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Study Guides > Mathematics for the Liberal Arts

The Mayan Numeral System

Learning Outcomes

  • Become familiar with the history of positional number systems
  • Identify bases that have been used in number systems historically
  • Convert numbers between bases
  • Use two different methods for converting numbers between bases

Background

As you might imagine, the development of a base system is an important step in making the counting process more efficient. Our own base-ten system probably arose from the fact that we have 10 fingers (including thumbs) on two hands. This is a natural development. However, other civilizations have had a variety of bases other than ten. For example, the Natives of Queensland used a base-two system, counting as follows: “one, two, two and one, two two’s, much.” Some Modern South American Tribes have a base-five system counting in this way: “one, two, three, four, hand, hand and one, hand and two,” and so on. The Babylonians used a base-sixty (sexigesimal) system. In this chapter, we wrap up with a specific example of a civilization that actually used a base system other than 10. Fig5_1_22The Mayan civilization is generally dated from 1500 BCE to 1700 CE. The Yucatan Peninsula (see figure 16[footnote]http://www.gorp.com/gorp/location/latamer/map_maya.htm[/footnote]) in Mexico was the scene for the development of one of the most advanced civilizations of the ancient world. The Mayans had a sophisticated ritual system that was overseen by a priestly class. This class of priests developed a philosophy with time as divine and eternal.[footnote]Bidwell, James; Mayan Arithmetic in Mathematics Teacher, Issue 74 (Nov., 1967), p. 762–68.[/footnote] The calendar, and calculations related to it, were thus very important to the ritual life of the priestly class, and hence the Mayan people. In fact, much of what we know about this culture comes from their calendar records and astronomy data. Another important source of information on the Mayans is the writings of Father Diego de Landa, who went to Mexico as a missionary in 1549. Fig5_1_23There were two numeral systems developed by the Mayans—one for the common people and one for the priests. Not only did these two systems use different symbols, they also used different base systems. For the priests, the number system was governed by ritual. The days of the year were thought to be gods, so the formal symbols for the days were decorated heads,[footnote]http://www.ukans.edu/~lctls/Mayan/numbers.html[/footnote] like the sample to the left[footnote]http://www.ukans.edu/~lctls/Mayan/numbers.html[/footnote] Since the basic calendar was based on 360 days, the priestly numeral system used a mixed base system employing multiples of 20 and 360. This makes for a confusing system, the details of which we will skip.
Powers Base-Ten Value Place Name
207 12,800,000,000 Hablat
206 64,000,000 Alau
205 3,200,000 Kinchil
204 160,000 Cabal
203 8,000 Pic
202 400 Bak
201 20 Kal
200 1 Hun

The Mayan Number System

Instead, we will focus on the numeration system of the “common” people, which used a more consistent base system. As we stated earlier, the Mayans used a base-20 system, called the “vigesimal” system. Like our system, it is positional, meaning that the position of a numeric symbol indicates its place value. In the following table you can see the place value in its vertical format.[footnote]Bidwell[/footnote] A chart showing the Mayan numeral system. 0 is represented by an oval with two lines in it. 1 is represented by a dot. 2 is two dots. 3 is three dots. 4 is four dots. 5 is a line. 6 is a dot above a line. 7 is two dots above a line. 8 is three dots above a line. 9 is four dots above a line. 10 is two lines. 11 is a dot above two lines. 12 is two dots above two lines. 13 is three dots above two lines. 14 is four dots above two lines. 15 is three lines. 16 is a dot above three lines. 17 is two dots above three lines. 18 is three dots above three lines. 19 is four dots above three lines. In order to write numbers down, there were only three symbols needed in this system. A horizontal bar represented the quantity 5, a dot represented the quantity 1, and a special symbol (thought to be a shell) represented zero. The Mayan system may have been the first to make use of zero as a placeholder/number. The first 20 numbers are shown in the table to the right.[footnote]http://www.vpds.wsu.edu/fair_95/gym/UM001.html[/footnote] Unlike our system, where the ones place starts on the right and then moves to the left, the Mayan systems places the ones on the bottom of a vertical orientation and moves up as the place value increases. When numbers are written in vertical form, there should never be more than four dots in a single place. When writing Mayan numbers, every group of five dots becomes one bar. Also, there should never be more than three bars in a single place…four bars would be converted to one dot in the next place up. It’s the same as 10 getting converted to a 1 in the next place up when we carry during addition.

