All of the mathematics of the Americas is very interesting, but in particular, Mayan mathematics is unique and very advanced given the lack of communication with European and Asian nations. Ronald Calinger claims that "the classic Maya had the most sophisticated number system in pre-Columbian America" (Calinger, 232). Much like ancient European civilizations, mathematics is seen in many areas of Mayan life. The Mayan people integrated mathematics into every part of their lives, from their very advanced calendar system to their architecture to their astrological tables to their religion. Based on twenty, the numerals in their vigesimal system (in contrast to our decimal system) had many representations. The Mayan people used zero as a placeholder in their numerals, the bar and dot representations specifically. The Mayan number system is quite simple to understand; yet these native peoples used it to calculate very complex things.

Mayan civilization extended from "southern Mexico (chiefly the Yucatan peninsula) to Belize, El Salvador, Guatemala, and Northern Honduras" (Calinger, 230). Although Mayan culture has lasted throughout the ages, the society flourished from "the late third century to the early ninth century of our Christian era," which is called the Maya classic age (Calinger, 230). It is important to note that unlike the Aztec civilization that simply disappeared, Mayan culture and civilization is very strong in today’s society in Central America. The Mayan culture was very apparent when I visited the Yucatan Peninsula last summer.

The Mayan numeral system is unique in that it is a vigesimal system, which means it is based on twenty instead of ten like our decimal system. There is speculation on why the Mayas chose a vigesimal system. Two possible reasons might be it fit their calendar nicely, or quite possibly, humans have 10 fingers and 10 toes, which add to 20 appendages to count upon. Proof of the second idea can be seen through one of the words for 20, which is uinic, which means man or human being. Michael Closs states that this may be in reference to "the totality of his digits" (Closs, 293). The Maya had names for the numbers 1-20 as seen below.

It is interesting to note that numbers 13-19 show that the Mayan people must have had some since of a decimal system because 13-19 are stated as threeten, fourten, fiveten, etc, which is very close to our thirteen, fourteen, fifteen, etc. Interestingly, 11 and 12 are not written in this manner, much like our decimal number names of eleven and twelve.

The simplest way the Mayans represented their numbers was with a bar and dot representation. Howard Eves compares the bars and dots to sticks and pebbles (Eves, 16). In this system, each dot represented one unit and each bar represented five units. This system assigns bar and dot representations for numerals 0 through 19. For example, one dot and two bars might represent eleven days if the day is the unit being counted. These numbers can be written either with the bars on the bottom and the dots above, which is done when doing arithmetic, or with the bars vertically and the dots to the left. Figure 1 shows how the numbers 0-19 can

Figure 1, Calinger 236

be represented. It is important to remember that just as the English language has spoken dialects, Mayan writings have dialects. Instead of having the vertical lines inside the ellipse in the number zero, some dialects have the ellipse empty. Another representation of zero is a half-open eye. Larger numbers in the bar and dot representation are written fairly easily. In some instances, c shaped or crescent shaped figures appear in the number where dots should be in relation to a bar as seen in Figure 1 for the number eleven. These represent zero so there is no confusion to what the number is. This representation of zero first appears in the third century (Calinger, 231). The Maya have a place value system much like our decimal system, but instead of reading the numbers left to right with the left being the largest place value spot, Mayan numbers are read from top to bottom with the top place being the largest place value spot. Four examples of numbers follow in Figure 2. Here is an explanation for the third example. You have one dot in the 8,000 space, one bar in space 400, two dots and a bar in space 20, and one bar in the 1’s space. One dot in the 8,000’s space equals 8,000 plus one bar in the 400’s space equals 5x400=2,000 plus two dots and a bar in the 20’s space equals 7x20=140 plus one bar in the 1’s space equals 5. The number represented is the sum of all the previous numbers (8000+2000+140+5) and is 10,145, as noted at the bottom of the figure. Also, the representation of zero is sometimes placed in the column as a placeholder where there is no number. For example, the representation of zero would sometimes be placed in the 1’s place, as in the first example of Figure 2, to represent 20.

Figure 2, Calinger 237

Another representation that the Mayans had for numbers was head variants. Much like the written numbers, the head variants have distinct heads for the numbers 1-12, but then combinations of ten and a corresponding number are used for 13-19. All of the head variants including the combinations for 14-19 are shown in Figure 3. The head variant for 13 in this figure is a reptile monster, which is different from both the 3 and the 10 (Closs, 337). This head variant was sometimes used instead of the combinations and is yet another example of Mayan dialect. Each head, 1-13 is an actual picture of something in the Mayan culture, and most of these head variants are pictures of their gods. For example, the first is a "young earth goddess," the third is the "God of wind and rain," the fourth is the "Sun god," and the eleventh is the "Earth God" (Closs, 335-336). Not all of the head variants have been positively identified. Researchers think they know which gods the numbers 2,6, and 9 stand for, but they are not positive about their findings (Closs, 335-336). Two important and interesting head variants are 10 and 12. Ten is the "Death god," while twelve has not been identified yet (Closs, 336). The jaw from the "Death god," 10, is used in combination with the entire head of the "Sun god," 4, to create the head variant for 14. The head variants for 15-19 are similar. Unlike the written words for the Mayan numbers, there is a head variant for 0. This head variant is different from all the others because it bares a hand across its jaw.

