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Caitlin Westjohn Professor Ji MATH 4388 Caitlin Westjohn Professor Ji MATH 4388 - History of Mathematics Mayan Numerals The Mayan peoples were one of the most advanced and culturally rich civilizations in Mesoamerica. They flourished in middle America, primarily in Southeastern Mexico, where they built sprawling city-states and mastered agriculture. The Mayans are known by many for their impressive architectural accomplishments and beautiful art, but they also excelled at mathematics. Their numeral system was the most advanced and long lasting in the pre-Columbian Americas, and one of the first to include a symbol for zero. The Mayans also developed a complex calendar system in close association to their numeral system, to the point The Mayan numeral system, in contrast to our current Hindu-Arabic numeral system, was vigesimal, or base 20. It is widely speculated that the reason for the vigesimal system is that the ancient Maya people had counted on both their fingers and their toes, as opposed to just their fingers.1 The Maya, as mentioned previously, are credited with one of the first explicit uses of zero as a placeholder, the Babylonians being the only civilization to potentially predate them.2 The Maya numeral system was composed of three symbols: a shell like symbol represented 0, a dot represented 1, and a horizontal bar represented 5. Numbers larger than 5 were denoted with the appropriate number of dots above the bar, and more bars were added needed. This was the structure used for number 1 through 19, as shown in Figure 1. 1 O'Connor, J. J., & Robertson, E. F. (2000, November). Mayan mathematics. Retrieved from ​ http://www-history.mcs.st-and.ac.uk/HistTopics/Mayan_mathematics.html 2Sharer, R. J., & Traxler, L. P. (2009). The ancient Maya. Stanford (Calif.): Stanford University Press. ​ ​ Figure 13 For numbers greater than 19, the basic 1-19 system was positioned in vertical levels according to powers of 20. Figure 2 shows a selection from the Dresden Codex, a Mayan codex detailing Maya history and some of their astronomical discoveries. The top left column, ending where there is a non-numeral glyph, can be converted to our decimal system as follows: Figure 24 3 • 200 + 0 • 20 1 + 6 • 20 2 + 3 • 20 3 + 2 • 20 4 = 3 + 0 + 2, 400 + 24, 000 + 320, 000 = 346, 403 . Modern researchers notate Maya numerals from left to right, where the most significant digit is leftmost and each place value is separated by a period.5 Then, our chosen selection would be 3Davies, D. (2019, May 15). Maya Number System and Mathematics - KS2. Retrieved from https://mayaarchaeologist.co.uk/2016/12/28/maya-numbers/ 4Ibid. 5 Freitas, P. J., & Shell-Gelasch, A. (n.d.). When a Number System Loses Uniqueness: The Case of the Maya - The ​ Mayan Number System. Retrieved from https://www.maa.org/press/periodicals/convergence/when-a-number-system-loses-uniqueness-the-case-of-the-maya written 2.3.6.0.3. In actuality, the numerals on this selection likely refer to days in the Long Count calendar, which is converted to decimal in a slightly different manner. The Long Count calendar is one of three calendars that the Maya regularly used. While it more or less is consistent with the Mayan numeral system described above, it differs in one 2 major way: rather than 20 ,​ they used 18•20 as the multiplier. As a result, the third place in Long ​ Calendar dates denoted the number of groups of 360s, as opposed to groups of 400s. This was because the Long Count calendar, which some researchers think actually predated the Maya6, utilized years of 360 days. The inscription above, then, more likely depicts the number 2.3.6.0.3 = 3 • 200 + 0 • 20 1 + 6 • 18 • 20 + 3 • 18 • 20 2 + 2 • 18 • 20 3 = 311, 763 , and refers to a date. The Long Calendar denotes dates by how many days have passed since a certain mythical creation day in Mayan culture. What day this is exactly is up to debate by researchers, though August 12, 3114 BCE is most widely agreed upon7. The numeral system is also easily seen reflected in the two Mayan calendars the Tzolkin ​ and the Haab. The Tzolkin was a ritual calendar with 13 months of 20 days, for a total of 260 ​ ​ ​ ​ days; the Haab was a civil calendar with 18 of 20 days and a 5 day “week” of bad luck called the ​ ​ Wayeb, for a total of 365 days.8 Both systems marked individual days within months from 0 to ​ 19, rather than the modern standard of starting at 1. This again highlights the prevalence of 0 in Mayan society. It simple to see how these calendar systems reflect their numeral system, given the recurrence of 20. 6Sharer, R. J., & Traxler, L. P. (2009). The ancient Maya. Stanford (Calif.): Stanford University Press. ​ ​ 7 Ibid. 8 O'Connor, J. J., & Robertson, E. F. (2000, November). Mayan mathematics. Retrieved from ​ http://www-history.mcs.st-and.ac.uk/HistTopics/Mayan_mathematics.html From what is known, there does not seem to be any indication that the Maya numeral system had notation for fractional numbers, or that they used their numeral system for multiplication or division operations.9 Despite this, the Maya were still able to make startlingly accurate astronomical and architectural calculations, so it does not appear to have held them back in any capacity. Their numeral system did lend itself to easy addition and subtraction; the simple pictographic nature of their numerals made it more intuitive than some other contemporary systems. Examples of addition and subtraction are shown in figure 3 and figure 4, respectively. Figure 310 Figure 411 The Mayan numeral system survived for centuries until the Spanish conquest of Central America eradicated it. Thankfully, enough of their monuments and codices survived that researchers today were able to discover the true extent of Maya’s mathematical accomplishments 9 O'Connor, J. J., & Robertson, E. F. (2000, November). Mayan mathematics. Retrieved from ​ ​ http://www-history.mcs.st-and.ac.uk/HistTopics/Mayan_mathematics.html 10 Davies, D. (2019, May 15). Maya Number System and Mathematics - KS2. Retrieved from ​ https://mayaarchaeologist.co.uk/2016/12/28/maya-numbers/ 11 Ibid. ​ and decipher many of their writings. Though the Maya are most notable for their astronomical accomplishments, they would not have been possible without the ingenuity of their numeral system. References Davies, D. (2019, May 15). Maya Number System and Mathematics - KS2. Retrieved from https://mayaarchaeologist.co.uk/2016/12/28/maya-numbers/ Freitas, P. J., & Shell-Gelasch, A. (n.d.). When a Number System Loses Uniqueness: The Case of the Maya - The Mayan Number System. Retrieved from https://www.maa.org/press/periodicals/convergence/when-a-number-system-loses-uniquen ess-the-case-of-the-maya-the-mayan-number-system O'Connor, J. J., & Robertson, E. F. (2000, November). Mayan mathematics. Retrieved from http://www-history.mcs.st-and.ac.uk/HistTopics/Mayan_mathematics.html Sharer, R. J., & Traxler, L. P. (2009). The ancient Maya. Stanford (Calif.): Stanford ​ ​ University Press. .
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