Incan and Mayan mathematics

Summary: The Incan and Mayan civilizations had a variety of mathematical achievements, including number systems and calendars.

The Inca Empire existed from 1438 until 1533 c.e., 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 written words, the Incas invented a clever method of recording numbers, usually for administrative purposes, using knotted cords called a quipu.

The Maya civilization flourished between 250 and 900 c.e. The homeland of the Mayans 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. Among the many remarkable accomplishments of Mayan culture were 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 colored, knotted cords. Every quipu has a main cord that 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 2000 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 colors of the cords and the topology of pendants and subsidiaries do not contribute to the numerical information but signify the item that 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 around 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 1, 2, 3, or 5 dots, which could be maize kernels. Acosta explicitly mentioned the numbers 1, 3, and 8. This has led to speculations that the Incas used so-called Fibonacci numbers in their calculations since 1, 2, 3, 5, and 8 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 1, lines representing 5, and a symbol for 0 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 represents multiples of 20.

The third place, however, does not represent multiples of 20×20=400 but multiples of 18×20=360.From then on, the fourth place represents multiples of 20×360=7200, the fifth place multiples of 20×7200=144,000, and so on. 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 the following:

9 × 144,000 + 12 × 7200 + 11 × 360 + 5 × 20 + 18 = 1,386,478

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 (357 c.e.). The earliest indisputable inscription using the Hindu–Arabic decimal system including a symbol for zero is from Cambodia (683 c.e.).

Mayan Calendars

The Mayas used three different calendars: the Tzolkin, the Haab, and the Long Count. A typical Mayan date looks like the following:

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 August 29, 683 c.e.

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 names of days:

AhauKanLamatEbCib

ImixChicchanMulucBenCaban

IkCimiOcIxEtznab

AkbalManikChuenMenCauac

Both the number and the day name change daily such that the calendar runs as follows: 1 Ahau, 2 Imix, 3 Ik, and so forth. Every possible Tzolkin date occurs once during the Tzolkin year of 13×20=230 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 18×20+5=365 days. The names of the months are the following:

PopTzecChenMacKayab

UoXulYaxKankinCumku

ZipYaxkinZacMuan

ZotzMolCehPax

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, and so forth. 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 73×260=52×365=18,980, which 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 13×144,000 days or approximately 5125 years, each era ending with a time of great change. A Long Count date is a five-digit Mayan number recording how many days have elapsed since the last transition of cycles. 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, and so forth. According to various Mayan sources, the previous era ended on the following date:

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 August 11, 3114 b.c.e. The Mayans thus expected the next cycle change upheaval to occur on 13.0.0.0.0 4 Ahau 3 Kankin, corresponding to December 21, 2012 c.e., 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 the Morning Star for 236 days, then it disappears on the far side of the sun for 90 days, then it reappears as the Evening Star for 250 days, and finally it disappears again for eight 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 because of a local difference between the eastern and western horizons.

It is a strange coincidence that 584=8×73 and 365=5×73 have the large common prime factor of 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 was 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 301×584 days, which gives a Venus period of exactly 583.92 days. This explanation, however, was questioned by the famous physicist, Nobel laureate, and amateur Mayanist Richard Feynman.

Bibliography

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

Feynman, Richard. Surely You’re Joking, Mr. Feynman! New York: Vintage, 1992.

O’Connor, J. J., and E. F. Robertson. “Mactutor History of Mathematics Archive: Mayan Mathematics.” http://www-history.mcs.st-and.ac.uk/HistTopics/Mayan‗mathematics.html

Teresi, Dick. Lost Discoveries: The Ancient Roots of Modern Science—From the Babylonians to the Maya. New York: Simon & Schuster, 2002.