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Inkas Math Calculators Yupana and Math Record Keeping Quipu

by Liliana Usvat

The Inca king appointed quipucamayocs, or keeper of the knots, to each town. These individuals would car out calculations on the Yupana and transfer this numeric information to quipus which are strings of knots representing a base ten counting system. The quipucamayocs, who were essentially government statisticians, kept official census records, official produce numbers of their town, and records of the towns animals and weapons. This and other information was sent annually to the capital city Cuzco 


Archaeologists describe yupanas as boards made of clay, stone, wood carved or simply plotted on land, divided into five rows and four columns, thus forming squares where stones were placed, grains of corn or quinoa acquired a value based numerical position.

yupana  is an abacus used to perform arithmetic operations dating back to the time of the Incas.

Several chroniclers of the Indies described, unfortunately approximately, the Incan abacus and its operation.

The first was Guaman Poma de Ayala that in 1615 approximately, wrote:

... They count through tables, numbering from one hundred thousand to one hundred and from ten thousand and ten, until the unit. They keep records of everything that happens in this realm: holidays, Sundays, months and years. These accountants and treasurers of the kingdom are found in every city, town, or indigenous village ..

The father Jesuit Josť de Acosta wrote:"

... they take the corn and put one here, three there, eight from another part; they move from a box and exchanged three other grains from one to another to finally get the result without error "

Father Juan de Velasco wrote:

... these teachers were using something like a series of tables, made of wood, stone, or clay, with different separations, in which they put stones of different shapes, colors and angular shapes


Triangular yupana

These yupana, made of stone, have 18 compartments of triangular shape, arranged around the table. On one side there is a rectangular tower with only one floor and three triangular compartments. In the central part there are four square compartments, coupled between them



Ancient quipus  from the Inca Empire are still a mystery. Quipus can be described as ancient books, but instead of have letters or symbols, they are made of strings and knots.The Inca Empire was one of the greatest empires by the time the Spanish Conquistadors discovered America. They lived in the Andes and developed a very advanced civilization.

In 1609, the Inca Garcilaso de la Vega published the first volume of his Royal Commentaries of the Incas in Lisbon. He wrote:;

The word quipu means both knot or to knot; it was also used for accounts, because they were kept by means of the knots tied in a number of cords of different thicknesses and colors, each one of which had a special significance. Thus, gold was represented by a gold cord, silver by a white one, and fighting men by a red cord. 

When their accounts had to do with things that have no color - such as grain and vegetables - they were classified by categories, and, in each category, by order of diminishing size. Thus, to furnish an example, if they had had to count the various types of agricultural production in Spain, they  would have started with wheat, then rye, then peas, then beans, and so forth. In the same way, in order to make an inventory of the arms of the imperial army, they first counted the arms that were considered to belong in a superior category, such as lances, then javelins, bows and arrows, hatchets and maces, and lastly, slings, an any other arms that were used. In order to ascertain the number of vassals in the Empire, they started with each village, then with each province: the firs cord showed a census of men over sixty, the second, those between fifty and sixty, the third, those from forty to fifty, and so on, by decades, down to the babes at the breast.

Occasionally other, thinner, cords of the same color, could be seen among one of these series, as though they represented an exception to the rule; thus, for instance, among the figures that concerned the men of such and such an age, all of whom were considered to be married, the thinner cords indicated the number of widowers of the same age, for the year in question: because, as I explained before, population figures, together with those of all the other resources of the Empire, were brought up to date every year.

According to their position, the knots signified units, tens, hundreds, thousands, ten thousands and, exceptionally, hundred thousands, and they were all as well aligned on their different cords as the figures that an accountant sets down, column by column, in his ledger. Indeed, those men, called quipucamayus, who were in charge of the quipus, were exactly that, imperial accountants.

The number of quipucamayus scattered throughout the Empire, was proportional to the size of each place. Thus the smallest villages numbered four, and others twenty, or even thirty. The Incas preferred this arrangement. even in places where one accountant would have sufficed, the idea being that, if several of them kept the same accounts, there was less risk that they would make mistakes.

Every year, an inventory of all the Inca's possessions was made. Nor was there a single birth or death, a single departure or return of a soldier, in all the Empire, that was not noted on the quipus. And indeed, it may be said that everything that could be counted, was counted in this way, even to battles, diplomatic missions, and royal speeches. But since it was only possible to record numbers in this manner, and not words, the quipucamayus assigned to record ambassadorial missions and speeches, learned them by heart, at the same time that they noted down the numbers, places and dates on their quipus; and thus, from father to son, they transmitted this information to their successors.

The speeches exchanged between the Incas and their vassals on important occasions, such as the surrender of a new province, were also transmitted to posterity by the amautas, or philosophers, who summarized them in simple, clear fables, in order that they might be implanted by word of mouth in the memories of all the people from those at court to the inhabitants of the most remote hamlets. The harauicus, or poets, also composed poems based on diplomatic records and royal speeches. These poems were recited for a great victory or festival, and every time a new Inca was knighted.

