The mathematics of the Incas (or of the Tawantinsuyu) was the set of numerical and geometric knowledge and instruments developed and used in the nation of the Incas before the arrival of the Spaniards. It can be mainly characterized by its usefulness in the economic field. The quipus and yupanas are proof of the importance of arithmetic in Inca state administration. This was embodied in a simple but effective arithmetic, for accounting purposes, based on the decimal numeral system; they too had a concept of zero, [1] and mastered addition, subtraction, multiplication, and division. The mathematics of the Incas had an eminently applicative character to tasks of management, statistics, and measurement that was far from the Euclidean outline of mathematics as a deductive corpus, since it was suitable and useful for the needs of a centralized administration. [note 1]
On the other hand, the construction of roads, canals and monuments, as well as the layout of cities and fortresses, required the development of practical geometry, which was indispensable for the measurement of lengths and surfaces, in addition to architectural design. At the same time, they developed important measurement systems for length and volume, which took parts of the human body as reference. In addition, they used appropriate objects or actions that allowed to appreciate the result in another way, but relevant and effective.
The prevailing numeral system was the base-ten. [2] One of the main references confirming this are the chronicles that present a hierarchy of organized authorities, using the decimal numeral system with its arithmometer: Quipu.
Responsible | Number of families |
---|---|
Puriq | 1 family |
Pisqa kamayuq | 5 families |
Chunka kamayuq | 10 families |
Pisqa chunkakamayuq | 50 families |
Pachak kamayuq | 100 families |
Pisqa pachakakamayuq | 500 families |
Waranqa kamayuq | 1000 families |
Pisqa waranqakamayuq | 5000 families |
Chunka waranqakamayuq | 10 000 families |
It is also possible to confirm the use of the decimal system in the Inca system by the interpretation of the quipus, which are organized in such a way that the knots — according to their location — can represent: units, tens, hundreds, etc. [3]
However, the main confirmation of the use of this system is expressed in the denomination of the numbers in Quechua, in which the numbers are developed in decimal form. This can be appreciated in the following table: [note 2]
Number | Quechua | Number | Quechua | Number | Quechua |
---|---|---|---|---|---|
1 | Huk | 11 | Chunka hukniyuq | 30 | Kimsa chunka |
2 | Iskay | 12 | Chunka iskayniyuq | 40 | Tawa chunka |
3 | Kimsa | 13 | Chunka kimsayuq | 50 | Pisqa chunka |
4 | Tawa | 14 | Chunka tawayuq | 60 | Suqta chunka |
5 | Pisqa | 15 | Chunka pisqayuq | 70 | Qanchis chunka |
6 | Suqta | 16 | Chunka suqtayuq | 80 | Pusaq chunka |
7 | Qanchis | 17 | Chunka qanchisniyuq | 90 | Isqun chunka |
8 | Pusaq | 18 | Chunka pusaqniyuq | 100 | Pachak |
9 | Isqun | 19 | Chunka isqunniyuq | 1000 | Waranqa |
10 | Chunka | 20 | Iskay chunka | 1 000 000 | Hunu |
The quipus constituted a mnemonic system based on knotted strings used to record all kinds of quantitative or qualitative information; if they were dealing with the results of mathematical operations, only those previously performed on the "Inca abacuss" or yupanas were cancelled. Although one of its functions is related to mathematics — as it was an instrument capable of accounting — it was also used to store information related to census, product amount, and food kept in state warehouses. [4] [5] Quipus are even mentioned as instruments the Incas used to record their traditions and history in a different way than in writing.
