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Mathematical Treasure: The Quipu

Frank J. Swetz (The Pennsylvania State University)


Quipus were knotted tally cords used by the Inca Civilization of South America (1400–1560). The system consisted of a main cord from which a variable number of pendant cords were attached. Each pendant cord contained clusters of knots. These knots and their clusters conveyed numerical information. In some complex instances, further pendant cords were attached to these primary pendants. The number, type of knots, and knot and cluster spacing, as well as the pendant array, all conveyed particular information. A further dimension of this system was use of color: different pendants were dyed different colors, conveying different meanings. One of the few existing records of quipu use is found in the Chronicle of Good Government (1615/1616), written in Spanish by the Inca author Guaman Poma de Ayala. Illustrations from his work are given below:

The chief account of the Inca Empire. Note “counting table” with
grains of maize used as counters in the lower left corner.

Accounting of storage goods at a collca or Inca warehouse.

This numerical storage and communication system was employed by Inca administrators and accountants. Quipus were mathematically efficient and readily portable. Some further illustrations of quipus are shown below:

Photo taken at the Museum of Anthropology and Archaeology, Lima, Peru, by Herb Kasube.

Photo taken at Museo de la Nacion, Lima, Peru, by Lisa Kolbe.

Note branching pendants.

Detailed discussions of the quipu and its uses can be found in:

Editor’s note: This article was published in 2008.

Index to more Mathematical Treasures

Index of Mathematical Objects


Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: The Quipu," Convergence (June 2010), DOI:10.4169/loci003206