IV. Wonders of Knotted Numerals.

IV : Wonders of Knotted Numerals
        [Contd. A Journey To the Wonderland of Math.by Ajay Kumar Chaudhuri]

It is often thought that mathematics can only develop after a civilization has developed some form of writing, although not easy for us to understand today.Surprisingly many civilizations reached highly advanced states even not developing  wring skill. Now of course it is difficult for us to know much about such civilizations since there were no written records to be studied today.

Such was the ancient South American civilization of the Inca Empire flourished in ancient Peru between 1400 and 1533 CE which does not appear to have found a need to develop writing. The Inca Empire, before the Spanish conquest in the 16^th century was vast. It spread over an area which stretched from what is now the northern border of Equator to Mendoza in West central Argentina and to the Maule River in central Chile. The Inca people numbered around 12 million but they were from different ethnic groups and spoke about 20 different languages. The civilization had reached a high level of sophistication with a remarkable system of roads, agriculture, textile design and administration. Though it was possible to achieve this sophistication of high level without writing, yet counting and recording of most necessary information were necessary. Surprisingly, the Incas had developed a unique method of recording numerical information without writing. They invented such a method which involved knots in strings, called ‘quipu’.

Quipus generally consisted of coloured, spun and piled threads or strings which were made from cotton, wool, or camelid fibre such as of Alpaca, Llama, Guanaco etc. The colours of the cords, their relative arrangements, spaces between the cords, the types on particular cords and relative position of the knots were all to indicate numerical recording on a logical basis. Actually the cords contained numeric as well as other values encoded by knots in a base of ten (as we do now?) positional system. Quipus could have only a few or up to 2000 cords. The combination of fibre types, dye colours and intricate knotting were actually novel form of what is now called ‘writing’. For example, a yellow strand might be used for recording amount of gold or amount of some food grains or a quipu representing population, the first set of strand represent number of men, the second set number of women while the third set for number of children. [Pic. No.1 : A Quipu in Larco Museum, Peru]    Pic.No1.

                                                  A Quipu in Larco Museum, Peru.

                      

From archeological evidence, it was also found that for sturdy recording, finely curved wood was used as a supplement, on which the colour coordinated cords would be attached.

One thing should be remembered that the quipu was not a calculator but a storage device. As the Incas had no written records, so the quipu played a major role in their administration by keeping numerical information.

Now let us see how quipus were read by the Incas. It has already been mentioned that numbers were represented by knots in a string using positional base 10 representations.If the number recorded be say,1421 then one knot was placed at the free end of the string (like unit position as in our decimal base system),a space was left and two touching knots were placed in the tens  position. Again a similar space was left and there were four touching knots for the hundreds position. Finally after leaving same space,a single knot was placed in the thousands position (which is very clear from the knots on the string,second from left in Pic. no 2, representing the number 1421)

For larger numbers more groups of knots were used, one for each power of 10, in the same way we use in decimal base system. Same knots were not used irrespective of the position. To represent a number in the unit position different styles of knots were used from those in other positions. In fact two different styles were used in the unit position, one style if the number be 1 and the other, if it is greater than one. The system had a zero position which was represented by no knot in that position. So the spacing had to be highly regular for the clear idea of the position of zero.

In the Inca regime there were experts of creating and deciphering quipus, called Quipu camayocs. They were actually the accountants of the Inca Empire, who could perform simple mathematical operation, such a addition, subtraction, multiplication and division. In absence of written records the quipus served as a means of recording history, a variety of information to the administration, and to the next generation, which [Pic. No. 2 : Decording of a Quipu] used them as reminders of the past. So, in such a way, these primitive computers – quipus had knotted their memory banks of information which tied together the Inca empire.

Pic.No 2.Decoding of quipu.

Relatively a small number of quipus have survived. Archeologists in Peru have found a ‘quipu’ on the site of the oldest city of Americas. The oldest known quipus, generally associated with the Incas of the vast South American empire dated about 650 A.D. But Ruth Shady, an archeologist, who was leading investigations into the Peruvial coastal city of Caral, said, ‘quipus were among a treasure trove of articles discovered at the site , which are about 5000 years old. She also said, ‘This is the oldest quipu and it shows us that the society also had a system of “writing” (which) would continue down the ages up to the Inca Empire and would last some 4500 years.’

There are many questions relating to quipus and their usages are still unanswered. Did the Incas apply their number system to solve mathematical problems? Was it merely for recording? Much information about quipus are found from the written records, written long after the Spanish conquest of the Incas. From such an old record, the picture like that of a counting board called “Yupana” was found. If this Yupana was essentially an abacus of the Incas, then it must have been used to solve problems. But it prompts the intriguing question of what these problems were? So, these are some fascinating topics which deserve much research to unknot the mysteries of the knots of the quipus.
Reference: Internet.
Image credit:A Quipu in Larco Museum.
(https://en.wikipedia.org/wiki/Quipu#/media/File:Inca_Quipu.jpg)

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