Computations using Maya Numbers

20th May 2023

Mexicolore contributor Ximena Catepillán

We are sincerely grateful to Dr. Ximena Catepillán for this illuminating introduction to the ancient Maya vigesimal (20-based) counting system. Millersville University of Pennsylvania mathematics Professor Emerita Ximena Catepillán has indigenous roots from South America and is passionate about the mathematics of the pre-Columbian Americas. Ximena taught mathematics at the Universidad de Magallanes in the Chilean Patagonia prior to coming to the USA to study her PhD in mathematics. She is the co-author of the textbook Mathematics in a Sample of Cultures.

The civilization of the Maya had a long history divided into three periods: The Pre-Classic period ca. 1200 BCE – 200 CE, the Classic period 200 CE – 900 CE, and the Post-Classic period 900 CE – 1519 CE. This civilization extended from what is now Belize, Central and Southern Mexico, Guatemala, El Salvador, and Honduras. The Maya developed a glyph writing system, a number system, and they mastered several areas of science, art, and architecture. Its most significant area was astronomy, and the mathematics they used for their calculations utilized a vigesimal number system, described in this section. For their calendrical computations they had an “almost vigesimal” system called chronological number system.
The Maya zero dates to 200 BCE – 100 BCE and was the key to develop their positional vigesimal number system. In figure 1 we have the Maya digits with the most common symbol used for zero, believed to represent a shell.

The Maya positioned the digits vertically and are read from bottom to top.
In figure 2, on the left we have the number 1,332, that is obtained by multiplying the digits from bottom to top by powers of 20 (vigesimal number system) and then adding the products. Hence, 12 x 20{superscript0 + 6 x 20{superscript1 + 3 x 20{superscript2 = 1,332, and, on the right, we have the number 96,022 corresponding to 2 x 20{superscript0 + 1 x 20{superscript1 + 0 x 20{superscript2 + 12 x 20{superscript3.

In the chronological Maya number system, the difference is at the third digit, which is multiplied by 18 x 20 as opposed to 20{superscript2, the fourth and higher digits continued to be multiplied by powers of 20. See figure 3 left, the number 1,212 and on the right the number 86,422. The chronological system was used to represent calendrical information in the codices, the books of Chilam Balam – compilations from older and now lost manuscripts, and in their inscriptions.

Four surviving Maya manuscripts (codices) have been found; these books have colored pages, inscribed on both sides and written on paper manufactured from the leaves of the maguey plant. The codices contain combinations of numbers, astrological and astronomical reckonings, information regarding wars, hurricanes, famines, and many other events.
There was a great number of these books, however, in 1562 the missionary Diego de Landa ordered the burning of a large number of books and religious images in an attempt to eradicate their native religion. In figure 4, we can see a portion of a page of the Dresden codex which contains information about Mars. The codex is called Dresden because it is at a library in Dresden, Germany.

On the right picture in figure 4, highlighted in red, bottom from right to left we can see in Maya chronological form, the numbers:
78 = 18 + 3 x 20,
156 = 16 + 7 x 20 = 78 x 2,
234 = 14 + 11 x 20 = 78 x 3,
312 = 12 + 15 x 20 = 78 x 4,
390 = 10 + 1 x 20 + 1 x 18 x 20 = 78 x 5, and
780 = 0 + 3 x 20 + 2 x 18 x 20 = 78 x 10.
On the top row we have from right to left the numbers:
468 = 78 x 6,
546 = 78 x 7,
624 = 78 x 8, and
702 = 78 x 9.

Why multiples of 78?
The number 78 has its origins on the 780-day synodic period of Mars, the time required for the planet to return to the same position as seen by an observer on earth. The Maya calculated the period to be 780 and the actual synodic period is 779.94, quite amazing.
The hanging animal-like figure is the Maya god for Mars, the god of the storm, also known as the “Mars beast” characterized by its ornamented nose.
On another page of the Dresden codex, we find information about lunar eclipses.