When the curacas and dignitaries of a province want to know some historical detail concerning their predecessors, they asked these quipucamayus, who were, in other words, not only the accountants, but also the historians of each nation. The result was that the quipucamayus never let their quipus out of their hands, and they kept passing their cords and knots through their fingers so as not to forget the tradition behind all these accounts. In fact, their responsibility was so great and so absorbing, that they were exempted from all tribute as well as from all other kinds of service.

All laws, ordinances, rites, and ceremonies throughout the Empire were recorded by these same means.

When my father's Indians came to town on Midsummer's Day to pay their tribute, they brought me the quipus; and the curacas asked my mother to take note of their stories, for they mistrusted the Spaniards, and feared that they would not understand them. I was able to reassure them by re-reading what I had noted down under their dictation, and they used to follow my reading, holding on to their quipus, to be certain of my exactness; this was how I succeeded in learning many things quite as perfectly as did the Indians.

Archeologists in Peru have found a "quipu" on the site of the oldest city in the Americas, indicating the device, a sophisticated arrangement of knots and strings used to convey detailed information, was in use thousands of years earlier than previously believed. Previously the oldest known quipus, often associated with the Incas whose vast South American empire was conquered by the Spanish in the 16th century, dated from about 650 AD.

But Ruth Shady, an archeologist leading investigations into the Peruvian coastal city of Caral, said quipus were among a treasure trove of articles discovered at the site, which are about 5,000 years old. "This is the oldest quipu and it shows us that this society ... also had a system of "writing" (which) would continue down the ages until the Inca empire and would last some 4,500 years,"

The main content of quipus are numbers, which are expressed by knots on a section of rope. Unlike our "Arabic" numbers which uses ten different symbols for each digit (0 to 9), quipu makers tied multiple knots in a tight sequence represent a "digit". Digits can range from no knots (empty space) representing zero, to nine knots representing nine. For example, seven knots in a sequence equals the digit 7.

Multiple sequences of knots represent "digits" that make up a number larger than ten. In other words, quipu was a positional ten-based numeric system that, instead of encoded in written symbols, is encoded in knots. In a positional number system, the position of where a "digit" occurs determines its actual value. For example, in the "Arabic" system, the digit 3 in the number 123 stands for the amount "three" because it is at the very end of the number. Mathematically, 3 x 100 = 3 x 1 = 3. On the other hand, in the number 321the digit 3 stands for 300 because it is the third to the last digit (3 x 102 = 3 x 100 = 300). The position of the 3 determines its multiplier's exponent.

Similarly, the number 321 would be represented as three sequences of knots, the first one with three knots, the second with two knots, and the last one with one knot. However, there is a twist (pardon the pun). Three different kinds of knots are used in quipu. Commonly, the single knot (S) is used to represent the value of one except in the very last position (or digit). In the last position, two different knot types are used. The figure eight knot (E) represents the value one in the last digit, where as multiple four-turn long knots (L) represent values higher than one in the last position. In other words, the figure eight knot and the four-turn long knot are both used to signal the end of a number.

Literary Quipu

In 1996 a manuscript called Historia et Rudimenta Linguae Piruanorum came to light in Italy among the family possessions of a Naples historian. This document was supposedly written in the early 17th century by Jesuits and contains a fragment of quipu as well as an explanation of how quipu was used to encode spoken language. According to the manuscript, "ideograms" or symbols with well-known meaning from Incan art were used as either phonograms (to represent sounds) or logograms (to denote words).

To represent a sound in this system, a symbol is woven at the beginning of a cord, followed by a number. The symbol is drawn from Andean iconography and would represent a well-known deity or a concept, and the number would point to which syllable of the word represented by the symbol to pronounce. One example given in the manuscript is a symbol for the god Pachamacac, which consists of the syllables pa,chaca, and mac. To represent the sound of pa, the quipu maker would weave the Pachacamac symbol followed by a knot for "one", telling the reader to only read the first syllable of the word Pachacamac. It is also possible to represent pa by weaving the symbol of Allpachamasca followed by two knots, meaning the second syllable should be read.

It is also possible to represent a logogram in this system. If the woven symbol does not have any accompanying knots, then the symbol serves as a logogram that represent the entire word of the symbol's meaning. Hence, for example, the Pachacamac symbol by itself on a quipu cord would read aspachacamac.

This system of mixing symbols with numbers does not exactly mean that quipu is a full writing system, since it relies on non-quipu symbols. However, the same manuscript also describes a translation of these symbols to distinct numeric values, meaning that it is possible to completely represent a phonogram or logogram with a group of two quipu numbers.




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