The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

Incan Quipus

Date: 09/30/1999 at 14:51:51
From: Ann Marie J.
Subject: Quipus

I need to know where I can find what the Incas used for numbers. I 
know that they are colored ropes, but we need more specifics.


Date: 10/03/1999 at 22:10:26
From: Doctor Twe
Subject: Re: Quipus

Hi Ann Marie!

According to Victor J. Katz in his book _A History of Mathematics: An 
Introduction, 2nd Edition_ (Addison-Wesley, 1998):

"A quipu is a collection of colored knotted cords, where the colors, 
the placement of the cords, the knots on the individual cords, the 
placement of the knots, and the spaces between the knots all 
contribute to the meaning of the recorded data. Every quipu has a main 
cord, thicker than the others, to which are attached other cords, 
called pendant cords, to each of which may be attached further cords, 
called subsidiary cords. Sometimes there is a top cord, a cord placed 
near the center of several pendant cords and tied so that when the 
quipu lies flat it falls in a direction opposite to the pendant cords. 
Data are recorded on the cords (other than the main cord) by a system 
of knots. The knots are clustered together in groups separated by 
spaces and represent numbers using a base 10 place-value system with 
the highest value place closest to the main cord. Thus the cord with 
three knots near the top and nine knots near the bottom represents the 
number 39. As additional help for reading the numbers, the knots 
representing units are generally larger knots than those representing 
higher powers of ten. The largest number so far discovered on a quipu 
is 97,357. Zeros are generally represented by a particularly wide 

"The pendant cords on quipus are themselves generally clustered in 
groups, sometimes with each group consisting of the same set of 
distinct colors. It is assumed that each color refers to a particular 
class of data being recorded on the quipu. In addition, one often has 
a top cord associated to a group of cords on which is recorded the sum 
of the numbers on the individual cords of the group. Sometimes certain 
of the pendant cords record sums of the numbers on other such cords. 
Sometimes it appears that the knots on a particular cord do not 
represent data at all but are simply labels. In any case, the quipus 
are not calculating tools, but only records. The calculations on which 
these records are based must have been done elsewhere, probably with 
some sort of counting board."

"What is not generally known is exactly what kinds of data a given 
quipu records, but one particular quipu is known to be a record of 
census data for a region of seven provinces. In this case, the people 
of each province were classified in one of two groups, each of which 
was further divided into two subgroups. Thus the individual pieces of 
data recorded on certain of the cords were the number of households in 
each province belonging to each of the subgroups. Other cords then 
represented the sums of the various pieces of this data, with one cord 
finally giving the grand total of the number of households in the 
entire region."

Katz references _Code of the Quipu: A Study in Media Mathematics, and 
Culture_ by Marcia Ascher and Robert Ascher, (University of Michigan 
Press, 1981), and says that the book has recently been reprinted under 
the title _Mathematics of the Incas: Code of the Quipu_ (Dover 
Publications, 1997). According to Katz, this work "provides a 
mathematical analysis of various techniques of quipu making and also 
provides exercises to help students learn the relevant mathematical 

As to Web sites, I used a few search engines and came up with the 

This site has lots of historical info, and lots of other links:

The Incas and their Descendants on the Web   

These sites have information on quipu. The last one has a good 

Talking Knots of the Inka   

The Quipu   

The Quipucamayu   

I hope this helps. Write back if you have any more questions.

- Doctor TWE, The Math Forum   
Associated Topics:
Elementary Math History/Biography
Middle School History/Biography

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.