Mayan Mathematics

    The Maya are commonly considered to be one of the most advanced ancient civilizations.  They occupied the eastern third of what is known as Mesoamerica.  This includes parts of modern day Mexico and Central America – most prominently, the Yucatan Peninsula (wwwG; wwwF; Richards, 1998).  The Mayan culture arose somewhere between 1500 and 800 BC (wwwC).  Some sources believe this date to be as early as 2600 BC (wwwA), but nearly all sources agree that the height of Maya civilization lasted from about 250 AD to 900 AD (Hammond, 1982; wwwA).

    The Maya are credited with developing many elaborate and sophisticated systems.  They had a highly evolved system of hieroglyphic writing.  It can be said that the Mayans developed the only true writing system native to the Americas (wwwB).  It is also believed that they invented the most sophisticated mathematical system ever to originate in the Americas (wwwH).

    The Mayans had a numeric system of base 20 called a vigesimal system.  Most of modern civilization works within the decimal system of base 10.  There are some interesting hypotheses as to why the Mayans chose a base 20 system.  The most probable is that the Mayans considered one whole group of numbers to be twenty, the total number of fingers and toes, whereas the base 10 system of counting probably originated by grouping the total number of just fingers, not toes.

    The entire system of written Mayan numbers is based on dots, lines, and a symbol said to resemble a shell.  Dots represent groups of one, lines represent groups of five, and the shell symbol stands for zero.  What is interesting to note is that the Mayans were one of the very few ancient civilizations with a concept and appreciation of the number zero (wwwH; Aveni, 1993).

    Shown below is every number from 1 to 19 with 0 at the end.  (Read left to right, top to bottom.)

    The Mayans did not have specific symbols for numbers greater than nineteen.  They instead used a system of position.  Much like how Arabic numbers are read from left to right where position denotes decreasing power of ten, Mayan numbers are read top to bottom where position denotes decreasing powers of twenty.
 

For example, 160,512 in Arabic numbering stands for 1´100000 + 6´10000 + 0´1000 + 5´100 + 1´10 + 2´1 (1´105 + 6´104 + 0´103 + 5´102 + 1´101 +2´100).  The same number in the Mayan system is written on the left.     This arrangement of numbers, read top to bottom, stands for 1´160000 + 0´8000 + 1´400 + 5´20 + 12´1 (1´204 + 0´203 + 1´202 + 5´201 + 12´200).

 

Another example is the representation of the very large number 3,510,940.  This is a good example of how important the zero symbol is when writing numbers.  Without the zero symbol, this number would only be read as 175,547, a difference of 3,335,393.  So a very important function of the zero symbol is to preserve position.  Another function of the zero symbol is to make addition and subtraction much easier.  As an example, try adding two Roman Numerals together; you’ll soon see the importance of zero as a placeholder (Aveni, 1993).

    Anthony F. Aveni has an interesting hypothesis as to how the Maya chose the written representation of numbers.  Building off of the idea that many number systems first began as hand gestures, he describes a simple way that dots, lines, and shell-like symbols relate to hand gestures.  The dot may represent the fingertip.  If the fingers are extended (while not touching one another) and viewed from the front, they look like simple dots and each dot represents one finger. The line may derive from an extended hand.  Again, if the fingers are extended (while touching one another) and viewed from the front or the side, they look like they form a line and each line therefore represents one full hand of fingers, or the number five.  Finally, the symbol for zero is often described as a shell, sometimes a leaf.  But it is possible that this symbol may represent a closed fist, which may have been the Mayan gesture for zero (Aveni, 1993).

    For the sake of simplicity in typing and reading Mayan numbers, most modern persons transcribe these numbers and dates into Arabic numbers (Hammond, 1982).  The accepted method of transcription is Arabic numbers separated by periods.  So the Mayan number shown above (3,510,940) would be written as 1.1.18.17.7.0 where decreasing orders of twenty are now read left to right.

    The base twenty system described above was used when counting tangible items.  The Mayan vigesimal system is slightly modified when counting units of time (Aveni, 1993).  The third place becomes 18´20 (360) instead of the usual 20´20 (400).  The fourth place then becomes 18´20´20 (7200) and the fifth place is 18´20´20´20 (144,000) and so on.  By changing the counting system in this way, it becomes easier to calculate years, for example.  One seasonal year is considered to be 365 days, so the Maya would write this number of days as 1.0.5 (1´(18´20) + 0´20 + 5´1) instead of the standard vigesimal way of 18.5 (18´20 + 5´1).  Remember that the notation of 1.0.5 is our translation of the Mayan number symbols.  If the Mayan numbers were written using their notation, you will see that 1.0.5 is simpler to write than 18.5.  Since these numbers would have been used often when writing dates and calendars, changing the base twenty system makes keeping track of dates simpler and makes these dates easier to write.

    By developing such a remarkable system of mathematics, the Maya were able to do so much more than other civilizations of their time.  The Maya endlessly searched to find patterns in past sky events I an effort to predict the future.  The Maya believed that “the future was present in the past” (Aveni, 1993). By making detailed observations of the heavens and by using their highly developed system of mathematics, the Maya were able to invent very accurate calendars to account for thousands of years.  The calendars chronicled and predicted many complex patterns seen in the sky; for example, the path and schedule of the sun and moon.  Or even more complex, visible planets orbits and even the schedule of observable eclipses.  Without their mathematical knowledge, the Maya may never have been able to understand the heavens and build such an accurate calendar system.  The Mayan mathematical system also allowed for a very widespread system of trade and commerce by being simple enough that even the uneducated Mayan people could add and subtract.  It is likely that there are other uses of the Mayan vigesimal system that haven’t even been discovered yet.  We will have to wait and see
 
 

Aveni, Anthony F.  Ancient Astronomers.  Montreal: St. Remy Press, 1993.

Hammond, Norman.  Ancient Maya Civilization.  New Brunswick: Rutgers University Press, 1982.

Richards, E. G.  Mapping Time: The Calendar and its History.  New York: Oxford University Press, 1998.

wwwA: http://www.civilization.ca/membrs/civiliz/maya/mmc01eng.html

wwwB: http://www.civilization.ca/membrs/civiliz/maya/mminteng.html

wwwC: http://www.magnet.ch/serendipity/hermetic/cal_stud/maya/boehm/boehm51.htm

wwwF: http://www.indians.org/welker/maya.htm

wwwG: http://www.criscenzo.com/jaguar/region.html

wwwH: http://www.civilization.ca/membrs/civiliz/maya/mmc05eng.html