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.
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
but multiples of
on, the fourth place represents multiples of ,200,736020
the fifth place multiples
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 220.127.116.11.18 means
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.
The Mayas used three different calendars: the Tzolkin, the Haab, and the Long Count.
A typical Mayan date looks like
18.104.22.168.18 6 Etznab 11 Yax.
Here, 22.214.171.124.18 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
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
days. The names of the
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
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
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
126.96.36.199.0 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 188.8.131.52.0 4 Ahau 8 Cumku corresponds to 11
August 3114 BC. The Mayas thus expected the next world destruction to occur on
184.108.40.206.0 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
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
days, giving a period of
583.93 days, or eight days after
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
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.
America, Central; America, South; Astronomy; Calendars; Zero.
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