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When a Number System Loses Uniqueness: The Case of the Maya
The Mayan Number System
The Mayan culture used a base \(20\) number system. It was an additive positional system that used two symbols, a dot for one, a horizontal bar for five, and a cowry shell for a place holder (it is unclear whether they also considered it a true numeric "zero''). Numbers were written vertically with the most significant digit at the top. They used a purely base \(20\) system for simple recording of commodities. However, their much more prevalent and culturally important calendar system used a modified base \(20\) system. The place values employed in this number system are tied to the Calendar Round and Long Count calendars. Each place represents the next order of numbers of days. So the first place, \(20^0,\) is days. The next, \(20^1,\) is months. The third would be years made up of \(20^2=400\) days. However, \(18\times20\) more accurately represents the year of 360 days (plus 5 Wayeb days) than does \(400,\) so here they used \(18\times20=360\) rather than \(20^2=400.\) Thus the Maya developed a modified base \(20\) system in which a date is represented by a number written (in modern Indo-Arabic notation) as: \[a.b.c.d.e=a(18\times20^3)+b(18\times20^2)+c(18\times20^1)+d(20^1)+e(20^0),\] where the third place value is not \(20^2\) but \(18\times20.\) After the third place, each higher place is \(20\) times the previous place value. So the system breaks only in the third place.
Please note that the calculation above is simply the modern Gregorian date for Isabel's birthday converted into Mayan numerals; it is not the date in the Mayan Long Count. Since the focus of this article is the uniqueness of the Mayan number system, and not their dating system, we preferred to focus on the conversion of a number and not how to convert dates. However, there are various websites that will convert a modern date into a Mayan Long Count date. Using the Maya Astronomy Page , the Mayan date of Isabel's birthday is \(220.127.116.11.8,\) or \[12 (18 \times 20^3) + 19 (18 \times 20^2) + 13 (18 \times 20^1) + 3 (20^1) + 8 (20^0) = 1,869,548\] days into the \(1,872,000\)-day Long Count cycle.
Below is an example of a Serpent Date found in the Dresden Codex.
Figure 2. Page from the Dresden Codex showing two Serpent numbers 
Shell-Gellasch, Amy and Pedro J. Freitas, "When a Number System Loses Uniqueness: The Case of the Maya," Loci (May 2012), DOI: 10.4169/loci003883