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Incan and Mayan Mathematics

The Inca Empire existed from AD 1438 until 1533 when it was conquered by the

Spanish, and the last Inca emperor, Atahualpa, was murdered. At its height, the Inca

Empire comprised most of present-day Peru, Bolivia, and Ecuador as well as parts of

Colombia, Chile, and Argentina. It was a culturally diverse but politically centralized

empire, based in the capital of Cuzco. Having no writing, the Incas invented a clever

method of recording numbers, usually for administrative purposes, using knotted

cords called quipus.

The Maya Civilization flourished between AD 250 and 900. The homeland

of the Mayas was the greater Yucatan Peninsula, including present-day Guatemala

and Belize as well as parts of Mexico, Honduras, and El Salvador. In contrast to the

Inca Empire, the Maya Civilization was never a political entity but consisted of a

multitude of independent city states. The many remarkable accomplishments of

Mayan culture include hieroglyphic writing, a vigesimal and duodevigesimal number

system, the invention of a symbol for zero, an elaborate system of calendars, and

highly accurate astronomical observations.

Incan quipus

A quipu is a bundle of coloured, knotted cords. Every quipu has a main cord which is

thicker than the others. Pendant cords are tied to the main cord, and subsidiary cords

are tied to pendant cords or other subsidiaries. Quipus have been found with as many

as 2,000 pendants and six levels of subsidiaries. The pendant and subsidiary cords

carry knots. Three types of knots are used: simple knots, figure-eight knots, and long

knots with two to nine turns. To record numbers, the Incas used a decimal number

system. Each digit other than the units is represented by a cluster of the appropriate

number of simple knots. The Incas did not have a special knot for zero but simply left

an empty space on the cord. Units are represented by a long knot with the appropriate

number of turns. If the unit is one, however, a figure-eight knot is used since a long

knot with only one turn is identical to a simple knot. For example, the number 701 is

represented by a cluster of seven simple knots, an empty space, and a figure-eight

knot. The digits are ordered with the units away from the main cord. Since the units

are distinguished from the other digits, the same cord can carry several numbers. The

colours of the cords and the topology of pendants and subsidiaries do not contribute

to the numerical information but signify what is being counted. There are about 800

quipus in museums today. The largest number found on a quipu is 97,357.

Quipus are not suitable for performing arithmetic. In 1590, Spanish Jesuit

missionary José de Acosta described how the Incas carried out difficult computations

by moving about maize kernels. A Peruvian drawing from about 1615 shows a tablet,

called a yupana, that might have been used for this purpose. This yupana is divided

into smaller squares each containing one, two, three, or five dots which could be

maize kernels. Acosta explicitly mentioned the numbers one, three, and eight. This

has led to speculations that the Incas used so-called Fibonacci numbers in their

calculations since one, two, three, five, and eight are the first such numbers.

Mayan numbers and the invention of zero

The Mayan number system is neither a pure grouping system like Roman or Aztec

numbers, nor a pure positional system like Hindu-Arabic numbers, but a mixture of

the two like Babylonian or Incan numbers. Numbers from 0 to 19 are written with

dots representing one, lines representing five, and a symbol for zero resembling an

eye. Thus, 17 is written as two dots and three lines. For numbers larger than 19, a

base-20 and, at one place, a base-18 positional system is used. The first place

represents units, and the second place multiples of 20. The third place, however, does

not represent multiples of

4002020

=

×

but multiples of

.3602018

=

×

From then

on, the fourth place represents multiples of ,200,736020

=

×

the fifth place multiples

of ,000,144200,720

=

×

etc. Mayan numbers were originally written vertically with

the units at the bottom. For convenience, Mayanists write them horizontally with the

units to the right. Thus, the Mayan number 9.12.11.5.18 means

.478,386,11820536011200,712000,1449

=

+

×

+

×

+

×

+

×

After the Babylonians, the Mayas, or possibly their Olmec predecessors, were the

first culture in the world to invent a symbol for zero. The earliest known occurrence

of this zero symbol is found on a stela in Uaxactun, Guatemala from AD 357. The

earliest indisputable inscription using the Hindu-Arabic decimal system including a

symbol for zero is from Cambodia, AD 683.

Mayan calendars

The Mayas used three different calendars: the Tzolkin, the Haab, and the Long Count.

A typical Mayan date looks like

9.12.11.5.18 6 Etznab 11 Yax.

Here, 9.12.11.5.18 is the Long Count date, 6 Etznab is the Tzolkin date, and 11 Yax

is the Haab date. This was the day of death of the great ruler Pacal of the city state

Palenque, corresponding to 29 August AD 683.

