Accounting without writing: how did the Inca Empire keep its accounts on knotted cords?

Does accounting need paper, ink and letters? The Inca Empire in South America answers this question with a "no". The Incas had no writing, yet they kept track of the population, harvests and livestock of a large state. For this they used cords.

Key points

1. What a quipu is. According to Britannica's definition, the quipu was an accounting device of the Andean peoples and was widely used especially in the period of the Inca state, until the empire fell in 1532. It consists of one main cord and many cords hanging from it; the knots on the hanging cords expressed numbers.
2. A state without writing, but with accounts. As the MacTutor history of mathematics page of the University of St Andrews states, the Incas had no written records, and the quipu was not a calculating device but a means of storing information. Officials called khipukamayuq kept official lists of the population, harvests, livestock and weapons on these cords.
3. Decimal places. Where a knot stood relative to the top cord showed its place (units, tens, hundreds), and a place without knots meant zero (Britannica). The Dumbarton Oaks museum also stresses that the Incas developed a decimal positional knot system. Researchers believe that the colour of a cord and the spacing between cords may also have carried meaning.
4. The tradition is older than the Incas. The Dumbarton Oaks museum dates Inca quipus to 1450–1534. According to the same museum, the oldest known quipus were made much earlier, in 600–1000 CE, by the Wari people.
5. The summary cord: an ancient "check sum". In one Inca quipu with 438 cords in the Dumbarton Oaks collection, with two exceptions, the hanging cords are grouped in tens and one summary cord is added to each group. According to the museum's note, the Incas used such summary cords to prevent errors and falsification. In today's language, it was the cord form of the "total" line in a summary statement: by comparing the parts with the total, an error could be found.
6. A report that is summed from the bottom up. In 2005 the Harvard researchers Gary Urton and Carrie Brezine analysed in the journal Science 21 quipus found at the Puruchuco administrative centre. As the Harvard press release says, 7 of them contained numerical data that officials summed up step by step, that is, the figures were gathered from a lower level to a higher level.
7. An archive not yet fully read. The researchers used a computer database covering nearly half of the roughly 650–700 surviving quipus. Even so, scholars admit that reading quipus precisely has not yet been fully solved: the numerical part is fairly well understood, while the non-numerical signs are still being studied.

Then and now

Example: counting the harvest in the "Bog'ishamol" orchard with a "quipu"

The example below was compiled by the author; the figures are hypothetical.

On the hypothetical "Bog'ishamol" farm in Tashkent region, three teams picked apples. As an experiment, the accountant Dilnoza opa made a record in the Inca style: one cord per team, the hundreds knots in the top row, the tens in the middle and the units at the bottom.

Check: 132 + 205 + 48 = 385. Note: the sum of the knots in the units row is 2 + 5 + 8 = 15, so 5 knots go into the units row of the summary cord, and 1 ten carries over to the tens row (3 + 0 + 4 + 1 = 8). If the warehouse manager had recorded 375 crates as received into the warehouse, the 10-crate difference between the summary cord and the warehouse record would stand out at once and would be grounds for an inventory count. The Inca accountant and today's accountant ask exactly the same question: "does the sum of the parts equal the total?"

Link with the previous period: if in Mesopotamia clay tokens "held" numbers in the form of objects, in the Andes this task was done by cords. At the next stage, in the Middle Ages, Hindu-Arabic numerals and ledger entries moved accounting onto paper.