The Mayan Number System
The Mayan number system dates back to the fourth century and was approximately 1,000 years
more advanced than the Europeans of that time. This system is unique to our current decimal
system, which has a base 10, in that the Mayan's used a vigesimal system, which had a base 20.
This system is believed to have been used because, since the Mayan's lived in such a warm
climate and there was rarely a need to wear shoes, 20 was the total number of fingers and toes,
thus making the system workable. Therefore two important markers in this system are 20, which
relates to the fingers and toes, and five, which relates to the number of digits on one hand or foot.
The Mayan system used a combination of two symbols. A dot (.) was used to represent the units
(one through four) and a dash (-) was used to represent five. It is thought that the Mayan's may
have used an abacus because of the use of their symbols and, therefore, there may be a
connection between the Japanese and certain American tribes (Ortenzi, 1964). The Mayan's
wrote their numbers vertically as opposed to horizontally with the lowest denomination on the
bottom. Their system was set up so that the first five place values were based on the multiples of
20. They were 1 (200), 20 (201), 400 (202), 8,000 (203), and 160,000 (204). In the Arabic form we
use the place values of 1, 10, 100, 1,000, and 10,000. For example, the number 241,083 would be
figured out and written as follows:
Mayan
Place Value Decimal Value
Numbers
1 times 160,000 = 160,000
10 times 8,000 = 80,000
2 times 400 = 800
14 times 20 = 80
3 times 1 =3
This number written in Arabic would be 1.10.2.14.3 (McLeish, 1991, p. 129).
The Mayan's were also the first to symbolize the concept of nothing (or zero). The most common
symbol was that of a shell ( ) but there were several other symbols (e.g. a head). It is interesting
to learn that with all of the great mathematicians and scientists that were around in ancient
Greece and Rome, it was the Mayan Indians who independently came up with this symbol which
usually meant completion as opposed to zero or nothing. Below is a visual of different numbers
and how they would have been written:
In the table below are represented some Mayan numbers. The left column gives the decimal
equivalent for each position of teh Mayan number. Remember the numbers are read from bottom
to top. Below each Mayan number is its decimal equivalent.
8,000
400
20
units
20 40 445 508 953 30,414
It has been suggested that counters may have been used, such as grain or pebbles, to represent
the units and a short stick or bean pod to represent the fives. Through this system the bars and
dots could be easily added together as opposed to such number systems as the Romans but,
unfortunately, nothing of this form of notation has remained except the number system that
relates to the Mayan calendar.
For further study: The 360 day calendar also came from the Mayan's who actually used base 18
when dealing with the calendar. Each month contained 20 days with 18 months to a year. This
left five days at the end of the year which was a month in itself that was filled with danger and
bad luck. In this way, the Mayans had invented the 365 day calendar which revolved around the
solar system.
Contributed by Mikelle Mercer
---------------------------------------------------------------------------------------------------------------------
the Mayan system of writing numbers was very simple. The way they wrote out numbers was
using bars and dots. Each dot represents a "1" and each bar represents a "5". Using this system,
you could write out any number from one to twenty by placinig bars and dots on top of each
other. Let's look at the number twelve as an example. It is made up of two bars,
each representing five; and two dots, each representing one. Added up, it
amounts to a sum of twelve.
We can also add two of these Mayan numerals together very easily by just
adding the dots together. When there is a total of five dots, those five dots
become one bar, and the remaining dots are placed on top. Look at this
example that adds "4" and "2" together. Five of the dots become a bar and the remainder is
placed above it. This is now the symbol for "6"
In the English counting system, there are ten symbols that are used to create numbers:
1,2,3,4,5,6,7,8,9, and 0. After we count past 9 we have to start using combinations of these
symbols. For example, the number forty-six(46) is represented by a "4" with a "6". The "4" is in
the tens place and represents four sets of ten. And the six is in the ones place, representing six
sets of one. With this system, we can create any whole number possibe.
The Mayan system was similar, but instead of having only ten symbols, like there are today,
there were twenty symbols. So instead of combing two symbols together after reaching nine,
they started after reaching nineteen. Because the Mayan system is based on 20 and not 10 like
the English system, instead of a tens place, there is a twenties place. Look at the example of the
number 46. There are two dots in the twenties place, indicating two sets of twenty. And there is a
bar with a dot in the ones place indicating six sets of one. Look at the diagram below to help
visualize where these numbers come from.
This base twenty system is still in use today by such tribes as the Hopi and the Inuits. We can
also add with these larger numbers almost as easily as the single digit ones. To do so, first add
together the bars and dots in the 1s place. If the sum is over twenty, there will be some carrying
involved, but we'll talk about that in just a little bit. Next, you sdd the bars and dots in together
that are in the 20s place. This gives us the number in Mayan numerals. To translate the number
into the kind of numbers that are in popular use today, there is just one trick to remember- Every
dot in the 20s place represents one set of 20. And each bar in the 20s place represents one set of
one hundred (or 5 sets of 20). Look at this example.
We start with the 1s place. First we add three to six to get nine. Then we move on to the 20s
place. We add the twenty to the sixty (3 x 20) add get eighty. So our final answer is eighty-nine.
Let's see if you can try one on your own. It's a little bit tricky, but I think you can get it anyway.
The Maya number system was a base twenty system.
Almost certainly the reason for base 20 arose from ancient people who counted on both their
fingers and their toes. Although it was a base 20 system, called a vigesimal system, one can see
how five plays a major role, again clearly relating to five fingers and toes. In fact it is worth
noting that although the system is base 20 it only has three number symbols (perhaps the unit
symbol arising from a pebble and the line symbol from a stick used in counting). Often people
say how impossible it would be to have a number system to a large base since it would involve
remembering so many special symbols. This shows how people are conditioned by the system
they use and can only see variants of the number system in close analogy with the one with
which they are familiar. Surprising and advanced features of the Mayan number system are the
zero, denoted by a shell for reasons we cannot explain, and the positional nature of the system.