Example

What is the value of this number, which is shown in vertical form? Fig5_1_25

Answer: Starting from the bottom, we have the ones place. There are two bars and three dots in this place. Since each bar is worth 5, we have 13 ones when we count the three dots in the ones place. Looking to the place value above it (the twenties places), we see there are three dots so we have three twenties. Fig5_1_26 Hence we can write this number in base-ten as: (3 × 201) + (13 × 200) = (3 × 201) + (13 × 1) = 60 + 13 = 73

 

Example

What is the value of the following Mayan number? A depiction of a Mayan number. The bottom has two lines and a dot above them, the middle has a circle with two diagonal lines, and the top has three lines and three dots above them.

Answer: This number has 11 in the ones place, zero in the 20s place, and 18 in the 20= 400s place. Hence, the value of this number in base-ten is: 18 × 400 + 0 × 20 + 11 × 1 = 7211.

Try It

Convert the Mayan number below to base 10. Fig5_1_28

Answer:

1562

Convert the base 10 number 561710 to Mayan numerals.

Answer: [latex]5617_{10} = 14,0,17_{20}[/latex]. Note that there is a zero in the 20’s place, so you’ll need to use the appropriate zero symbol in between the ones and 400’s places.

 
In the following video we present more examples of how to write numbers using Mayan numerals as well as converting numerals written in Mayan for into base 10 form. https://youtu.be/gPUOrcilVS0 The next video shows more examples of converting base 10 numbers into Mayan numerals. https://youtu.be/LrHNXoqQ_lI

Adding Mayan Numbers

When adding Mayan numbers together, we’ll adopt a scheme that the Mayans probably did not use but which will make life a little easier for us.

Example

Add, in Mayan, the numbers 37 and 29: Fig5_1_30[footnote]http://forum.swarthmore.edu/k12/mayan.math/mayan2.html[/footnote]

Answer: First draw a box around each of the vertical places. This will help keep the place values from being mixed up. Fig5_1_31 Next, put all of the symbols from both numbers into a single set of places (boxes), and to the right of this new number draw a set of empty boxes where you will place the final sum: Fig5_1_32 You are now ready to start carrying. Begin with the place that has the lowest value, just as you do with Arabic numbers. Start at the bottom place, where each dot is worth 1. There are six dots, but a maximum of four are allowed in any one place; once you get to five dots, you must convert to a bar. Since five dots make one bar, we draw a bar through five of the dots, leaving us with one dot which is under the four-dot limit. Put this dot into the bottom place of the empty set of boxes you just drew: Fig5_1_33 Now look at the bars in the bottom place. There are five, and the maximum number the place can hold is three. Four bars are equal to one dot in the next highest place. Whenever we have four bars in a single place we will automatically convert that to a dot in the next place up. We draw a circle around four of the bars and an arrow up to the dots' section of the higher place. At the end of that arrow, draw a new dot. That dot represents 20 just the same as the other dots in that place. Not counting the circled bars in the bottom place, there is one bar left. One bar is under the three-bar limit; put it under the dot in the set of empty places to the right. Fig5_1_34 Now there are only three dots in the next highest place, so draw them in the corresponding empty box. Fig5_1_35 We can see here that we have 3 twenties (60), and 6 ones, for a total of 66. We check and note that 37 + 29 = 66, so we have done this addition correctly. Is it easier to just do it in base-ten? Probably, but that’s only because it’s more familiar to you. Your task here is to try to learn a new base system and how addition can be done in slightly different ways than what you have seen in the past. Note, however, that the concept of carrying is still used, just as it is in our own addition algorithm.

 

Try It

Try adding 174 and 78 in Mayan by first converting to Mayan numbers and then working entirely within that system. Do not add in base-ten (decimal) until the very end when you check your work.

Answer: A sample solution is shown.

In the last video we show more examples of adding Mayan numerals. https://youtu.be/NpH5oMCrubM In this module, we have briefly sketched the development of numbers and our counting system, with the emphasis on the “brief” part. There are numerous sources of information and research that fill many volumes of books on this topic. Unfortunately, we cannot begin to come close to covering all of the information that is out there. We have only scratched the surface of the wealth of research and information that exists on the development of numbers and counting throughout human history. What is important to note is that the system that we use every day is a product of thousands of years of progress and development. It represents contributions by many civilizations and cultures. It does not come down to us from the sky, a gift from the gods. It is not the creation of a textbook publisher. It is indeed as human as we are, as is the rest of mathematics. Behind every symbol, formula and rule there is a human face to be found, or at least sought. Furthermore, we hope that you now have a basic appreciation for just how interesting and diverse number systems can get. Also, we’re pretty sure that you have also begun to recognize that we take our own number system for granted so much that when we try to adapt to other systems or bases, we find ourselves truly having to concentrate and think about what is going on.

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