Ancient Maya Arithmetic

maya arithmetic

The ancient Maya used a mathematical system that is “vigesimal.”

A vigesimal counting system is based on 20 units (0 – 19), instead of the 10 unit (0 – 9) based counting system that we use today called the decimal system.

The decimal mathematical system widely used today is believed to have possibly originated by counting the number of fingers that the average person has. Counting with our fingers gives us our ten unit based metric system. It is believed that the Maya possibly began counting with both their fingers and toes, which gave them their twenty based ‘vigesimal’ system. Their counting system is based on groups of twenty, instead of our modern ten.

When we count using our decimal system, we count to ten and then we add one value to the next tier, or level with a value of ten (10, 20, 30, etc.) for each cycle we reach on tier one (0-9). Using the decimal system, the count of 11 is the value of ‘1 unit’ from “tier 2” and ‘1 unit’ from “tier 1,” making the total value equal to 11 ( 10 + 1 = 11).

Each tier in the decimal system counts up in multiples of ten. 1 x 10=100 x 10=1,000 x 10=10,000 and so on.

maya arithmetic
Decimal tiers. 1 value from tier 2 is 10 and 1 value from tier 1 is 1, One value each from tier 1 and 2 equals 11 (10 + 1 = 11 ).

Although we users of the modern decimal system are not used to thinking of our counting in tiers or levels, it does help us understand how the count using the Maya vigesimal system. Just as the decimal system goes by tiers of ten: 1, 10, 100, 1000, 10000, etc., the Maya vigesimal system goes by tiers of twenty: 1, 20, 400, 8000, 160000, etc..

Maya Math
Comparison of decimal and vigesimal tiers.

The modern decimal system has ten possible digits for each placeholder in a tier, numbering from 0 to 9. In the Maya vigesimal system, the first tier’s placeholders has twenty possible digits, numbering from 0 to 19.

Linear comparison of first decimal and vigesimal tiers.

When the last number of a tier is reached, the count in each system proceeds up to the next tier. Adding a value of 10 in the decimal system or 20 in Maya vigesimal system.

Linear comparison of second decimal and vigesimal tiers.

For the number “21,” both systems used the twenty from tier 2 and the one from tier 1 to make: 20 + 1 = 21. Except with the Maya vigesimal system, they used 1 value on tier 1 and 1 value on tier 2. The decimal system uses 1 value on tier 1 and 2 values on tier 2.

To make “11,” the decimal system uses 1 value from tier 2 equaling 10 and 1 value from tier 1. (10 + 1 = 11). The difference in the Maya vigesimal system that value of 11 is actually calculated with two ‘fives’ from tier 1 and an ‘one’ also from tier 1. On their system, if the count hasn’t reached or passed the value of 20, it remains on the first tier. Additionally, the value of 20 was divided by 4 values of 5.

To represent these values, the Maya came up with two symbols of which they use, a dot and a bar. The Maya used a system of bars and dots for their numbers, instead of representing them by different symbols as it is done in our numbering system. A system the we adapted from Latin and Arabic. Each ‘dot’ in the Maya system represents the value of 1 unit on a given tier. Each bar represents the value of 5 dots of the tier below it.

1 dot equals 1 value and 1 bar equals 5 dots.

So when counting to the value of 11, the Maya counted two bars and one dot from tier one ( 5 + 5 + 1 = 11 ).

Two bars and one dot on tier 1 equals eleven. The dots go about the bars,

The Maya used a either a cocoa seed pod or a shell to represent the value of 0. They used a system of stacked bars and dots to represent the first 20 numbers.

Table showing first 20 Maya numbers and their Arabic equivalents. Centro de Estudios del Mundo Maya. Yucatan, Mexico. Maya World Studies Center.

It is believed that because the cacao bean was commonly used as currency throughout Mesoamerica, the Maya symbolized it with a dot to represent one bean. Another reason to believe that the Maya got the idea of using a dot from the cacao bean, is the fact that they packaged their cacao beans in quantities of 8000 to a bag.

The sum of 8000 is one of the place values on the Maya vigesimal system (203). The Maya name for the value of 8000 is called a “pic.” This is also the name for the sack they used to pack cacao beans in. It’s understanding why the Maya found it convenient to call value of 8000 a “pic” and the sack they used that head 8000 cacao beans. A set standard among the Maya for an amount in trade.

