × 7.0 cm × 1.7 cm

Original artifact location

Babylonia (historical name), Iraq (current name)

Current artifact location

Peabody Museum of Natural History, New Haven

Catalog number

YPM BC 024321 (current), YBC 10529 (historical)


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

  • Bruins, E. M. "Tables of Reciprocals with Irregular Entries." Centaurus, Vol. 17, pp. 177–188, September 1973.
  • Jones, A. "Before Pythagoras: The Culture of Old Babylonian Mathematics: Numbers on Clay." Brochure accompanying the exhibition Before Pythagoras: The Culture of Old Babylonian Mathematics. On view at ISAW from November 12th–December 17th, 2010.
  • Neugebauer, O. and Sachs, A. (Ed.). "Reciprocals of Regular and Irregular Numbers: YBC 10529." In Mathematical Cuneiform Texts. New Haven, CT: American Oriental Society and the American Schools of Oriental Research, p. 16, 1945.
  • Ore, O. Fig 8-1 in Number Theory and Its History. New York: Dover, pp. 173–174, 1988.

Image Credits

Yale Peabody Museum of Natural History