How the Aztec Mathematical System Came to be and Contributed to us Today
by Destiny Harrison, Delaney Garcia, Jaysiya Norman, Jewel Samson, Raquel Cruz, Katelyn Woodley, Kalyna Mai and Olivia Nixon
For the competition, we were tasked with studying Aztec mathematics. Aztec mathematics was one of the most complicated mathematical writings of any of the pre-Columbian people. Aztecs used hand, heart and arrow symbols to represent their numbers. The researchers put immense effort into studying Aztec agricultural manuscripts trying to understand how the genius-seeming people arrived at area calculations. [1] Only when they factored in the important and well-used glyphs did the figures make sense. The term Aztec comes from the ethnic groups that were both politically and socially dominant in Mexico during the 1330s through the early 1500s.
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Geographer Barbara Williams of the University of Wisconsin and math genius Maria del Carmen Jorge Y Jorge of the Universidad Nacional Autonoma de Mexico studied long and hard over two documents that described the farming properties owned and controlled by citizens in the city of Tepetlaoxtoc for about four years in the 1540s[2]. There are only a few records written in the pre-Mexico period that includes perimeter and area data; shown by Dr. Jorge’s studies. Most of the documents from this time were lost. The researchers tried to copy the area predictions and originally had issues. When the Aztecs realized that the arrow, hand, and head drawings showed ground distances they were finally were able to come up with some predictions. [3]
The system they use to record the areas is confusing, even for modern mathematical geniuses, according to what Dr. Jorge told LiveScience during an interview. She was very excited to see that actually using the symbols made it easier to discover the areas.
Each symbol stood for a distance that was less than the standard area unit called a land rod. These arrow, heart and hand symbols were similar to what we now call fractions, according to the studies Dr. Jorge conveyed. [4]
They called them units of measure, smaller than the length unit. [5]The Aztecs had their own forms of calculation. The Aztecs used a base-20 number system, and labeled the ones with dots and 20s with bars. Aztec math has more numbers than we do now or at least symbols labelling numerical theories. The land holding documents were written for the use of tax. “The type of the mathematics the Aztec used to calculate land holdings was made to be constant with the calendar mathematics which are well known for. The ability to make calculations while using proportions was spread across cultures at the time” (Moskowitz[6]). Usually a finger was used to show the number. Going by the dots, one would be represented as one dot;
1 = á§
And so forth with the number system;
2 = á§á§
3 = á§á§á§
The Aztecs were dominant citizens in Mexico for several hundred years. The Aztecs used a number system that had been around for a long time. It is a vigesimal system as the apposed to our decimal system. The Aztecs used a dot number system. They left behind their comprehensive math writings, and fascinating scholars until this day. The Aztecs had a intelligent idea of what zero was and there is no proof that they had a symbol representing it. The lines and dots look very simple but was actually very complicated. The Aztecs were one of few groups to completely measure out many things in life. The Codex Vergara, from 1540, has representable drawings and altitude of individual fields. Previous study has showed the fact that they had knowledge of multiplication and division and they even had some information about geometry.
The mathematics that the Aztecs used was highly based upon earlier versions of Egyptian mathematics; mirroring their use of images as numbers. The images had no order and could be written in any way to still represent the same singular number, just as the Egyptian symbols.
The Aztecs used body parts to measure. For example, if you stretched out your left the measure from your heart to the tip of your finger would be one singular measure (much like our current foot, inch, and mile). The system is puzzling, used to record area and perimeter during architecture and other building methods. Still, the measurement system was complex for its time period.
Aztecs recorded only the total area. Dr. Williams has analyzed the Codex Vergara (which is what they called the Aztec math). Aztecs used several types of algorithms to calculate areas. Some involved simply multiplying length by width.
The Aztecs came up with many different ideas such as multiplying the average of two different sides by an adjacent side. When a measurement did not match an exact number of land rods they used their standard unit of normal measurement, which went to about 2.5.
