

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. This was
essential for their
commerce
and Calendrical
calculations.
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 normalform
(baranddot) glyphs, or as glyphs known variously as headvariant
glyphs or portrait glyphs. Baranddot, or normalform, notation
is the more common form of numeric notation and is much simpler than
headvariant glyph representation, being composed of just three basic
items:
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 020,
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.
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.
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 20days month. It is noteworthy that the names for the numbers 110
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 fullfigure 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,
fullfigure glyphs are:



