Mathematics of the Aztecs/Mexica

Aztec migration and empire
The ancestors of the Aztecs (“Aztec” will refer here generically to several cultural groups who inhabited the Valley of Mexico and central Mexico during the 13th through 16th centuries, and spoke variations of Nahuatl. This is imperfect but will avoid lengthy digressions) migrated into the basin of Mexico from a place their oral histories refer to as Aztlan. Although not located precisely, Aztlan was probably a region in northwest Mexico (and possibly extended into the southwestern U.S.) These movements began in the late 12th century and continued until around 1250. Beginning with a settlement on an island in Lake Texcoco, they slowly expanded in size and influence. The Aztecs founded the city of Tenochtitlan on the island in 1325, which remained their main city as Aztec power grew.
In 1372, the Aztecs came into their own as a major power in the region, and this is generally ascribed as the date of the founding of the dynasty of Aztec rulers. In 1428, after a series of wars, the Aztecs formed an alliance called the Triple Alliance, which ruled over most of central Mexico until 1521, when they were defeated by a combination of Spanish arms and European diseases for which they had no immunity.

Pic 1: Extent of the Aztec empire and tributary states c. 1519

The style of Aztec writing is mainly pictographic, that is, it consists of a series of pictures that represent certain words or objects. The pictographs themselves are not a complete written language; rather, they serve more as an outline to communicate the major elements of a narrative. Occasionally, these signs are used in combination to convey more complex words or names. There are very few surviving codices from precolonial times, since the Spanish confiscated and burned many of them for various reasons.

Numbers, number symbols, and counting
When the Aztecs arrived in the Valley of Mexico, they settled in a land that had been populated by various other groups for centuries. The existing cultures had been in close contact with the Maya for centuries, and had adopted many Mayan practices, which the Aztecs then apparently took up. Although it can only be inferred that the Aztecs borrowed much of their number and calendar systems from the Maya, the similarities are so strong that a coincidence is highly unlikely.

Pic 3: Page from Codex Borgia showing two ‘trecenas’ and associated deities

From the image above (pic 4), we can see that at times the Aztecs copied the Mayan practice of replacing five dots by one bar, which certainly makes things easier at times. You can also see that at times a symbol for a fraction of 400 was used instead of drawing five or ten flags. This incomplete standardization is another feature of earlier mathematics symbolism (In Roman numerals, it used to be common practice to write XIIII or XIV interchangeably. If you can find an antique book where the chapters are numbered with Roman numerals you’ll probably see XIIII for Chapter 14); usually these were cultures where only the elite were literate, so the potential audience for the writing is small. In those cases, you can often take shortcuts and rely on others to understand what you meant. In their writing, Aztecs frequently used number symbols as a label to indicate a certain number of particular things (see pic 5).

Pic 5: Part of a tribute record from Codex Mendoza (detail from fol. 52r). The number symbol for 400 atop each pictogram indicates that 400 of that type of cloth was given as tribute

The second excerpt (pic 7) gives the calculated area of each field, in order (now the fields are represented by abstract rectangles without regard for scale). Now the units are represented by the tick marks in the tab at upper right, and the number written in the center or at bottom has to be multiplied by 20. For instance, in the first field at top, the tab has 9 units; the number written horizontally in the center is 2x400+7X20, which gives a total area of 949 square units for the first field. It is unknown exactly how the Aztecs calculated the areas; the results given in the codex are consistent with use of several different methods, depending on the shape of the field (see the Science article in the references for full details).

Pic 7: Excerpt from the Codex Vergara (2)

This device was to have been used as a base-20 abacus, with the beads in the lower part representing units, and in the upper part representing fives. Unfortunately, no surviving nepohualtzitzin has been found, and even if the Aztecs used an abacus-like device, there is no assurance that it would use four beads in the lower half and three in the upper half. The Japanese soroban and the Chinese suan pan both come in variations, with many having five beads at bottom and two in the top half. The word itself does not appear in early Nahuatl dictionaries compiled by the Spanish priests, and earlier historians who reported on their visits to the region in the 18th and 19th centuries state that nepohualtzitzin is the Nahuatl word for a thing the Incas call a quipu or khipu. Alexander de Humboldt (pic 10), in his 1814 report on the region, wrote:-

Pic 9: Possible model of the ‘nepohualtzitzin’

Boturini goes on to state that in New Spain (Mexico), the use of knotted cords had been replaced entirely by pictorial representations, and the only surviving one he could locate was in the possession of an Indian (probably Tlaxcaltec) nobleman, and was so badly degraded from age that it was too fragile to handle; furthermore, the possessor himself could not do more than give Boturini a very general idea of their use. So, it seems that, at least in the colonial era, the term nepohualtzitzin referred to a version of the Incan quipu that for whatever reason was no longer in use in Mexico.
Regardless of whether it was via an abacus or something else, the Aztecs must have had some way of doing very sophisticated calculations, often involving rather large numbers. It would be unheard of for any society to carry on at the level of complexity that the Aztecs maintained over a large area without having ways of doing involved calculations. They kept records of land ownership, censuses of tributary states, and very detailed accounting of tribute owed and paid by all parts of the empire. None of this would be possible without a cadre of officials who were very skilled at computation and accounting.

Pic 11: A fraction of the Aztec agricultural system, including fields, waterways and roads - illustration by Alberto Beltrán

In addition to the tonalpohualli, the Aztecs used a second calendar, known as the xihuitl. This consisted of eighteen “months” of 20 days each, along with a set of five days at the end, called the nemontemi. This yields a xihuitl calendar of 365 days. The tonalpohualli and xihuitl calendars both return to the same day simultaneously every 18,980 days (for the math-inclined among us, this is because 18,980 is the least common multiple of 260 and 365), or 52 xihuitl years, at which point the entire calendrical cycle begins again. Within a 52-year cycle, any particular day can be specified by giving both its day in the tonalpohualli and in the xihuitl, for example (Two Xocotluetzi, One Snake), which specifies the date Two Xocotluetzi in the xihuitl calendar, and One Snake (or One Coatl) in the tonalpohualli. The completion of a 52-year cycle was a momentous event for the Aztecs, partly because it was literally a once-in-a-lifetime event for most Aztecs. This was also considered a perilous time because Aztec religious beliefs held that humans would be exterminated at some later time corresponding to the end of one of these cycles. It required elaborate rituals lasting several days to ensure that the gods would allow the next cycle to commence.

Pic 13: A stone Aztec sculpture of a ‘xiuhmolpilli’ (‘bundle of years’), the all-important 52-year cycle

Conclusion
Although much of the pre-Columbian writings of the Aztecs have been lost, by studying the surviving codices and inscriptions, as well as accounts from early Spanish sources, we have an overall picture of an Aztec civilization that was literate and sophisticated, and was able to carry out complex calculations for calendar-keeping, censuses, taxation, surveying, and tribute collection. Unfortunately there is little surviving physical evidence for how the Aztecs (or the Mayas for that matter) did their computing, and so until something more concrete comes along we are left mainly to conjecture. That should not prevent us from acknowledging that Aztec society and government depended on highly-developed computational ability, in whatever form it took on, and were able to develop ways to master the numeracy skills needed to manage an extensive governance system and trade network.

Pic 15: Aztec number symbols for 10, 20, 80, 400 and 8,000