The Aztecs also used mathematics for astronomy. Although most of their information in math astronomy was lost there is a pretty good synthesis of the remaining information in the book “Skywatchers” by Anthony Aveni. For example the Aztecs used their system to find out that the cycle of Venus is 584 days. The Aztecs even did the math to workout out the eclipse season although they didn’t know the shape of the earth or the size.They had found out when the eclipse would happen but they were not total for sure the eclipse would be the total eclipse.
They would use the number system to keep track of natural disasters As of what we know the Aztecs are the only early Americans to leave technical documents. Aztecs like to get their measurements right, definitely when it came to taxation. The Aztecs documents were very organized and detailed.
The Codex Vergara was created in about 1540, and it involved schematic drawings and the exact measurements of individual fields. It has been revealed that an understanding of multiplication and division,principles of geometry was needed.
Although the Aztecs are only early Americans that have these kinds of documents, it’s reasonable to consider that other groups such as the complex Mayans used similar systems. There is an observation that the Aztecs were living off their religion and that their science and facts were all guided by it. Studies have shown that the Aztecs probably liked to get their measurements right and definitely when it comes to tax. Previous studies have shown that they had the ability of doing multiplication, division and even had some laws of geometry.
Aztec math is very valuable because it was one of the first mathematical systems that was made. It was also an ancient tax time. Aztec math is very intriguing; it’s different unlike a lot of math people do today. The Aztec mathematics helped make the calendar and help make a writing system that uses hieroglyphics.
The Aztec civilization blossomed after the come down of the Toltecs around the 11th century AD. By the year 1325 they had found the city of Tenochtitlan in the valley of Mexico and became one of the strongest military groups. The Aztec empire as it was known, was very powerful, lead several other groups, grand taxes, taking sacrifice prisoners and gripping their achievements. A diamond represents 10, a flag represents 20, feathers represent the hundreds, a bag of cacao beans represents 8,000.
The Aztec empire was a collection of small states that were controlled by chiefs. Their alliance with the neighboring city states of Tetzcoco and Tlacopan made them so powerful that they came to control most of what is now called Mexico. The Aztec king claimed to be born from the line of the gods and ruled through a council of nobles and officials. Noble males served in the military, in government, and in priesthood. The lower classes worked as slaves and commoners; this category was most of the Aztec people.
The Aztec recorded the area of shapes once they found the area. Aztec people were very intelligent, they worked hard to get their math where it is today. Astronomy is the intelligence of planets and stars. The Aztecs developed an elaborate calendar that reflected the cosmic cycles of their religion, called the Sun Calendar.
The Aztecs were very smart and intelligent thinkers as well. The Aztec education was valued in their culture. At this time the math they had came up with was more advanced than the other systems. The Aztecs were not just good at math but was also good at science; that helped them with making some of their choices. Their math relied on their science sometimes.
Today we use math for so many different reasons like to buy what we want or need but one of the reasons why the Aztecs made their math system was so when they built anything they would have the measurements right and also for taxes.
The Aztec Empire was known to be very powerful and dominated several other cultures , applying taxes, taking sacrifice prisoners, and taking in their advancements. This is how they were influenced by the Mayans, not directly but by the Toltecs who had gotten that knowledge years before, in comparison to their math and their calendars.
Even though there is no precise Aztec math, it is believed that the Aztecs were taught a Mayan-based math, using the same view and symbols, so it turned out the same way. It was also used in the marketplace for the trading of goods, but more significantly it was implemented by astronomers in the important search of the skies and space, through the calendars.
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The Aztecs were very religious and led themselves through offerings, human and animal sacrifices to the gods; the Aztec stone was a sacrificial stone made sometime in the 1470’s in the temple of the war god Huitzilopochtli, but the stone also represented the Aztec cosmos. There were people charged with the responsibility of studying the cosmos, like the priests who were the astronomers as well. It was obvious that these people needed to posses a strong mathematical knowledge in order to take control and study the skies, and interpret the calendar, so we can see that math was an integral part of the future priests’ education.
The calendars created by the Aztecs were made up of a combination of the earlier calendars and Aztec’s general religious beliefs. The Aztecs believed that each and every era of the world was labelled by the end of the 52nd year or life cycle, and that post that either a new era was about to begin or the end of the world was drawing nearer.
