After we talked about Babylonian mathematics and how they used a base-60 system, it got me thinking about different ancient cultures and the numbering systems that they used. My little brother is currently on an LDS mission in Guatemala, and he sent me some pretty cool pictures of Mayan ruins. Also, I remember back to 2012 when many people were hysterical about the “end of the world” because the Mayan calendar had stopped on a specific day in December. But man, if I was part of a civilization from thousands of years ago, I would NOT have made a calendar that went even that far. Had I been in charge, people might have thought the world was ending around the time Columbus made it to the Americas. A lot of what we know about the Mayan people was lost when they were invaded by the Conquistadors from Spain. A Spanish missionary, Diego de Landa, had a great respect for the Mayan people, but despised their religious customs. He ordered most of their religious icons, texts, and other documents to be destroyed, but of those that survived remain the Dresden Codex, the Madrid Codex and the Paris Codex. (Side note: I think it’s interesting that these documents are named after those that found them rather than those that created them.) The Dresden Codex, which will be discussed hereafter, is probably the most well-known. The Mayan people were a city-building and innovative society. There were 15 large cities (some of 50,000 or more) in the Yucatan peninsula. The people were governed by “astronomer-priests” that manipulated others with their religious instructions. (True case of where math and knowledge is power!)

The Mayans had one of the most advanced number systems in the world at its time. It was a base-20 system (kind of) that also relied quite heavily on the number five. Some think that this is due to five fingers and five toes on each hand and foot. Their system relied on three different symbols. A “pebble” (small black circle) was used to represent the ones place. A “stick” (straight black line) was used to represent the fives place, and a shell was used to represent the number zero. In a report about the Mayan numbering system, JJ O’Connor and EF Robertson explained why the system was not exactly a base twenty system, but one that had been slightly modified: “In a true base twenty system the first number would denote the number of units up to 19, the next would denote the number of 20’s up to 19, the next the number of 400’s up to 19, etc. However although the Maya number system starts this way with the units up to 19 and the 20’s up to 19, it changes in the third place and this denotes the number of 360’s up to 19 instead of the number of 400’s. After this the system reverts to multiples of 20 so the fourth place is the number of 18 × 20^{2}, the next the number of 18 × 20^{3} and so on.” (O’Connor & Robertson, 2000)

The Dresden Codex contains written evidence of the use of this numbering system. Ifrah claims that the system was used in astronomy and calendar calculations. He states, “Even though no trace of it remains, we can reasonably assume that the Maya had a number system of this kind, and that intermediate numbers were figured by repeating the signs as many times as was needed.” (Ifrah, 1998)

One of the reasons the Mayan numbering system was not a true base-20 system was because it was partially a base-18 system as well. The reason for this seems to be the calendars that they maintained. Robertson and O’Connor state: “The Maya had two calendars. One of these was a ritual calendar, known as the Tzolkin, composed of 260 days. It contained 13 “months” of 20 days each, the months being named after 13 gods while the twenty days were numbered from 0 to 19. The second calendar was a 365-day civil calendar called the Haab. This calendar consisted of 18 months, named after agricultural or religious events, each with 20 days (again numbered 0 to 19) and a short “month” of only 5 days that was called the Wayeb…” (O’Connor & Robertson, 2000).

It amazes me what the Mayans were able to do with the knowledge that they had available to them. Much of what we know of the Mayans comes from the few documents that have been preserved and the ruins that have been interpreted, but I wish that more information was available on how they used these systems in their societies and the other ways in which they used mathematics.

References

Ifrah, G. (1998). A universal history of numbers: From prehistory to the invention of the computer. London.

O’Connor, J., & Robertson, E. (2000, November). Mayan mathematics. Retrieved from The MacTutor History of Mathematics Archive: http://www-history.mcs.st-and.ac.uk/HistTopics/Mayan_mathematics.html