Math Use During Maya and Aztec Civilizations
by Araceli Leon

amleon@ucsd.edu

Written on 97-05-26

Presented on 97-06-02

Published here on 99-09-24

Have you ever heard of the Mayan Numeration System or the Aztec Calendar? With no doubt this calendar, and of course the numeration system, have mathematics involved in their development and its application in the everyday life of these cultures; we might not know to what extent or involvement, but I believe it is a subject worth looking into and even though there is still a lot to be uncovered and analyzed, I'll present to you what I learned from these cultures and their math knowledge applied to their daily life.

Mayan Civilization was located in the region of Veracruz, Campeche, Chiapas, Yucatan and Tabasco in Mexico, and in parts of Guatemala, Belize and Honduras in Central America. This civilization was made up of several groups, that's why they occupied such a large territory, even though they were independent of each other, they shared many common things, such as the language, their math & astronomy knowledge, and their beliefs. This is why they are considered one big civilization, as opposed to individual groups or cultures.

The way they achieved this sharing was by means of congresses or reunions, in which they told each other of their own group's discoveries or achievements in a specific area, so when the reunion was over each delegate will go and share/implement another group's ideas and so on until all of them had basically the same knowledge about a specific problem or area, one example could be the calendar (Von Hagen, 364).

The Mayas came up with a system of numbers of base-20, unlike ours which is of base-10 or decimal, they used this numeration in their daily life from something so common as trading in the marketplace to more advance aspects such as astronomy and the development of their calendar, which many years later another great civilization would take advantage of as well. But, why 20? Well, these people were very religious and had many "superstitions" or beliefs, which led how they lived and how they acted. They viewed the number 20 as a perfect, special or magic number since a person has 10 fingers and 10 toes, giving us 20 in total!!, so they tried to keep numbers as lucky or special as possible.

In their numeric system, the Mayas didn't have a place for fractions only whole numbers, and they were able to do addition and subtraction very easily, but is not known with certainty if multiplication and division were possible or not.

zero . .. ... ....

0 1 2 3 4

. .. ... ....

------- ------- ------- ------- -------

5 6 7 8 9

. .. ... ....

------- ------- ------- ------- -------

------- ------- ------- ------- -------

10 11 12 13 14

. .. ... ....

------- ------- ------- ------- -------

------- ------- ------- ------- -------

------- ------- ------- ------- -------

15 16 17 18 19

As you can see the Mayas used three symbols to represent their numbers: a shell-like figure to represent nothing or zero, a dot for one unit, and a bar that corresponded to five units. It is believed that they had a symbol for the number twenty, but is not used as we will see in a minute. Also, the value of a number increased from bottom to top in vertical columns (Coe, 184), and not from right to left as we are used to.

The way to represent a number in this system is as follows:

On the first level we represent numbers from 0-19 using the three symbols we have, to represent the number 20 we go up one "level" and use a dot (having putting a "zero" in our first level), in this level we can represent numbers from 20-380, (multiples of 20^1), using the same procedure we can represent the number 400 or any of its multiples in the third level (20^2), and so on always going on to higher value levels (20^n, n is the level and n=0,1,....). So to add or subtract is just a matter of adding dots, bars and shells, keeping in mind that the value goes up from bottom to top.

Here is an example of how you apply all this I just said:

To add: And similarly, to subtract:

.. .. .... . ..

.. + .. = ----- - = -------

------- -------

.... ------- .... -------

------- ------- 80 - 25 = 55

-------

449 + 55 = 504

Maya merchants used this vigesimal system when trading, using mainly cacao beans and the like, such as maize grains, to do their counting, (besides using them as medium of exchange), by performing computations on the ground or on a flat surface (Coe, 185). Another very important use for this numeric system, was the development of the calendar, but for this implementation they introduced a slight change in one of our levels of value. Before giving you this change, I'll guide you through a brief introduction of this calendar, making it a little easier to understand their change.

The Mayas had two calendars: 1) a ritual calendar, which consisted of 260 days; and 2) a solar or Classic calendar, with 365 days.