However, the system was not a truly positional system as we shall now explain.
In a true base twenty system the first number would denote the number of units up to 19, the next
would denote the number of 20's up to 19, the next the number of 400's up to 19, etc. However
although the Maya number system starts this way with the units up to 19 and the 20's up to 19, it
changes in the third place and this denotes the number of 360's up to 19 instead of the number of
400's. After this the system reverts to multiples of 20 so the fourth place is the number of 18 ×
202, the next the number of 18 × 203 and so on. For example [ 8;14;3;1;12 ] represents
12 + 1 × 20 + 3 × 18 × 20 + 14 × 18 × 202 + 8 × 18 × 203 = 1253912.
As a second example [ 9;8;9;13;0 ] represents
0 + 13 × 20 + 9 × 18 × 20 + 8 × 18 × 202 + 9 × 18 × 203 =1357100.
Both these examples are found in the ruins of Mayan towns and we shall explain their
significance below.
Now the system we have just described is used in the Dresden Codex and it is the only system
for which we have any written evidence. In [4] Ifrah argues that the number system we have just
introduced was the system of the Mayan priests and astronomers which they used for
astronomical and calendar calculations. This is undoubtedly the case and that it was used in this
way explains some of the irregularities in the system as we shall see below. It was the system
used for calendars. However Ifrah also argues for a second truly base 20 system which would
have been used by the merchants and was the number system which would also have been used
in speech. This, he claims had a circle or dot (coming from a cocoa bean currency according to
some, or a pebble used for counting according to others) as its unity, a horizontal bar for 5 and
special symbols for 20, 400, 8000 etc. Ifrah writes [4]:-
Even though no trace of it remains, we can reasonably assume that the Maya had a number
system of this kind, and that intermediate numbers were figured by repeating the signs as many
times as was needed.
Mayan Numeration
The pre-Columbian Mayans developed a fairly sophisticated system
of
numeration, primarily for the purpose of making calenders and
keeping
track of time. (A concern for quantifying the passage of time,
and
minding the calender, seems to have been a characteristic of many
primitive peoples, and prompted much of the early record-
keeping.)
An example of a Mayan representation of a number is shown below:
The Mayans wrote their numbers vertically, with each
"digit" being
represented by either a set of dots and horizontal lines or else
a
symbol that looks (to me) like an empty bowl, which denotes zero
(an
impressive invention of the Mayans, considering how many millenia
it
took the people of the other hemisphere to think of it). For the
non-zero digits, each horizontal dash represents 5, and each dot
represents 1, and these are simply added together to give the
value
of the digit. Thus each non-zero digit consists of from 0 to 4
dots,
and from 0 to 3 lines, and these arrangements, along with the
"empty
bowl", give representations for every number from 0 to 19.
Then they used a "place" system (another impressive invention),
with
the lowest digit signifying 1's, and the higher places signifying
more powers of the base which was nominally always 20. However,
the
system had one anomaly in that the denomination increased by a
factor
of 18 instead of 20 when rising from the second to the third
digit.
The presumed explanation for this is simply that since the Mayans
were mainly interested is counting days, and their basic annual
calender had 360 days, it was most convenient for the
denomination
of the 3rd least significant digit to be (20)(18) = 360 instead
of
(20)(20) = 400.
Of course, one consequence of this anomaly is that the possible
representations of a given number are not necessarily unique.
For
example, suppose we tip the Mayan numbers over, so the digits are
horizontally arrayed, and we use our numerals to signify the
digit.
Then the decimal number 360 could be represented in the Mayan
system
as either (1 0 0) or as (18 0). A nice feature of our more
conventional fixed-base representations is that they give a
strict
one-to-one correspondence between the natural numbers and all the
possible permutations of a fixed set of digits.
How Mayan numbers worked
The Mayan system is interesting as they developed it without any contact with the other systems
on this website. It is similar to the Babylonians but the Mayans chose different numbers as their
bases. They used dots to represent numbers under five, so four is four dots. Five is represented
by a line. So six is a line and a dot, and seven is a line and two dots, and thirteen is two lines and
three dots. This is a unary system, but using five as a base rather than the more common ten.
This works up to nineteen, but rather than twenty being four lines, they started a new count
above the first one. Zero is represented by a shell. So twenty is a single dot above a shell. This
stacking of numbers rather than having them in a line is a little disconcerting at first! You have to
talk about rows rather than columns. This second system, for counting the twenties and powers
of twenty, is positional, and even has a zero, to show that you have no digit in this row.
To calculate a Mayan number, you need to divide the number into powers of twenty.
The great advantage of the positional system is that you need only a limited number of symbols
(the Mayans only had two, plus their symbol for zero) and you can represent any whole number,
however big. The Mayans had a sophisticated number system, but a little complex. Presumably
the Mayans chose five and twenty as the two bases of their system as there are five fingers on
one hand, and twenty fingers and toes on one person. Although we think of other systems using
base ten, in fact the Romans had almost a base five within their base ten, as they had separate
symbols for five, fifty, and five hundred. An eastern abacus has beads for five as well as beads
for units. It makes sense, as it is hard to instantly recognise groups of symbols more than five.
Some peoples, like the Babylonians arranged them into neat patterns to make it easier, but then it
became harder to draw. The Mayan system is quick to write, and simple to understand. However,
multiplication tables need to be learned up to twenty, rather than just ten!
The Mayan had a second Number System, used for dating buildings and on Calendars, etc. This
would be a more formal system, rather than a number system used for calculation. The following
was sent to me by Zoe Anne (Hemphill) Tom, who drew it.
