A great many descriptions of ancient Maya mathematical notation read something like this:
The Maya made use of a base-20 (vigesimal) system with the units of 1, 20, 400, 8,000, 160,000, etc.. To write a number, a scribe would show multiples of these units in a set columnar order, moving down from highest to lowest, and add them accordingly. “32” for example would be written as single dot for 1, representing one unit of 20, above the two bars and two dots for 12, corresponding to the “ones” unit (1×20 + 12×1 = 32). A larger number such as 823 would be written in three places as two dots followed by one dot followed in turn by three dots, standing for the necessary multiples of 400, 20, and 1 respectively (2×400 + 1×20 + 3×1 = 823).
Similar descriptions of Maya math pervade the literature, textbooks and the internet. For example Michael Coe writes in the latest edition of The Maya (p. 232):
Unlike our system adopted from the Hindus, which is decimal and increasing in value from right to left, the Maya was vigesimal and increased from bottom to top in vertical columns. Thus, the first and lowest place has the value of one; the next above it the value of twenty; then 400; and so on. It is immediately apparent that “twenty” would be written with a nought in the lowest place and a dot in the second.
The illustration accompanying this text provides many examples of this purely vigesimal system:
Maya mathematical notation is described the same way in a number of other influential books widely read in classrooms and seminars, such as The Ancient Maya: New Perspectives (McKillop 2004:277) or the venerable The Ancient Maya (Sharer and Traxler 2006:101). In the latter work, two types of counts are represented (see below) – the purely vigesimal or base-20 count (with units of 1, 20, 400, and 8,000) alongside what’s called the “chronological count” (with units of 1, 20, 360, 7,200). The second is of course the basis for the familiar Long Count system.
A big problem exists with all of these seemingly straightforward descriptions of Maya mathematical notation. As far as I am aware no purely vigesemal place-notation system was ever written this way. It’s true that in Mayan languages numbers are base-20 in their overall structure, just as in most Mesoamerican languages. In Colonial Yukatek, for example, we have familiar terms for these units: k’al (20), bak’ (400), pik (8,000), and so on. However, ancient scribes never represented these units in a columnar place notation system, as is so commonly described in the textbooks. That format was instead always reserved for a for the count of time, in what we know as the Long Count. That system is mostly vigesimal, but it is skewed in one of its units (the Tun, of 360 days) in order to conform as much as possible to the number of days in the solar year (365). To reiterate: the columns of numbers we find in the pages of the Dresden Codex or painted on the walls of Xultun (stay tuned, folks…) are all day counts; the positional notation system was never used for reckoning anything else.
In the ancient inscriptions non-calendrical counts using large numbers are quite rare, mostly found in connection to tribute tallies, such as the counting of bundled cacao beans. But in those settings the scribes always seem to show nice rounded numbers (as in ho’ pik kakaw, “5×8,000 [40,000] cacao beans,” shown in the murals of Bonampak) without all the place units we know from the Long Count. In the Dresden and Madrid codices, counts of food offerings are given as groupings of WINIK (20) signs with accompanying bars and dots for 1-19. In this way a cluster of four such elements (4×20) with 19 writes 96 (See Love 1994:58-59; Stuart, in press).
There is a good deal we still don’t know about the ways the Maya wrote quantities, especially of non-calendrical things. The pattern nonetheless seems clear that the place notation system of the Long Count was restricted to time reckoning, and never applied to the purely vigesimal counting structure we see reflected in Mayan languages. The descriptions of written numbers found in the many texts about the ancient Maya therefore need to be corrected.
Sources Cited:
Coe, Michael. 2011. The Maya (8th edition). Thames and Hudson, New York.
Love, Bruce. 1994. The Paris Codex: Handbook for a Maya Priest. University of Texas Press, Austin.
McKillop, Heather. 2006. The Ancient Maya: New Perspectives. W.W. Norton, New York.
Sharer, Robert, and Loa Traxler. 2005. The Ancient Maya (6th edition). Stanford University Press, Stanford.
Stuart, David. In press. The Varieties of Ancient Maya Numeration and Value. To appear in The Construction of Value in the Ancient World, ed. by J. Papadopolous and G. Urton. Cotsen Institute of Archaeology, UCLA, Los Angeles.