The Maya represented written values in a vertical manner, whereas our modern representations of number values are expressed horizontally. For example, we write the number 27 horizontally with the number two then the number seven to the right of it. As the numeric value increases, the number representations are added to the left, continuing horizontally.

The Maya, however, would write the value representation of 27 vertically. Their symbol for seven (a line representing five units with two dots over it) would be on the bottom and the symbol for 20 (a dot on the line above) would be directly over it. The same applies for other numbers. The Maya script or sculptor would fashion their own style of glyph, but the marked value would still be written from bottom to top.

Maya horizontal positional value representations.

Maya positional counting displayed values of 20 or more by placing a value symbol over another value symbol. When writing with the Maya vertical vigesimal positioning system, the value of 20 is written with a shell representing zero placed at the base on the bottom position and a single dot, which represents the value of twenty is placed over it in the second vertical position.

A single dot in this position over a zero means one unit of the second tier equaling 20. To write 21, the zero on the first tier would be changed to a single dot (1 unit) and for the subsequent numbers up to 19, counting up to 39.

As they reach the count of 39 again another dot is added to the second position. Any number higher than 19 units in the second position is written using units of the third position.

A unit of the third position is worth 400 (20 x 20), so to write 401 a dot goes in the first position, a zero in the second and a dot in the third. Positions higher than the third are also multiplied by twenties from the previous ones.

Examples of Maya horizontal positional value representations.

The Maya only made one exception to this mathematical order of tiers and that was with their calendar calculations. For example, the Haab’ calendar’s third position only has a value of 360 instead of 400. This is because the calendar only calculates 18 values of 20, or more accurately, the eighteen 20-day uinals (months) of the Haab’ year of 360 days (K’in). 

Examples of Maya horizontal calendric value representations.

The Maya names for their numbers are as following:

0 – xix im 10 – lahun

1 – hun 11 – buluc 20 – hun kal 400 – hun bak

2 – caa 12 – lahca 40 – ca kal 800 – ca bak

3 – ox 13 – oxlahun 60 – ox kal 1200 – ox bak

4 – can 14 – canlahun 80 – can kal 1600 – can bak

5 – hoo 15 – hoolahun 100 – hoo kal 2000 – hoo bak

6 – uac 16 – uaclahun 120 – uac kal 8,000 – pic

7 – uuc 17 – uuclahun 140 – uuc kal 160,000 – calab

8 – uaxac 18 – uaxaclahun 200 – ka hoo kal 3,200,000 – kinchil

9 – bolon 19 – bolonlahun 300 – ox hoo kal 64,000,000 – alau

Each set counted by twenties:

21 = hun-tukal = 1 + 20

22 = ca-tukal = 2 + 20

23 = ox-tukal = 3 + 20

24 = can-tukal = 4 + 20

25 = ho-cakal = 5 to 2 x 20

26 = ua-ctukal = 6 + 20

27 = uuc-tukal = 7 + 20

28 = uaxac-tukal = 8 + 20

29 = bolon-tukal = 9 + 20

30 = lahun-cakal = 10 to 2 x 20

31 = buluc-tukal = 11 + 20

32 = lah-ca-tukal = 12 + 20

33 = ox-lahun-tukal = 13 + 20

34 = can-lahun-tukal = 14 + 20

35 = ho-lahun-cakal = 15 to 2×20

36 = uac-lahun-tukal = 16 + 20

37 = uuc-lahun-tukal = 17 + 20

38 = uaxac-lahun-tukal = 18 + 20

39 = bolon-lahun-tukal = 19 + 20

Numbers held great significance in the Maya culture. For example, the number 20 signifies the total number of digits a person has: 10 fingers and 10 toes, or five digits on four limbs.  As all five digits on a single limb is 1/4th the value of a whole of 20, or the value of tier 1.

The number 13 refers to the number of major joints in the human body where the Maya believed disease and illness entered the body. These joint locations were: one neck, two shoulders, two elbows, two wrists, two hips, two knees and two ankles for a total of thirteen.

It’s these two numbers, 20 and 13, that are used to make up the Tzolk’in calendar. The Tzolk’in is believed to be the first calendar used by the Mayas. The number 13 is also the number of levels in heaven where the Maya believed the Sacred Lords ruled the Earth.

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written by Njord Kane © 2016 Spangenhelm Publishing

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