In figure 5, the numbers are representing number of days and are listed next to a sequence of three consecutive calendar dates of the Tzolkin calendar. The predicted eclipse would occur in one of the three dates. The highlighted number in red on the left corresponds to 9,183 days, using the chronological method, 3 + 9 x 20 + 5 x 18 x 20 + 1 x 18 x 20{superscript2 = 9,183.
The second highlighted number from the left is 0 + 0 x 20 + 6 x 18 x 20 + 1 x 18 x 20{superscript2 = 9,360
the other highlighted numbers from left to right are: 9,537, 9,714, 9,891 and 10,039. Notice that the number of days between these numbers are 148 or 177 days, the number of days between the eclipses:
9,183 + 177 = 9,360
9,360 + 177 = 9,537
9,537 + 177 = 9,714
9,714 + 177 = 9,891, and
9,891 + 148 = 10,039.

Why the numbers 177 and 148?
The synodic period of the moon is 29.530588 days or 29 days 12 hours 44 minutes and 3 seconds, and the interval between any two successive lunar eclipses can be either 1, 5, or 6 synodic periods.
Note that 5 synodic periods = 5 x 29.530588 = 147.65294 days, and 6 synodic periods = 6 x 29.530588 = 177.18353 days. The Maya calculated the periods as 148 and 177 days.

In the same figure 5, highlighted in calypso, we have Tzolkin dates, as part of three consecutive dates associated to each of the number of days.
The Tzolkin calendar was one of over 21 calendars in use in Mesoamerica upon the arrival of Hernán Cortéz in 1519 in what Mexico is today. The 260-day Tzolkin, or sacred calendar, is still in use by some of the shamans in the Highlands of Guatemala for divination and ceremonial purposes. It uses 20 names of Maya gods with 13 numbers, hence, a Tzolkin calendar cycle has 260 days and begins with 1 Imix and ends with 13 Ahau. The Maya wrote the dates using the Maya number together with the glyph representing each god.

Going back to figure 5, highlighted in calypso, we have from left to right the Tzolkin dates for the predicted lunar eclipses, 5 Eb, 13 Muluc, 8 Cimi, 3 Akbal, 11 Ahau and 3 Lamat, see figure 7.
Note that:
5 Eb + 177 = 13 Muluc
13 Muluc + 177 = 8 cimi
8 Cimi + 177 = 3 Akbal
3 Akbal + 177 = 11 Ahau
11 Ahau + 148 = 3 Lamat.

In figure 8, we have the 260 Tzolkin dates, starting with 1 Imix and continuing with 2 Ik, 3 Akbal, 4 Kan, 5 Chicchan, etc. until you reach the last date 13 Ahau. Use the calendar table to locate 5 Eb and then count 177 days to reach 13 Muluc, these dates correspond to two consecutive eclipse dates.

Which operations did the Maya use to perform their calendrical computations?
While they used a vigesimal system to write the numbers, this system was never used in connection with days. No inscriptions use vigesimal numbers but rather chronological numbers. They probably used successive sums to perform the product of any quantity by a single digit. However, there are no records that indicate how they multiplied numbers with more than one digit. There is some supporting evidence that they used a grid on the ground or a stucco floor to perform the computations with beans, corns kernels and sticks.

Multiplication example using vigesimal numbers
We will use a grid to multiply 66 and 22. We begin writing the numbers using Maya digits (see Step 1, above).

Next, we locate the numbers outside a grid (see Step 2, left).

We continue by multiplying the corresponding cells 3 x 1, 3 x 2, 6 x 1, and 6 x 2, and entering the digits on the cells (see Step 3, right).

Next, we add the numbers within the grid in each one of the diagonals, obtaining 12, 12, and 3 (see Step 4, left).

The numbers we got in the diagonals from right to left, are the digits of the answer, which is the number 12 + 12 x 20 + 3 x 20{superscript2 = 1,452 (see Step 5, right).

See figure 9 (above) for a summary of the multiplication steps.

All images supplied by and courtesy of the author

A Maya limerick (Ode to vigesimal counting):-
In today’s world we still count in 10s -
To the Maya that doesn’t make senz.
Of digits we’ve plenty
- we should count on all 20:
More natural, they’d say. No offenz!