A Comparison of the Organization and Use of Chinese and Mesoamerican Abaci

by David B. Kelley

The Mesoamerican bar-and-dot number system has similarities to the Chinese solid-and-broken-bar system.

Bar-and-dot System: The numbers 1 to 9 represented using the Mesoamerican bar-and-dot system; a dot represents "1" and a bar represents "5".

The remains of a 3/4 vigesimal¹ (base twenty number system) abacus were purportedly found in Mexico. Whether or not this is true, the fact remains that the Mesoamerican bar-and-dot number signs appear to fit very systematically into such an arrangement, as demonstrated below. Additionally, this arrangement, involving seven rows and thirteen columns, also matches the most common arrangement for the Chinese decimal/hexadecimal (base sixteen) abacus is something that cannot be ignored. The evident similarities in the design of the Chinese solid-and-broken-bar and the Mesoamerican bar-and-dot number symbols is suggestive of some sort of relationship between the two systems. What that relationship may be is not clear, but it is hoped that further research will reveal its nature.

Terminology

The 2/5 Chinese Abacus: The components of a 2/5 (indicating 2 beads in the Upper Deck and 5 beads in the Lower Deck) Chinese abacus are identified in this image. The beads are arranged to show the decimal number 1,999.

The Comparisons

Example Decimal Abacus: The abaci showing simple column values and bead values appearing in the examples below, display column-values along the top and and the bead-values on the left.

In the following set of comparisons, the same number is represented in the left image using the Chinese solid-and-broken-bar system and in the right image using the Mesoamerican bar-and-dot system.

3/4 Abacus: simple column values with Chinese solid-and-broken-bar signs and number values below. This is NOT a functional Duodecimal (base 12) Abacus, and although each lower deck bead has a value of "1", each Broken Bar sign below, actually has a numeric value of "2".	3/4 Vigesimal Abacus: simple column values with Mesoamerican bar-and-dot signs and number values below. This is a fully functional Vigesimal Abacus, but owing to a 13-column limitation, only 11 bar-and-dot signs and values, and two uses of the "zero" sign are shown.
3/2 Duodecimal (base 12) Abacus: simple column values with Chinese solid-and-broken-bar signs and number values below. This is a fully functional Duodecimal Abacus, and to make it functional, the value of the Broken Bar sign has been changed to "1", and the signs and values re-arranged.	3/2 Abacus: simple column values with Mesoamerican bar-and-dot signs and number values below. This is NOT a vigesimal abacus.

Examples of Place-Value Use

3/2 Duodecimal Abacus: place-value column values with Chinese solid-and-broken-bar signs and number values below showing the decimal number 1,999.

7 X	1 (12⁰, 1st column) =	7
10 X	12 (12¹, 2nd column) =	120
1 X	144 (12², 3rd column) =	144
1 X	1728 (12³, 4th column) =	1728
Total:		1999

3/4 Vigesimal Abacus: Regular place-value column values with Mesoamerican bar-and-dot signs and number values below showing the decimal number 1,999.

19 X	1 (1st column) =	19
19 X	20 (2nd column) =	380
4 X	400 (3rd column) =	1600
Total:		1999

3/2 Duodecimal Abacus: place-value column values with Chinese solid-and-broken-bar signs and number values below.

7 X	1 (1st column) =	7
10 X	12 (2nd column) =	120
1 X	144 (3rd column) =	144
1 X	1728 (4th column) =	1728
Total:		1999

The solid-and-broken-bar signs are markedly similar to the Mesoamerican bar-and-dot signs, especially those associated with the Calendric use of the Mesoamerican signs. In the case of the example above, we see a total of five solid-bar signs and four broken-bar signs each consisting of two sub-parts, are very similar to the five Mesoamerican bar-signs and eight dot-signs.

3/2 Vigesimal Abacus: Calendric place-value column values with Mesoamerican bar-and-dot signs and number values below.

19 X	1 (1st column) =	19
9 X	20 (2nd column) =	180
5 X	360 (3rd column) =	1800
Total:		1999

The abacus shown above has been modified to reflect the Calendric use of the Mesoamerican vigesimal number system. This means that the third column has a limit of three upper-deck beads (with each bead = "5") and two lower-deck beads ((with each bead = "1"). Accordingly, all columns to the left of the second column have different values from those derived from the Regular vigesimal abacus presented earlier.

Final Comments

The solid-and-broken-bar (I-Ching) number signs represent one of the several variant Chinese number-symbol systems. The Chinese I-Ching number symbols are based on a combination of the two Yao (

) 'Elementary Forms'. And, the two Yao form the basis of the Pa-Kua (

) 'Eight Diagrams' found in the I-Ching (

) 'Book of Divination'. On the two Yao, the I-Ching specifically states that, "The number 3 was assigned to heaven, 2 to earth, and from these came the (other) numbers."; the original line in Chinese is, "

"². This has been interpreted to mean that the Solid-Bar had a numeric value of "3" and the Broken-Bar a value of "2". However, as the above analyses demonstrate, with those values the two Yao cannot generate a complete series of numbers, because there is no sign with a value of "1". And so, the pertinent line from the I-Ching was perhaps mis-interpreted; the line appears to describe an actual 3/2 duodecimal abacus, and was referring to its two decks: the upper one ("Heaven") with three beads per column, and the lower one ("Earth") with two beads per column. Finally, in order for the duodecimal abacus to be fully functional, each Upper Deck bead and associated Solid Bar sign would have necessitated a value of "3", and each Lower Deck bead and associated Broken Bar sign, a value of "1".

¹Remnants of the vigesimal system used by the Mayans still remain today; e.g. when counting in French, "quatre vingt" is translated "four twenty", meaning "eighty"; see: The Numerals and Mayan Numerals.

² The Sacred Books of China. The Texts of Confucianism, Part II: The Yi King (Chapter 1, Paragraph 2). James Legge, trans. 1882.

For more information about Mayan Calendars see: Calendopaedia, The Classic Maya Calendar and Day Numbering System and the corresponding Wikipedia entry.