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In numeric quipus, the types of knots and their positions indicate numeric value. Single knots may occur alone or in clusters. Each single knot stands for 10, 100, 1000 or greater powers of 10. The larger powers are positioned closer to the primary cord. Multiple single knots in a cluster indicate multiples of the designated power of 10. For instance, a group of three single knots in the hundreds position would mean 300. Unit values are denoted by long knots with two to nine turns or a figure-eight knot indicating the value 1. As shown below, many numeric quipus contained an additional cord recording the total:

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First cord 1989 Second cord 2021 Third cord 1342

Computational Explanation

The Inca state (which included regions of modern-day Peru, Chile, Ecuador, Bolivia and Argentina) was the largest empire of the Americas. With a population of nearly 10 million, its people built massive administrative centers, temples and extensive road and canal systems over an area of more than 900,000 square kilometers. They did so in the inhospitable, extreme terrain of the Andes mountains:

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Inca Empire

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In much of the empire, the Incas organized male heads of household into units of 10, 50, 100, 500, 1000, 5000 and 10,000. This decimal administration hierarchy was used to tabulate labor for both civil and military duties, including farming, herding and artisanry, as well as portage, guard duty and war service:

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Fine Incan weavings were usually produced from alpaca wool and cotton. Clothing was emblematic of social status and political rank, and Inca elites wore special tunics decorated with rows of distinctive geometric designs. Many of the textiles produced under imperial supervision were standardized, and their designs had symbolic meanings related to prestige, cosmology and ethnic identity:

Since textiles were the most precious items to the Incans, it was important to know if different flocks of llamas and alpacas were increasing or decreasing in size. In order to store and transmit such valuable numbers, the Incas used a practical mathematical device consisting of strings and knots known as a quipu. Quipus were used all over the vast Incan empire in a standardized decimal number system. Here are some illustrations of Incans using quipus as appearing in the writings of Fray Martín de Murúa:

Reading complex quipus was the job of an expert, the quipucamayoc, who by observing and touching each of the cords and knots interpreted the meanings of its colors, the thickness of the cords, the twisting of the threads and the kinds and positions of knots on them. The senses of touch and sight were essential for this kind of interpretation.

Quipus Illustrated by Poma de Ayala

In 1615, Felipe Guaman Poma de Ayala, a Quechua nobleman and chronicler known for denouncing the ill treatment of the Incas under Castilian rule, wrote a 1200-page manuscript, New Chronicle and Good Government, organized around his 398 pen-and-ink drawings of Incan daily life. The manuscript contains 12 pages and 7 drawings concerning the use of quipus at every level of Inca bureaucracy.

The following pages depict a quipucamayoc, chief accountant and treasurer in charge of the quipus of the kingdom:

The following illustrations depict from left to right the native administrator of resources with the book and quipu he uses for accounting, the secretary and accountant who records the dispositions of the royal lords and the Incan state administrator:

The following pages show an Incan astrologer carrying a quipu and a fork-like post and a quipu messenger who could speedily send important information along Incas' network of post roads:

Finally, the following page depicts an encounter at a collca (Incan warehouse) in which the 10th Incan ruler, Topa Inca Yupanqui, interviews his accountant or warehouse-keeper, who extends a quipu that contains records of goods in the storage chambers:

Numeric Quipus

Quipus were usually made of cotton, and more rarely of alpaca or llama fiber yarn, and consisted of a thick primary cord from which dangled a series of thinner pendant cords. The quipu system allowed quipucamayocs to store decimal numbers using positional notation in the same way we write numbers nowadays—that is, where the number of ones is represented by the numeral in the first digit, the number of tens by the numeral just to the left of the first digit, hundreds by the next to the left, then thousands and so on. For example, with the number 2357, the 7 is in the ones place, the 5 is in the tens place, the 3 is in the hundreds place and the 2 is in the thousands place:

In numeric quipus, the types of knots and their positions indicate numeric value. Single knots may occur alone or in clusters. Each single knot stands for 10, 100, 1000 or greater powers of 10. The larger powers are positioned closer to the primary cord. Multiple single knots in a cluster indicate multiples of the designated power of 10; for instance, a group of three single knots in the hundreds position would mean 300. Unit values are denoted by long knots with two to nine turns or a figure-eight knot indicating the value 1. Many numeric quipus contained an additional cord recording the total. For example, if there were three different flocks of alpacas of 753, 370 and 1092 individuals, the quipucamayoc of that region would have included a fourth cord in his quipu recording the total number of alpacas as 2215:

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Numeric quipus were important and useful for storing and transferring exact numerical data across the Incan empire. To perform arithmetical computations, quipucamayocs transferred numbers encoded by knots into beans, corn seeds or pebbles across a yupana board. The yupana enabled the Incans to perform large computations effortlessly following a set of simple rules consisting of moving, adding or subtracting numerical tokens:

The Khipu Database Project

The Khipu Database Project (https://khipukamayuq.fas.harvard.edu) began in the fall of 2002 with the goal of collecting all known information about quipus into one centralized repository. Having the data in digital form allows researchers to ask questions about quipus that would previously have been very difficult, if not impossible, to answer. The project currently contains detailed data on more than five hundred individual quipus.