The Tzolkin calendar is based on two independent cycles of 13 and 20 days,

respectively. A Tzolkin date consists of a number from 1 to 13 followed by one of the

following 20 day names:

Ahau Kan Lamat Eb Cib

Imix Chicchan Muluc Ben Caban

Ik Cimi Oc Ix Etznab

Akbal Manik Chuen Men Cauac

Both the number and the day name change daily such that the calendar runs as

follows: 1 Ahau, 2 Imix, 3 Ik, etc. Every possible Tzolkin date occurs once during the

Tzolkin year of

2602013

=

×

days. This follows from the so-called Chinese

Remainder Theorem—which the Mayas must have known at least in some special

cases—and the fact that 13 and 20 have no common divisors.

The Haab calendar consists of 18 months of 20 days, followed by five extra

days. The length of the Haab year is thus

36552018

=

+

×

days. The names of the

months are:

Pop Tzec Chen Mac Kayab

Uo Xul Yax Kankin Cumku

Zip Yaxkin Zac Muan

Zotz Mol Ceh Pax

The days of each Haab month are numbered from 0 to 19. The Haab calendar thus

runs as follows: 0 Pop, 1 Pop, 2 Pop, etc. The final five days, called Uayeb, are

numbered from 0 to 4; these days were considered unlucky. The least common

multiple of 260 and 365 is

.980,183655226073

=

×

=

×

This means that the combined Tzolkin-Haab calendar repeats itself after 73 Tzolkin

years, or 52 Haab years, or 18,980 days.

The Mayas believed in a cycle of eras of 000,14413

×

days or

approximately 5,125 years, each era ending with the destruction of the world. A Long

Count date is a 5-digit Mayan number recording how many days have elapsed since

the last such destruction. There is a unique correspondence between the last digit of

the Long Count date and the Tzolkin day name: If the last digit is 0, the day name is

Ahau; if the last digit is 1, the day name is Imix, etc. According to various Mayan

sources, the previous era ended on

13.0.0.0.0 4 Ahau 8 Cumku.

The problem of translating Long Count dates into dates in the Gregorian calendar is

known as the Correlation Problem and has been a topic of considerable controversy.

Today, most Mayanists believe that 13.0.0.0.0 4 Ahau 8 Cumku corresponds to 11

August 3114 BC. The Mayas thus expected the next world destruction to occur on

13.0.0.0.0 4 Ahau 3 Kankin, corresponding to 21 December AD 2012, when the

present Long Count cycle ends.

Mayan astronomy and the Dresden Codex

The Dresden Codex is one of only four original Mayan books that have survived to

the present day. It contains astronomical tables in which the number 584 figures

prominently. This is the best integer approximation to the average period of Venus, as

seen from the Earth, of 583.92 days.

In the Codex, 584 is divided into parts of 236, 90, 250, and 8, reflecting the

phases of Venus: first Venus appears as Morning Star for 236 days, then it disappears

on the far side of the Sun for 90 days, then it reappears as Evening Star for 250 days,

and finally it disappears again for 8 days while it is between the Earth and the Sun.

The difference between 90 and 8 is explained by the fact that, as seen from the Earth,

Venus moves more slowly relative to the Sun when it is on the far side of the Sun.

The difference between 236 and 250 is thought be due to a local difference between

the eastern and western horizons.

It is a strange coincidence that

738584

×

=

and

735365

×

=

have the large

common prime factor 73. This implies that five Venus periods correspond very

closely to eight Haab years, and indeed the Codex contains a Venus table of this

length of time. The Mayas knew, however, that this correspondence is not exact. To

compensate, they subtracted either four days after

58461

×

days, giving a period of

583.93 days, or eight days after

58457

×

days, giving a period of 583.86 days. It has

been suggested that the Mayas used the first correction four times and the second

correction once, thus subtracting a total of 24 days after

584301

×

days, which gives

a Venus period of 583.92 days exactly. This explanation, however, was questioned by

the famous physicist, Nobel laureate, and amateur Mayanist Richard Feynman.

SEE ALSO:

America, Central; America, South; Astronomy; Calendars; Zero.

FURTHER READINGS:

Coe, Michael D. Breaking the Maya Code. New York: Thames and Hudson, 1992.

Feynman, Richard. Surely You're Joking, Mr. Feynman!. London: Vintage, 1992.

Katz, Victor J. A History of Mathematics. New York: Harper Collins, 1993.

David Brink, Ph.D.

University College Dublin, Ireland