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Figure 3, Closs 335

The most impressive thing that results from the Mayan numerical system is their system of calendars. The Mayan culture followed two different calendars, the Sacred Round and the Vague Year, that combined into one calendar, the Calendar Round. The Sacred Round is made up of 260 days divided into 13 weeks of 20 days each (Calinger, 234). Another explanation of the Sacred Round is 13 day numbers and 20 day names (Calinger, 238). The numbers and days of this calendar are matched up 1-1 through the 13^th day. When the 14^th day occurs, the numbers start over, while the day names continue. So, if the 13^th name is Ben and the 14^th name is Ix, then the dates would be expressed 13 Ben and 1 Ix. When the names run out, they start over with the next number in the sequence. If Ahau is the 20^th name and 7 Ahau is a date, then 8 Imix is the next date because 8 is the next number in the sequence and Imix is the first name in the cycle (Closs, 295). This continues through 13 cycles of the 20 day cycle, which results in 260 days. The second calendar or the Vague Year is comprised of 365 days. This calendar has "18 named months of 20 days each and a residual period of 5 days" (Closs, 295). This calendar works much like our present day calendar. Each month has exactly the same amount of days except one. The18 months and the five day residual period have Mayan names, while the days are just represented by numbers. Very interestingly, sometimes the last day of each month is represented by the glyph signifying end, while other times, the last day of each month is signified by the "seating" or "installation" glyph representing the zeroeth day of the next month (Closs, 298).

The two calendars combine to create the Calendar Round. It is called the Calendar Round because if one imagines each of the calendars on wheels or turning gears, the current date is the represented by the combination of both calendars where the wheels or turning gears meet. In other words, each date is represented by both its date on the Sacred Round and its corresponding Vague Year date. Closs points out the since "the lowest common multiple of 260 [number of days in the Sacred Calendar] and 365 [number of days in the Vague Year] is 18,980, the Sacred Round and the Vague Year combine to form 18,980 (=73x260=52x365) Calendar Round dates. In the Calendar Round dates, the Sacred Round date is portrayed first, then the corresponding Vague Year date. Notice the bar and dot numbers to the left of each of the glyphs in Figure 4 (below). When using the Calendar Round dates, the vigesimal system of the Maya was modified to fit the calendar. Instead of the traditional 1, 20, 400 sequence, the sequence took a new form 1, 20, 360. This is because of the 18 month calendar (18x20=360).

The Mayans used their calendars for many things, but the most impressive would have to be their astronomical tables. These tables found in the Dresden Codex predict numerous astronomical events. There is an eclipse calendar, which specialists think predict both the lunar and solar eclipses (Makemson, 187). This table uses a lunar calendar along with the Sacred Round to predict these eclipses. It should be noted that these eclipses could occur anywhere on the earth and are not limited to only visible eclipses for the Mayan people (Grattan-Guiness, 146). Also impressive is the table in the Dresden Codex that shows the motion of both Venus and Mars (Makemson, 209). In particular, this table told when Venus would appear as both the morning star and the evening star (Grattan-Guinness, 146). The Dresden Codex tells of the movement of Venus with relation to the Calendar Round.

Figure 4, Closs 302

Mayan mathematics was fully integrated in their culture. This was especially evident when reviewing the glyphs they chose to represent numbers. As stated before, all of the head variant number representations except 13 were also representations of gods. Also, by using numbers in the "names of many gods and goddesses," the Mayans further integrated mathematics and religion (Closs, 301). Most of these names are very straight forward such as "Bolon Yocte," which stands for Nine Strides, or "Lahun Chan," which translated is Ten Sky. Each of these is a known deity. Oddly enough, some god glyphs are unknown, so one of the only things known about them is the number associated with them. Two of these gods along with the first two can be seen in Figure 4. It is known that God Q is associated with death, and God R is associated with benevolence, but besides that and their numbers, nothing else is known. The last two figures in Figure 4 are the Moan Bird, which is thought to represent "the 13 heavens of Mayan cosmology," hence the 13 in front of both symbols (Closs, 301). One last mix of mathematics and culture is the drawing of Pauahtun, the Mayan patron god of all scribes, teaching mathematics on a vessel (Grattan-Guinness, 144).

Mayan mathematics is very interesting and unique. It is very evident that the Mayan culture embraced mathematics in everything that they did. Their unique vigesimal system allowed them to use their calendar system to predict many astronomical things, including eclipses and the movement of Venus and Mars. Yet, the system was simple enough for the common people of the society to use it for "commercial records, taxes or levies of tribute, and census" (Calinger, 231). The use of numbers in the naming of their gods and the use of gods in the glyphs for their numbers shows the integration of their religion and mathematics. Finally, their use of zero shows an understanding of something that was beyond many of their contemporaries. Clearly, mathematics was very important to the Mayan people.