The Mexica had a sophisticated mathematical system that combined elements of a vigesimal, or base-20, system with pictographic representations. The system used a series of signs for the numbers 1 to 19, and it combined these signs to represent larger numbers. The Mexica used a flag-like symbol for 20, a feather-like symbol for 400, and a bag-like symbol for 8000, and they could represent very large numbers with these signs. The system was used for the calculation of tribute, for the regulation of the calendar, and for the recording of historical information. The Mexica were skilled mathematicians, and they used their system to make sophisticated calculations in astronomy, in land surveying, and in the regulation of the markets.
The Vigesimal Number System
The Mexica number system was based on a vigesimal, or base-20, system, which means that it counted in groups of twenty rather than in groups of ten. The vigesimal system was common in Mesoamerica, and it may have been related to that the Mexica used their fingers and toes to count. The Mexica counted from 1 to 19 using a series of pictographic signs, and they reached 20 by using a special sign.
The Mexica had a series of signs for the numbers 1 to 19. The sign for 1 was a dot or a small circle. The sign for 2 was two dots, the sign for 3 was three dots, and so on up to 19, which was represented by 19 dots. The Mexica also had a sign for 20, which was a flag-like symbol called pantli. The sign for 20 was used as a base for the higher numbers, and the Mexica would write a number of dots above the flag to indicate multiples of 20. For example, the sign for 40 would be two flags, and the sign for 60 would be three flags.
The Mexica also had signs for 400 and 8000, which were the next two orders of magnitude in the vigesimal system. The sign for 400 was a feather-like symbol called tzontli, and the sign for 8000 was a bag-like symbol called xiquipilli. The Mexica could combine these signs to represent very large numbers, and they used them to record the tribute owed by the subject provinces and the populations of the major cities.
The Pictographic Signs
The Mexica mathematical system used pictographic signs to represent the numbers, and the signs were combined in a variety of ways to express complex numerical information. The most common form of the signs was the dot, which was used to represent the numbers 1 to 19. The dots were typically arranged in a row, and the number was read from left to right. The Mexica also used a variety of other signs, including the flag for 20, the feather for 400, and the bag for 8000.
The Mexica signs were combined in a number of ways to represent complex numbers. The simplest combination was the addition of dots, which was used to represent the numbers 1 to 19. The next combination was the addition of flags, which was used to represent multiples of 20. The Mexica could also combine the flag, the feather, and the bag to represent very large numbers. For example, the number 8420 would be represented as one bag, one feather, one flag, and one dot (8000 + 400 + 20 + 1 = 8421, with some adjustments for the conventions of the system).
The Mexica had a sophisticated understanding of the use of zero, and they represented zero with a special sign — most commonly a stylized shell glyph, although flowers and other forms were also used. The use of zero allowed the Mexica to perform complex calculations, and it was central to their mathematical system. The Mexica use of zero, like that of the Maya, was one of the independent inventions of the concept in world history.
The Calculation of Tribute
The Mexica mathematical system was used extensively in the calculation of tribute, and the tribute lists, recorded in the Codex Mendoza and other sources, are among the most sophisticated examples of the use of the system. The tribute lists typically used a grid format, with the provinces listed in the rows and the goods listed in the columns. The amounts owed by each province were recorded using the Mexica numerical signs, and the lists were used by the calpixqui to verify the collections.
The Mexica had a sophisticated system of accounting, and they used their mathematical system to make complex calculations. The Mexica could calculate the total tribute owed by all the provinces, the share of the tribute that was due to each of the three members of the Triple Alliance, and the share of the tribute that was due to the various officials and the priesthood. The Mexica could also calculate the value of the tribute in standardized units, and they used this information to plan the activities of the state.
The Calendar and Astronomy
The Mexica mathematical system was also used in the calculation of the calendar, and the calendar calculations required sophisticated mathematical skills. The Mexica had to calculate the position of the days in the Tonalpohualli and the Xiuhpohualli, the day of the week, the month, the year, and the calendar round. The Mexica also had to calculate the position of the sun, the moon, and the planets, and they used their mathematical system to predict eclipses and other astronomical events.
The Mexica had a special interest in the planet Venus, and they kept careful records of its appearances and disappearances. The Mexica calculated the synodic period of Venus, the time it takes for the planet to return to the same position in the sky, and they used this information to determine the timing of the most important rituals. The Mexica calculated the synodic period of Venus to be 584 days, which is very close to the modern value of 583.92 days. This is a remarkable achievement, and it testifies to the sophistication of Mexica astronomy.
The Markets and Trade
The Mexica mathematical system was also used in the regulation of the markets and the conduct of trade. The Tlatelolco market was one of the largest in the pre-Columbian Americas, and it required a sophisticated system of weights, measures, and money. The Mexica used cacao beans as a form of money, and they developed a system of standard weights for the measurement of goods. The Mexica also used cotton cloaks as a higher denomination of money, and they used copper axe-blades for the largest transactions.
The Mexica market officials were responsible for the verification of the weights and measures, and they used the mathematical system to detect fraud. The market officials had the power to fine, imprison, and even execute those who violated the market regulations, and they shaped the Mexica administrative system. The Mexica were also skilled surveyors, and they used their mathematical knowledge to measure and allocate land, particularly the chinampa plots around Tenochtitlan.
The Survival of Mexica Mathematics
Many elements of the Mexica mathematical system survived the Spanish conquest and continued to influence Mexican culture. The use of the vigesimal system, the pictographic signs for numbers, and the calculation of the calendar are all elements of the Mexica tradition that have survived in modified form in Mexican culture. The modern Mexican counting system, which uses vigesimal elements in some of its features, reflects the long history of Mexica mathematical practice.
The Spanish conquest brought the European decimal system and the Arabic numerals to Mexico, and these gradually replaced the Mexica system. Yet the Mexica system continued to be used in many indigenous communities, particularly in the calculation of the traditional calendar and in the conduct of traditional trade. The persistence of the Mexica mathematical tradition in the face of intense Spanish pressure ranks among the great survivals of Mexican culture.
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See also:
- What numbers and counting did the Aztecs use? — the basic signs
- How did the Aztecs use the vigesimal system? — the base-20 system
- What mathematical knowledge did the Aztecs have? — the broader knowledge
- The Tribute System and Imperial Wealth — the use in tribute
- The Aztec Calendar and Timekeeping — the use in astronomy
- Aztec Art, Architecture, and Science — the broader context