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Incan Yupana

Ancient Incan abacus

Yupanas were ancient calculating device of the Incas. The term yupana derives from the indigenous Andean Quechua language in which yupay means "to count." The most important historical document concerning the yupana is a sketch made by Felipe Guaman Poma de Ayala in 1615, which was originally lost then rediscovered at the Royal Library of Copenhagen in 1916. While the yupana depicted by Poma de Ayala is laid out in a 5 × 4 grid, examples shaped as polygons with different numbers of sides are also common.

Incan Yupana

The earliest known yupanas have been dated to 200–600 CE. These early yupanas are morphologically distinct from the later ones used for computation and have variously been interpreted as game boards or architectonic models. Musical instruments with similar shapes were also used by Andean peoples. However, the type of yupana described by Poma de Ayala in 1615 consisted of a tablet upon which stones, grains or beans were placed and manipulated to perform calculations. While there are many historical accounts of these calculating devices, not many details concerning their functioning have survived. However, it is still possible to infer much about how these devices were likely used.

Artifact format


Artifact origin

Modesto Omiste, Potosi, Bolivia

Current artifact location

Museo de Instrumentos Musicales de Bolivia, La Paz, Bolivia


Arithmetic timeline Rhind Papyrus Egyptian Mathematical Problem Papyrus Salamis Tablet Bakhshali Manuscript Pacioli's Summa Incan Yupana Napier's Rabdologiæ

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Computational Explanation

Other Resources

Additional Reading

  • Acosta, J. La Historia Natural y Moral de Las Indias. 1586. Reprinted Mexico: Fondo de Cultura Economica, 1940.
  • Apaza Luque, H. J. "La yupana, material manipulativo para la educación matemática." Ph.D. thesis. Madrid, Spain: Universidad Autónoma de Madrid, 2017. [link]
  • Ascher, M. "Mathematical Ideas of the Incas." Ch. 10 in Native American Mathematics (Ed. M. P. Closs). Austin, TX: University of Texas Press, pp. 261–289, esp. pp. 264–266, 1996.
  • Burns Glynn, W. "La tabla de cálculo de los Incas." Boletín de Lima, Vol. 11, pp. 57–70, 1981.
  • Casas, G. P. "Los Sistemas Numericos Del Quechua y el Aymara." Revista andina, Vol. 40, pp. 149–178, 2005.
  • Day, C. Quipus and Witches' Knots: The Role of the Knot in Primitive and Ancient Cultures. Lawrence: University of Kansas Press, 1967.
  • Depaulis, T. "Inca Dice and Board Games." Board Games Studies, Vol. 1/1998, pp. 26–49, 1998.
  • Leonard, M. and Shakiban, C. "The Incan Abacus: A Curious Counting Device." Journal of Mathematics and Culture, Vol. 5, pp. 81–106, 2010. [link]
  • Locke, L. L. "The Ancient Peruvian Abacus." Scripta Mathematica, Vol. 1, pp. 36–43, 1932.
  • Pareja, D. "Instrumentos Prehispanicos de Calculo: el Quipu y la Yupana." Integracion, Vol. 4, pp. 37–56, 1986.
  • Pereyra, H. "El Antiguo ábaco peruano según el manuscrito de Guaman Poma." In Quipu y Yupana. Consejo Nacional de Ciencia y Tecnología. Lima, Perú, 1990.
  • Poma de Ayala. G. F. The First New Chronicle and Good Government: On the History of the World and the Incas up to 1615 (Trans. and ed. R. Hamilton). Austin, TX: University of Texas Press, 2009.
  • Prem, D. Yupana Inka: Decoding the Inka's Math. English Edition Kindle, 2019. [link]
  • Tun, M. "Yupana." In Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (Ed. H. Selin). Dordrecht, Netherlands: Springer Science+Business Media, 2014. [link]
  • Valera, B. Exsul Immeritus Blas Valera Populo Suo e Historia et Rudimenta Linguae Piruanorum Indios. Napoli: Archivio Miccinelli-Cera, 1618.
  • Wassén, H. "The Ancient Peruvian Abacus." In Origin of the Indian Civilizations in South America (Ed. E. Nordenskiold). Goteborg, Sweden: Elanders Boktryckeri Aktiebolag, 1931.