The Ritual Calendar, (also known as tzolkin), was used for religious purposes and to name children. It was made up of 13 months with 20 days each (20*13=260), the number 13 was also special since they believed that their gods were created or existed in groups of 13; they used a combination of the numeric system with glyphs or little drawings, to represent each day of the year; so every day had a unique representation. To name the days, they would start with the number 1 followed by the representation of the first day (all days and months had names, here I'm representing months with the numbers 1-13, for simplicity), for the second day it would be 2 followed by the glyph of the second day, the third day a 3 followed by the glyph of the third day, and so on until you get to the 14th day which was represented with the number 1 followed by the glyph of the 14th day, and continuing with the same system until you get to the end of either the days or months count in which you'll start from the first glyph or number, as I did with the 14th day.

The Classic or Solar Calendar, (also known as haab), was used to represent important dates, such as conquests, ascendanceÌs, and the like, as well as relative dates of importance to the astronomers and farmers. This calendar was divided into 18 months of 20 days each plus a 5-day month, to give a total of 365 days, with this calendar was necessary to introduce a change in their number system, but this Ïnew systemÓ was only used for calendaric purposes. Since the days in the calendar were not a multiple of 400 (20^2), what they did was that they give a maximum value of 360 or 18*20 to the third level, and they continued with this value on up instead of the 400, that is:

4th level/order (360*20) . . = 2*360*20 = 14 400

3rd level/order (20*18) = 5*20*18 = 3 800

2nd level/order (20) . = 1*20 = 20

1st level/order (1) ... = 1*3 = 3

Now, with this modification they were ready to implement their day counting.

These two calendars were combined to form a new greater cycle, known as Round Calendar, with 260*365 days totaling 19980 or 73 tzolkin and 52 haab, a unit to measure a cycle of 52 years, (Ciudad, 102). This is known to us as Long Count, this gives the total days that had passed since the Maya zero date in 3114 BC, (Calendar Notes). With the aid of a conversion table, they were able to record dates in this new calendar.

Conversion Table:

1 Baktun (20 katuns) = 1* 144 000 days

1 Katun (20 tuns) = 1* 7 200 days

1 Tun (18 uinals) = 1* 360 days

1 Uinal (20 kins) = 1* 20 days

1 Kin = 1 day

An example of how we will write theses dates in modern notation could be:

12.18.2.13.16, where that will mean, 12 baktuns, 18 katuns, 2 tuns, 13 uinals and 16 kins, (Calendar Notes). You can do the math and tell me how many days are those.

The Mayas constructed ceremonial centers as well, which included pyramids, ball courts (with an impressive acoustic), temples, etc., all of which only their ruins remain. Here is where you will encounter all this dates engraved in stone along with a form of writing, to depict their surrounding.

One of the mysteries left behind is the disappearance of this civilization, and why they abandoned their cities, one of the causes could have been invasion of other groups like the Toltecs, who took over and eventhough they preserved some of the Mayas characteristics and knowledge, like the number system and the calendar, created a new group or culture.

With the Toltecs possessing all this knowledge and adapting it to their own lives, there was no great loss of all the great things the Mayas had created and worked on for so much time, and in fact it helped to keep it alive and in use, so that another great Civilization could take advantage of it as well, that civilization being The Aztecs.

The Aztec Civilization flourished after the downfall of the Toltecs around the 11th century AD. By the year 1325, they had founded the city of Tenochtitlan in the Valley of Mexico and had become one of the stronger military groups. The Aztec Empire as it was known, was very powerful and dominated several other groups, imposing taxes, taking sacrifice prisoners, and absorbing their advancements. This is how they were influenced by the Mayas, not directly but by the Toltecs whom had acquired that knowledge years before, in relation to their math and their calendars.

Eventhough there is no explicit Aztec math, it is believed that they were taught a maya-derived math, using the same concept and symbols, so it worked the same way. It was also used in the marketplace for trading of goods, but more significantly it was implemented by astronomers in the important analysis of the skies, through the calendars.

The Aztecs had two calendars as well, a solar year and a sacred calendar. These were: the tonalpohualli or Sacred Calendar, which was of religious character and was made up of 260 days or 13 months of 20 days each; and the Solar Calendar which was made up of 18 months of 20 days each with a 5-day month considered to be unlucky, and destined to rest and nothing else. These two calendars come together in the Round Calendar which, the same as with the Mayas, resulted in a 52 year cycle. The difference between the Maya and Aztec calendars lies in different names for the days, but basically are the same.

