Mathematics of the Incas

It is often thought that mathematics can only develop after a civilisation has developed some form of writing. Although not easy for us to understand today, many civilisations reached highly advanced states without ever developing written records. Now of course it is difficult for us to know much about such civilisations since there is no written record to be studied today. This article looks at the mathematical achievements of one such civilisation.

The civilisation we discuss, which does not appear to have found a need to develop writing, is that of the Incas. The Inca empire which existed in 1532, before the Spanish conquest, was vast. It spread over an area which stretched from what is now the northern border of Ecuador 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 many different ethnic groups and spoke about 20 different languages. The civilisation had reached a high level of sophistication with a remarkable system of roads, agriculture, textile design, and administration. Of course even if writing is not required to achieve this level, counting and recording of numerical information is necessary. The Incas had developed a method of recording numerical information which did not require writing. It involved knots in strings called quipu.

The quipu was not a calculator, rather it was a storage device. Remember that the Incas had no written records and so the quipu played a major role in the administration of the Inca empire since it allowed numerical information to be kept. Let us first describe the basic quipu, with its positional number system, and then look at the ways that it was used in Inca society.

The quipu consists of strings which were knotted to represent numbers. A number was represented by knots in the string, using a positional base 10 representation. If the number 586 was to be recorded on the string then six touching knots were placed near the free end of the string, a space was left, then eight touching knots for the 10s, another space, and finally 5 touching knots for the 100s.

586 on a quipu

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