Several chroniclers also mention the use of quipus to store historical news. [note 3] However, it has not yet been discovered how this system worked. In the Tahuantinsuyo, it was specialized personnel who handled the strings. They were known as quipucamayoc and they could be in charge of the strings of an entire region or suyu. Although the tradition is being lost, the quipus continue to be used as mnemonic instruments in some indigenous villages where they are used to record the product of the crops and the animals of the communities. [5]
According to the Jesuit chronicler Bernabé Cobo, the Incas designated to certain specialists the tasks related to accounting. These specialists were called quipo camayos, in whom the Incas placed all their trust. [6] In his study of the quipu sample VA 42527 (Museum für Völkerkunde, Berlin), Sáez-Rodríguez noted that, in order to close the accounting books of the chacras, certain numbers were ordered according to their value in the agricultural calendar, for which the khipukamayuq — the accountant entrusted with the granary — was directly in charge. [7] [8]
In the case of numerical information, the mathematical operations were previously carried out on the abacuss or yupanas. These could be made of carved stone or clay, had boxes or compartments that corresponded to the decimal units, and were counted or marked with the help of small stones or grains of corn or quinoa. Units, tens, hundreds, etc. could be indicated according to whether they were implicit in each operation.
Recent research regarding the yupanas suggests that they allowed to calculate considerable numbers based on a probably non-decimal system, [9] but based in relation to the number 40. If true, it is curious to note the coincidence between the geometric progression achieved in the yupana and the current processing systems; [10] on the other hand, it is also contradictory that they based their accounting system on the number 40. If the investigations continue and this fact is confirmed, it would be necessary to compare its use with the decimal system, which according to the historical tradition and previous investigations, was the one used by the Incas. [11]
In October 2010, Peruvian researcher Andrés Chirinos with the support of the Spanish Agency for International Development Cooperation (in Spanish, Agencia Española de Cooperación Internacional para el Desarrollo, AECID), reviewed drawings and ancient descriptions of the indigenous chronicler Guaman Poma de Ayala and finally deciphered the riddle of the yupana — that he calls "pre-Hispanic calculator" — as being capable of adding, subtracting, multiplying, and dividing. This made him hopeful to finally discover how the quipus worked as well. [12]
There were different units of measurement for magnitudes such as length and volume in pre-Hispanic times. The Andean peoples, as in many other places in the world, took parts of the human body as a reference to establish their units of measurement. There was not a single system of units of obligatory and uniform use throughout the Andean world. Many documents and chronicles have recorded different systems of local origin that remained in use until the 16th century.
Among the units of length measurement, there was the rikra (fathom), which is the distance measured between a man's thumbs with arms extended horizontally. [13] The kukuchu tupu (kukush tupu) was equivalent to the Spanish codo (cubit) and was the distance measured from the elbow to the end of the fingers of the hand. [14] There was also the capa (span), and the smallest was the yuku or jeme, which was the length between the index finger and the thumb, separating one from the other as much as possible. The distance between two villages would have been evaluated by the number of chasquis required to carry an errand from one village to the other. They would have used direct proportionality between the circumference of a sheepfold and the number of chacra partitions.
The tupu was the unit of measurement of surface area. In general terms it was defined as the plot of land required for the maintenance of a married couple without children. Every hatun runa or "common man" received a plot of land upon marriage and its production had to satisfy the basic needs of food and trade of the spouses. It did not correspond to an exact measurement, since its dimensions varied according to the conditions of each land and from one ethnic group to another. [15] The quality of the soil was taken into consideration and the necessary rest time was calculated accordingly, which had to be considered after a certain number of agricultural campaigns. After that time, the couple could claim a new tupu from their curaca.
Among the units of measurement of capacity there is the pokcha, which was equivalent to half a fanega or 27.7 liters. Some crops such as corn were measured in containers; liquids were measured in a variety of pitchers and jars. There were boxes of a variety of cántaros and tinajas, and straw or reed boxes in which objects were kept. These boxes were also used in warehouses to store delicate or exquisite products, such as dried fruits. Coca leaves were measured in runcu or large baskets. Other baskets were known as ysanga. Among these measures of capacity there is the poctoy orpurash (almozada), which is equivalent to the portion of grains or flour that can be kept in the concavity formed with the hands together. [16] The ancient inhabitants of the Andes knew the scales of saucers and nets as well as the huipe, an instrument similar to steelyards. [17] Apparently, its presence is associated with the works of jewelry and metallurgy, trades in which it is necessary to know the exact weights to use the right proportions of the alloys.