Garcilaso de la Vega, a 17th-century Spanish chronicler, reported the following about quipu practices:

"Although the quipucamayus were as accurate and honest as we have said, their number in each village was in proportion to its population, and however small, it had at least four and so upwards to twenty or thirty. They all kept the same records, and although one accountant or scribe was all that would have been necessary to keep them, the Incas preferred to have plenty in each village and for each sort of calculation, so as to avoid faults that might occur if there were few, saying that if there were a number of them, they would either all be at fault or none of them."

This passage clearly implies that multiple quipus with the same information existed in one area, though it is not precisely known how matching quipus were created or used. Urton (2003) suggested several possibilities:

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Duplicate records could have been made as a part of a system of checks and balances.

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Duplicates might have been made for distribution: one copy for the town, one copy for the capital of Cusco.

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In some close matches, all values on one quipu are higher than the values on the second quipu. This could indicate a change through time: an increase or decrease in what was being recorded.

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There are a couple of instances of three matching quipu. There is some evidence that the data from a small quipu may have been assimilated into a larger one.

Most ancient quipus from the database survived the passage of time thanks to the extremely dry desert conditions in which they were found. By contrast, quipus from humid places have almost certainly been lost forever

The Khipu Database Project contains 74 quipus whose exact origin locations are not known. To recover that piece of missing information, there is an ongoing computer-assisted research project trying to infer origin by looking at the structural similarity with the 266 quipus mapped below:

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These efforts, together with new archaeological discoveries, give a better understanding and highlight the impressive intellectual and administrative accomplishments of the indigenous peoples of the Americas.

Other Resources

Additional Reading

• Ascher, M. and Ascher, R. Mathematics of the Incas: Code of the Quipu. New York: Dover, 1997.
• 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. 266–288, 1996.
• d'Altroy, T. N. The Incas, 2nd ed. Chichester, England: Wiley Blackwell, 2015.
• Day, C. Quipus and Witches' Knots: The Role of the Knot in Primitive and Ancient Cultures. Lawrence: University of Kansas Press, 1967.
• Gaenger, S. Relics of the Past: The Collecting and Study of Pre-Columbian Antiquities in Peru and Chile, 1837–1911. Oxford, England: Oxford University Press, 2014.
• Gilsdorf, T. E. Introduction to Cultural Mathematics: With Case Studies in the Otomies and Incas. New York: Wiley, 2012.
• Joseph, G. The Crest of the Peacock: Non-European Roots of Mathematics. Princeton, NJ: Princeton University Press, 2011.
• Locke, L. "The Ancient Quipu, a Peruvian Knot Record." American Anthropologist, Vol. 14, No. 2, pp. 325–332, 1912.
• Miccinelli, V. and Animato, C. Quipu: il nodo parlante dei misteriosi incas. Genoa, Italy: Edizioni Culturali Internazionali Genova, 1989.
• Minelli, L. L. and Domenici, D. Per Bocca D'altri:Inca, Gesuiti E Spagnoli Nel Peru' Del XVII Sec. Alma Digital Library, Università di Bologna, 2007.
• Pappas, T. The Joy of Mathematics: Discovering Mathematics All Around You. San Carlos, CA: Wide World Publishing/Tetra, pp. 14–15, 1989.
• 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.
• Stone, R. Art of the Andes: from Chavín to Inca, 2nd ed. London: Thames & Hudson, 2002.
• Urton, G. Signs of the Inka Khipu: Binary Coding in the Andean Knotted-String Records. Austin, TX: University of Texas Press, 2003.
• Valera, B. Exsul Immeritus Blas Valera Populo Suo e Historia et Rudimenta Linguae Piruanorum Indios. Napoli: Archivio Miccinelli-Cera, 1618.

Additional Links

• Atlas Obscura: "The College Student Who Decoded the Data Hidden in Inca Knots" by Katherine Davis-Young (14 Dec. 2017)
• Dumbarton Oaks Research Library and Collection: The Role of Knots
• Ethnologisches Museum der Staatlichen Museen Khipu V A 16148
• Go Geometry: The Incas: The Quipu
• The Guaman Poma Website: A Digital Research Center of the Royal Library, Copenhagen, Denmark
• Jean-Jacques Quisquater: Khipus
• Khipu Database Project
• Mathematical Treasures: The Quipu
• National Geographic Threads That Speak: How the Inca Used Strings to Communicate
• Peabody Museum of Archeology and Ethnology at Harvard University Quipu 41-52-30/2938
• Princeton University Art Museum Quipu with three figures y1990-73
• Smithsonian National Museum of Natural History Quipu A365240-0
• Spurlock Museum Quipu 1998.19.2769A
• Spurlock Museum Quipu 1998.19.3355
• Wikipedia: El primer nueva crónica y buen gobierno
• Wikipedia: Felipe Guaman Poma de Ayala
• Wikipedia: Martín de Murúa
• Wikipedia: Quipu