The Maya Mathematical System

The Maya Civilization, in and around Guatemala, discovered and used the concept of Zero, before any other culture in the world, (except that of  the Hindu that used it for astronomical calculations only), they based its system counting al the fingers and toes in the human body, our system use only the fingers in the hand, Zero is represented with a shell (see below). The oldest Mayan inscriptions known with a Zero are the Stelas  18 and 19 from Uaxactún, del 357 AC. (Michael Closs). The oldest Long count date is from El Baúl in Cotzumalguapa, in the Pacific Lowlands of Guatemala, dated from 32 BC.

 European cultures obtained the zero only because Arab scholars in Bagdad in the seventh century AD translated a Hindu text on astronomy and thus rediscovered the zero. Subsequently, an Arab mathematical treatise employing it was translated into Latin, and Eureka! Our Culture gained this vital idea, although it did not come into general use in the Western Civilization until many centuries later.

 Also they use a positional system, making easier to calculate and write big numbers, long before any other culture. A few Sumerian tablets show the faint beginnings of calculations based on a positional system, but no more.

The numerical glyphs can be seen on monuments and codices as normal-form (bar-and-dot) glyphs, or as glyphs known variously as head-variant glyphs or portrait glyphs. Bar-and-dot, or normal-form, notation is the more common form of numeric notation and is much simpler than head-variant glyph representation, being composed of just three basic items:

1.             

   a bar , , or for five;

   a dot , , or for 1 (note the extra decoration sometimes used to fill space); and

   a shell , a lobed symbol , a hand  , or other  glyphs  for zero.

   Portrait glyphs are just that; portraits of the gods that are the integers. They’re also called head variants, because only the head is shown. In the vast majority of cases, only a portion of the head is shown, although full heads do exist. A couple of samples include (the glyph for nine) and (the glyph for fifteen).

A table lists the Mayan numbers 0-20, in both normal and portrait glyph form. (view in new window)

The decimal mathematical system widely used today goes by 1, 10, 100, 1000, 10000, etc., the Maya vigesimal system goes 1, 20, 400, 8000, 160000, etc. While in the decimal system there are ten possible digits for each placeholder 0 - 9, in the Maya vigesimal system each placeholder has a possible twenty digits 0 - 19. For example, in the decimal system 33 = 10 x 3 + 3 while in the vigesimal system 33 = 20 + 13. It only uses three symbols, alone or combined, to write any number. These are: the dot - worth 1 unit, the bar - worth 5 units and the zero symbolized by a shell.
 It also uses a vigesimal positioning system, in which numbers in higher places grow multiplied by 20´s instead of the 10´s of our decimal system

This table shows the first 20 numbers and their Arabic equivalents.

Symbols 1 5 0 Numerals 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Numbers in the Maya system can be written vertically or horizontally. In vertical writing, the bars are placed horizontally and the dots go on top of them, in this case the vigesimal positions grow up from the base. When written horizontally, the bars are placed vertically and the dots go to their left and higher vigesimal positions grow to the left of the first entry.

25 is written this way:

1 X 20 = 20   5 X 1 = 5

Examples with larger quantities:

Each number is Multiplied by:  20	[(1 x 20) +(0 x 20) = 0] = 20	[(3 x 20) + 6)] = 66	[(400 x 5) + 0 +14)]    =  2,014	[8,000 + (400 x 3) + (20 x 1) + 19] =  9,239	[(160,000 x 1) +( 8000 x 2) +  (400 x 3) + (20 x 1) + 15] = 177,235
160,000 (8000 x 20)
8,000 (400 x 20)
400 (20 x 20)
20 (1 x 20)
0 (0 x 20)

Thus when writing vertically the vigesimal positioning system, to write 20 a zero is placed in the first position (base) with a dot on top of it, in the second position. The dot in this place means one unit of the second order which equals to 20. To write 25, the zero would change to a Bar (5 unit) and for the subsequent numbers the original 19 number count will follow in the first position. As they in turn reach 19 again another unit (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 also grow multiplied by twenties from the previous ones. Examples of the numbers mentioned:

(Note: the Maya made one exception to this order, only in their calendaric calculations they gave the third position a value of 360 instead of 400, the higher positions though, are also multiplied by 20. This was to make it easier to incorporate the solar cycle)

Mayan Names For Numbers (view in new window)

Link to Maya mathematic operations (Sp): http://www.enlacequiche.org.gt/centros/cecotz/TECNOLOGIA/matematicas.htm

It is interesting to compare this list with the words for 1 up to 20 in Dutch, English and French, and to look for traces of an ancient vigesimal system. The table clearly shows a mathematical peculiarity. For the names of the numbers bigger than 20 and having a 10 or a 5, the count is directed towards the next multiple of 20 and the linking word -tu- disappears. This is very normal for the way the Maya think, since as far as the fives are concerned, they do not take into account the units that have already been counted. Instead they take into account the following quantity, for example: as far as the calendar is concerned, they do not count in terms of past time but in terms of the future, directed towards the next unit of time.

The first day of the month does not get number 1 but 0, while number 19 instead of number 20 is allocated to the last day of the 20-days month. It is noteworthy that the names for the numbers 1-10 are approximately the same in all languages.

Fractions: It is necessary to say that, contrary to what is claimed in some publications, the Maya were familiar with a certain notion of fractions. To indicate parts in general, they used the term tzuc, which literally means "part". Subsequently tu, can, tzucil, ban cah, equals the 4 parts of the world (cah) or the 4 quarters of the world. For the notion "1/4" I found the expression heb: to open:

• heb u = 1/4 opening; and, U = moon;

• hun heb u = 1/4 moon; moon opening of 1/4;

• ca heb u = 2/4 moon; moon opening of 1/2;

• ox heb u = 3/4 moon; moon opening of 3/4.

For the notion "1/2", 2 possible applications could be found:

• Tan coch = half, in the middle; and,

• lub = legua (5,5 km);

• tan coch lub = half a legua;

• tan coch tu cappel lub = in the middle of the second legua (5.5 km), or, 1 + 1/2 "legua";

• tan coch kin tu cappel = in the middle of the second day = 1 1/2 day.

Xel = dividing the unit in two and subtracting one part; Xel is in fact a negative fraction:

• xel u ca kin bé = -1/2 + 2 days = 1 1/2 days;

• xel u ca cuch = -1/2 + 2 loads = 1 1/2 loads;

• xel u cappel lub = -1/2 + 2 leguas = 1 1/2 legua;

• xel u yox katun = -1/2 + 3 katun = 2 1/2 katun;

• xel u ca kal = -1/2 x 20 + 2x20 = -10 + 40 = 30;

• xel y yox bak = -1/2 x 400 (bak) + 3 x 400 = 1300.

Infinity

We find the following interesting terms in regard of the notion of infinity:

 Hun tso'dz’ceh,  to count the hairs that a deer has. Maxocbin, infinite in number. Hunhablat, countless; Picdzaac(ab), long number, countless; Ox’lahun D’zakab, eternal thing. Hunac,  countless times.

In addition to the normal and portrait forms, however, we also have a very, very few samples of what are known as full-figure glyphs only  in Quiriguá, Guatemala. These represent the logical conclusion of the portrait glyphs, as they also show the bodies and their accoutrements. As example,  full-figure glyphs are: