The Maya Abacus

The Mesoamerican abacus is called a “Nepohualtzintzin.”

The arrangement of the Maya Abacus, or Nepohualtzintzin, is with seven beads (or balls or cacao beans) of two colors per level.  Four of the beads are of one color, representing single units; and three of the beads are another color, representing five single units on the same level. The value of one of the four beads are single unit sums which are equal to the sum of all the units of one level below it.  On the Maya abacus, the first level’s sum is twenty units (vigesimal).  <see Maya Arithmetic>

The single unit values on the Maya abacus/ counting system are represented in illustration 1 below.
The white lines represent levels and are labeled: A, B, C, D, E, and F. 

illustration 1

The represented value of the beads on each level in these examples are:

Blue beads = 1 unit.

Red beads = 5 units.

Each level’s unit increases in value each level up (as represented by white lines in illustration 1) and are as follows:

Line A = 1, (the value of each single blue bead)

Line B = 20,

Line C = 400,

D = 8000, E = 160000, etc.. as shown above in illustration 1.

Using the Nepohualtzintzin is much like using any other of the World’s various abacuses through the span of time.

Using the Maya Abacus, you begin the count on the first line, line A (level or tier 1), by sliding a blue bead to the middle of line, counting to 1 (see illustration 2 below).

illustration 2

The absence of any beads in the middle of any line would be zero and represented with a shell in written Mayan.

A representation of the value of 1 with a single dot on level one and a shell, representing zero(0) over it, on level two.

Continue counting up with your Maya Abacus by adding blue beads.

Illustration 3
Illustration 4

When you’ve used all four blue beads counting to 4,

Illustration 5

Slide all the blue beads back to their original position and slide one red bead, value 5, to the middle to make the count at 5. as displayed in illustration 6.

Illustration 6

The count continues by adding a blue bead next to the red bead, making the sum 6.

Illustration 7

Continue adding on line A with blue beads, counting up 7, 8, and 9. Ten is then counted by returning the blue beads to their original position and sliding an additional red bead (value is 1 red unit = 5 blue units) to the middle (5 + 5 = 10).

Illustration 8

An additional red, adds 5 to the count, making the sum 15.

Illustration 9

The count continues by adding blue beads, counting up 16, 17, 18, and then with all red and blue balls in center equals 19. (illustration 10).

Illustration 10.

Continue counting up to 20 by sliding all the beads on line A back to their original start positions and slide 1 blue bead on line B (valued at 20 per unit) to the middle.

Illustration 11

A single blue bead in the middle of the second level (line B) brings the count to 20. When writing this number, you would add a shell on the bottom tier to represent zero as a place holder.

Increase the count by adding 1 blue bead on the first line, line A (illustration 12).

Illustration 12

Some additional examples of using the Maya Abacus count:

Illustration 13

Remember, each blue bead is the value of 1 unit and each red bead is the value of 5 single units, with the value of each unit starting at 1 on the first level and increased 20 times the value of a single unit value of the line (1, 20, 400, 8000, etc.).

In illustration 14, one red bead ( value = 5) on line two, representing 5 blue beads (value 1 unit = 20), shows the total of 100.

Illustration 14.

We then increase the count to 111 by adding two red beads on line A, valued at 5 units each and 1 blue bead on Line A, valued at 1 unit (100 + 11 = 111).

Illustration 15.

We set the count at 400 by adding 1 blue bead on line C (3rd level). We know that the value of the single blue unit bead on the third level are worth 400 because the sum of a units on the 3rd level are equal to the sum of all the units from the previous level, which were valued at 20 per unit, which is the sum of all the units on the 1st line which are valued at 1.

Illustration 16.

When we write 400 in Mayan, we write a single dot on the 3rd level and a single shell on each level under it, on the 1st and 2nd levels, as seen under illustration 16 above.

Illustration 17.

( 1 x 400 ) Line C


(13 x 20 ) Line B


( 7 x 1 ) Line A

= 667

The development of the Nepohualtzintzin, or Mesoamerican abacus explains the magnitude of understanding that the Mesoamericans had in mathematics.

Knowledge in these mathematics made it possible that they were able to make such exact calculations of universal cosmogony. The Nepohualtzintzin, which essentially was a pre-Hispanic computer, was not only able to make mathematical calculations, but also astronomical and gestation interpretations. The Nepohualtzintzin as an instrument that is similar to other abacus in different cultures, such as the Japanese soroban.

The abacus helps make it possible to perform not only basic operations such as: addition, subtraction, multiplication and division, but it also can be used for complex operations like roots, powers and integral and differential calculus operations.

The word “Nepohualtzintzin” comes from the Nahuatl language and is formed by the roots; Ne – personal -; pohual or pohualli – the account -; and tzintzin – small similar elements. This roughly translates into: counting with small similar elements by somebody.

The knowledge of the Nepohualtzintzin and its use was passed on to students, whom dedicated their entire lives from childhood to mastering and calculating the events and movements of the skies.

Unfortunately, the Nepohualtzintzin and its teachings were among the victims of the evangelizing paranoia of the Spanish Conquest. The Nepohualtzintzin proves that Mesoamercian cultures already had great capabilities in scientific and technological developments prior to the arrival of the Europeans.

A typical Nepohualtzintzin usually has 13 rows with 7 beads in each row.

A Nepohualtzintzin has 13 rows with 7 beads in each row, which makes for a total 91 beads.

  • The 91 beads in the Nepohualtzintzin represent the number of days in a season within the year.

  • Two Nepohualtzitzin make a total of 182 beads, which is the number of days of corn’s cycle from sowing to harvest.

  • Three Nepohualtzintzin making for a total of 273 beads and is the number in days of a human baby’s gestation time from conception to birth.

  • Four Nepohualtzintzin complete the cycle of a year’s time, minus a day and a quarter.

The Nepohualtzintzin accounts for the absolute precision needed for the higher scientific and mathematical levels that the Mesoamericans had developed many years before the arrival of the Spanish conquistadors.

David Esparza Hidalgo’s had rediscovered the Nepohualtzintzin upon finding diverse engravings and paintings of the ancient Mesoamerican math instrument in Mexico. Very old Nepohualtzintzin that were discovered could have also been attributed to the Olmec culture. Some of the ancient abacus that have been found were usually in the shape of bracelets, especially those discovered in the Maya area. One of the ancient Nepoualtzitzin abacus appeared on a painted vase in Guatemala known as the “Nejar Vase.”

Maya Nepoualtzitzin as a Bracelet.

The existence of the Mesoamerican abacus gives us insight that the ancient Maya already had the sufficient knowledge to devise and handle a device of mathematical complexity and also extend its usage into their daily lives and activities.

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