Especially the volume of their colcas (trojas) and their tambos (state warehouses, located in key points of the Qhapaq Ñan). They used the runqu (rongos: bales), portable containers or ishanka (baskets) or the capacity of a chacra. They would have handled the proportionality of the volumes of prisms with respect to their heights — without varying the bases. [18]
To measure time, they used the day (workday), which could include a morning, even an afternoon. Time was also useful, indirectly, to appreciate the distance between two cities; for example, 20 days from Cajamarca to Cusco was the accepted time measurement.
Months, years, and the phases of the moon — much consulted for the tasks of sowing, aporques and harvests and in navigation — were also measured in days. [18]
The Inca Empire, officially known as the Realm of the Four Parts, was the largest empire in pre-Columbian America. The administrative, political, and military center of the empire was in the city of Cusco. The Inca civilization rose from the Peruvian highlands sometime in the early 13th century. The Spanish began the conquest of the Inca Empire in 1532 and by 1572, the last Inca state was fully conquered.
Quipu are recording devices fashioned from strings historically used by a number of cultures in the region of Andean South America.
Pachacuti Inca Yupanqui, also called Pachacútec, was the ninth Sapa Inca of the Chiefdom of Cusco, which he transformed into the Inca Empire. Most archaeologists now believe that the famous Inca site of Machu Picchu was built as an estate for Pachacuti.
A chasqui was a messenger of the Inca empire. Agile, highly trained and physically fit, they were in charge of carrying messages in the form of quipus, oral information, or small packets. Along the Inca road system there were relay stations called chaskiwasi, placed at about 2.5 kilometres (1.6 mi) from each other, where the chasqui switched, exchanging their message(s) with the fresh messenger. The chasqui system could be able to deliver a message or a gift along a distance of up to 300 kilometres (190 mi) per day.
Inca education during the time of the Inca Empire was divided into two principal spheres: education for the upper classes and education for the general population. The royal classes and a few specially-chosen individuals from the provinces of the Empire were formally educated by the Amawtakuna (philosopher-scholars), while the general population were passed on knowledge and skills by their immediate forebears. Since the Incas did not have a written language, but instead had Quipus to record, it is difficult to determine the type of educational system the Incas did have.
The Quijos-Quichua (Napo-Quichua) are a Lowland Quechua people, living in the basins of the Napo, Aguarico, San Miguel, and Putumayo river basins of Ecuador and Peru. In Ecuador they inhabit in the Napo Alto as well as the rivers Ansuy and Jatun Yacu, where they are also known as Quijos Quechua. The Quijos Original Nation (NAOKI) has an extension of community territory of approximately 13,986, 78 hectares. It was recognized as such on March 13, 2013, by Codenpe. It is made up of dozens of groups, communities and organizations, according to their status.
A panaca or panaqa, or panaka was a family clan of the Sapa Inca, the kuraka or emperor of the Inca Empire. The panacas were formed by the descendants of a Sapa Inca or his wife. The basic social institution of the Incas is the ayllu. An ayllu is a group of families that descended from a common ancestor, united by culture and religion, in addition to the agricultural work, livestock and fishing of the same territory. The ayllu concept transcended into nobility, so that the royal kinship could establish a lineage, called panaca or royal house.
