# Mayan Mathematics

Image from MacTutor History of Mathematics.

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 × 202, the next the number of 18 × 203 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