The Aztec Calendar or Sunstone, is believed to represent the world age or eras of it. It combines the mythology of the Aztecs with a precise count of time, which they used to lead their lives and how to view the world around them. This is one of the majestic relief pieces left behind by the Aztecs, it is 13.5 ft in diameter and weighs 25 tons (Aztec Calendar Stone), it has captivated thousands of people around the world to try to decipher what it means or what is supposed to tell, the truth is, it is still a mystery.

The Aztecs were also very religious and guided themselves through offerings and sacrifices to the gods, the Aztec stone was a sacrificial stone dedicated sometime in the 1470's in the temple of the war god Huitzilopochtli, but also represented the Aztec cosmos (Fagan 32). There were people responsible of studying the cosmos, like the priests who were the astronomers as well. It is obvious that these people needed to posses a strong mathematical knowledge in order to study the skies, and interpret the calendar, so we can see that math was an integral part of the future priests' education.

The Calendar itself was made up of a combination of the earlier calendars and aztec's beliefs. The aztecs believed that every era of the world was marked by the end of the 52nd year or cycle, and that after that either a new era was about to begin or the end of the world was near. Every era was marked by a unique process, like Ïthe Era of the Earth Sun, the Age of Great Winds, the Age of Fire, the Age of Floods, and the present one the Age of EarthquakesÓ. These were the five Suns, all of which are depicted in this calendar: the present one in the center and all other four, with their date of destruction, surrounding the fifth sun, which is symbolized as one of the Aztec gods, probably the sun god, (Brundage 27). So the stone didn't commemorate the birth of the fifth sun but its destruction by earthquakes, and the Aztecs believed that in order to keep the gods happy they had to supply them with fresh human hearts, so as to preserve continuity of life (Fagan 35).

The ring following these four suns, represented as rectangles, depicts the representation of the 20-days of the tonalpohualli; and the second outermost ring depicts the xiuhmolpilli with its 52 years. As I said before the complete meaning of the sunstone is ignored, but that it is an impressive archeological piece there is no doubt about it.

Unfortunately, all this is just a brief overview of their capacity and knowledge, since very little was left intact by the Spaniard conquest and the natural evolution of time, that only very limited is known for a fact and everything is based on manuscripts or codices left behind along with recent discoveries, which can be very misleading.

With these calendars the Mayas and Aztecs were able to trace important dates as well as to deeply investigate the skies and all the constellations/stars present. They were capable of astonishing math and astronomic knowledge that can be compared to the ancient Egyptians, Hindus and Babylonians, of course each gave humanity a unique and different gift of knowledge, but there is still a lot to be learned from them and it might take centuries to really understand how they developed all this knowledge and why time was so important for them, so that they devoted so much time to its study.

#### Exercises

Perform the following operations using the Mayan numeration system given:

1.- 13,587 + 33 + 240 = ???

2.- 875 + 96 = ???

3.- 694 - 321 = ???

Do the following conversion to total days, according to the conversion table given:

17.6.12.3.10

#### References

Bancroft, Hubert. The Works of Hubert: The native Races. vol 2 San Francisco: The History Company, 1882.

Brundage, Burr C. The Fifth Sun. Austin: University of Texas Press, 1991.

Calendar Notes. Homepage. http://www.halfmoon.org/calendar.html

Ciudad, Andres. Los Mayas: El Pueblo de los Sacerdotes Sabios. Madrid: Ediciones Anaya, 1988.

Coe, Michael D. The Maya. Fifth Edition. New York: Thames & Hudson, 1993.

Fagan, Brian M. The Aztecs. New York: W.H. Freeman and Company, 1984.

Gann, Thomas. The History of The Maya. New York: Charles Scribner's Sons, 1931.

Garces, Guillermo C. Los Codices Mayas. Mexico: SEP/SETENTAS, 1975.

Microsoft Encarta Encyclopedia, 1996. Mayas, Aztecs.

Van Hagen, Victor W. Ancient Sun Kingdoms. New York: The World Publishing Company, 1961.