Every era was marked by a unique process, like Ãthe Era of the Earth Sun, the Age of Great Winds, the Age of Fire, the Age of Floods, and the present one the Age of Earthquakes. These were the five Suns, all of which are shown in the Aztec calendar: the current one in the center and all other four, with their date of ending, surrounding the fifth sun, which is symbolized as an Aztec god (one of the Aztec gods), probably the sun god. The stone didn’t commemorate the start of the fifth sun but the destruction of it by earthquakes. The Aztecs believed that in order to keep the gods happy they had to supply them with fresh human hearts, to keep on living.
With the Aztec calendars, the Aztecs were able to track important dates with their calendars, as well as use them to study and investigate space and the Earth. They used the calendars to track when they could study the constellations and stars best (see them best). They were good at their knowledge in astronomy and mathematics they used to help them; their system could be compared to the Egyptian system (as mentioned), but also the Hindus and Babylonians.
Our opinion on the Aztec culture is immense; we think very highly of them and their mathematical skills. We think that the mathematics is helpful to our modern world today. They contributed to our society by helping develop our own, modern mathematics. Without the Aztec’s mathematics, we may never have had developed our own. We might not have started using decimals, zeros as placeholders, time, money, or even numbers in general!
We think that Aztec mathematics is very helpful to us. We use Aztec mathematics today in school, teaching, and our jobs. Our future careers may depend on mathematics that directly root from the Aztec developments.
Delaney thinks it is interesting to her because the developments of Aztec mathematics helps you with math equations today. On the contrary, Jaysiya thinks it is very useful because it helps us in the 21st century (career, historian, and etc wise). Raquel thinks it is entertaining because it started long ago and it helps us even modernly. Olivia thinks that it is very creative because of the many different symbols used in the system. Kalyna thinks that the Aztec mathematics were a great development in mathematics for the time period it was used in because of the complex equations and calculations. Katelyn thinks that Aztec math is very interesting to learn about, she would love to learn more about it. Destiny says that she thinks it is an interesting topic to learn and that she thinks that it could have been a difficult system to come up with.
Overall, our conclusion to this essay follows;
- How did Aztec mathematics affect us today? Without the contributions of the knowledge passed on through Aztec culture, we might never have developed our modern mathematics that led to great inventions such as the car, keys, TVs, the internet and computers, and so many more commonly used household items that we depend on in the 21st century.
- How did the Aztec mathematics develop? The Aztecs used a complex complication of developing forward previous systems; the Egyptian hieroglyphs, the Mayan dot-and-bar-system, and so forth. But they developed so much on their own, including but not limited to calendars, decimals, and even the late development of zeros as place holders.
Works Cited
Andrei, Mihai. “Science ABCs – How Aztecs did the Math.” http://www.zmescience.com, 9 Apr. 2008, www.zmescience.com/other/science-abc/science-abc-how-aztecs-did-the-math/
Holden, Constance. “How Aztecs did the Math.” http://www.sciencemag.org, 3 Apr. 2008
Leon, Araceli. “Math Use During Maya and Aztec Civilizations.” http://www.hermetic.ch, www.hermetic.ch/cal_stud/maya/mathuse.htm
Moskowitz, Clara. “Amazing Aztecs were Math Whizzes Too.” www.livescience.com, 2 Apr. 2008, www.livescience.com/2427-amazing-aztecs-math-whizzes.html
Siddiqui, Ajaz. “Aztec Number System.” www.math.temple.edu, 8 Feb. 2004, www.math.temple.edu/~zit/Native%20American/9%20Aztecs_num.pdf
[1]Sources? I know it’s tedious, but it’s worth it.
[2]Makes it sound like Maria and Barbara worked on the documents in 1540. I know what you meant, but you might want to clarify. “they studied two documents from the 1540’s for 4 years.” or something along those lines.
[3]cite source please.
[4]Not sure conveyed is the right term for this sentence.
[5]They called what units of measure? A bit unclear. You guys are doing well. I see the hard work. You got this. Keep it up.
[6]In text citations are (last name, page #).
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