The Incas were most notable for establishing the Inca Empire which was centered in modern-day South America in Peru and Chile. It was about 4,000 kilometres (2,500 mi) from the northern to southern tip. The Inca Empire lasted from 1438 to 1533. It was the largest Empire in America throughout the Pre-Columbian era. The Inca state was known as the Kingdom of Cuzco before 1438. Over the course of the Inca Empire, the Inca used conquest and peaceful assimilation to incorporate the territory of modern-day Peru, followed by a large portion of western South America, into their empire, centered on the Andean mountain range. However, shortly after the Inca Civil War, the last Sapa Inca (emperor) of the Inca Empire was captured and killed on the orders of the conquistador Francisco Pizarro, marking the beginning of Spanish rule. The remnants of the empire retreated to the remote jungles of Vilcabamba and established the small Neo-Inca State, which was conquered by the Spanish in 1572.
The term Peruvian literature not only refers to literature produced in the independent Republic of Peru, but also to literature produced in the Viceroyalty of Peru during the country's colonial period, and to oral artistic forms created by diverse ethnic groups that existed in the area during the prehispanic period, such as the Quechua, the Aymara and the Chanka South American native groups.
The Inca army was the multi-ethnic armed forces used by the Tawantin Suyu to expand its empire and defend the sovereignty of the Sapa Inca in its territory.
The Kingdom of Cusco, also called the Cusco confederation, was a small kingdom based in the Andean city of Cusco that began as a small city-state founded by the Incas around the start of 13th century. In time, through warfare or peaceful assimilation, it began to grow into the Inca Empire (1438–1533).
The official flag of Cusco has seven horizontal stripes of color: red, orange, yellow, green, sky blue, blue, and violet. This rainbow flag was introduced to Peru in 1973 by Raúl Montesinos Espejo, in recognition of the 25th anniversary of his Tawantinsuyo Radio station. As the flag's popularity grew, Cusco mayor Gilberto Muñiz Caparó declared it an official emblem in 1978. Since 2021, the official flag has also featured the golden Echenique's sun.
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A yupana is a counting board used to perform arithmetic operations, dating back to the time of the Incas. Very little documentation exists concerning its precise physical form or how it was used.
William Burns Glynn was a British engineer and researcher of the Pre-Columbian era in Peru, noted for his work investigating possible Pre-Columbian writing.
Sabine Hyland is an American anthropologist and ethnohistorian working in the Andes. She is currently Professor of World Christianity at the University of St Andrews. She is best known for her work studying khipus and hybrid khipu-alphabetic texts in the Central Andes and is credited with the first potential phonetic decipherment of an element of a khipu. She has also written extensively about the interaction between Spanish missionaries and the Inca in colonial Peru, focusing on language, religion and missionary culture, as well as the history of the Chanka people.
The economy of the Inca Empire, which lasted from 1438 to 1532, was based on local traditions of "solidarity" and "mutualism", transported to an imperial scale, and established an economic structure that allowed for substantial agricultural production as well as the exchange of products between communities. It was based on the institution of reciprocity, considered the socioeconomic and political system of the Pre-Columbian Andes.
Polo Ondegardo was a Spanish colonial jurist, civil servant, businessman and thinker who proposed an intellectual and political vision of profound influence in the earliest troubled stage of the contact between the Hispanic and the American Indigenous world. He was born in Valladolid, when the city was the capital of the kingdom of Castile, to a prominent noble family that had strong ties to the royal family. He spent his entire adult life in South America in what is now Peru and Bolivia. He was involved in the political and economic management of the Spanish colony and based on his good knowledge of the laws as licenciado (licentiate) acquired a deep knowledge and practical experience of the Native Americans in the southern Andes, being an encomendero, visitador and corregidor in the provinces of Charcas and Cusco. His administrative reports, well known and appreciated by his peers and contemporaries, have had wide repercussions in the field of Andean studies up to the present time.
Incan animal husbandry refers to how in the pre-Hispanic andes, camelids played a truly important role in the economy. In particular, the llama and alpaca—the only camelids domesticated by Andean men— which were raised in large-scale houses and used for different purposes within the production system of the Incas. Likewise, two other species of undomesticated camelids were used: the vicuña and the guanaco. The guanacos were hunted by means of chacos.