The Maya Arithmetic |

INTRODUCTION

**The history and
civilization of ancient peoples has always intrigued modern man.
Today we are visiting the Maya and looking at their mathematics,
especially their number system. We find it to be sophisticated,
logical, and yes, even beautiful.
**

Did the Maya have a numerical system, and if so, how did it work?

**THE MAYA
COUNTING SYSTEM**

**The Maya of Central
America understood the concept of zero and place notation
hundreds of years before its earliest known use in India and
medieval Islam. When Europeans arrived in the Americas, they
found that the abacus was in use in both Mexico and Peru. **

**The Maya number
system is in some respects very similar to ours but** **instead
of the decimal system we have today, the Maya used the vigesimal
system for their calculations - a system based on 20 rather than
10. This means that instead of the 1, 10, 100, 1 000 and 10 000
of our mathematical system, the Maya used 1, 20, 400, 800 and 16
000. Base twenty was also used in their calendar, developed by
astronomers for keeping track of time. They used a notation
with bars and dots as "shorthand" for counting. A dot
stood for one, a bar stood for five and a shell represented zero.
The numbers could be written from bottom to top or from right to
left. Most of the time they were combined with their head symbols
: the beautiful Maya glyphs (discussed and shown later). **

**Some numbers were
considered more sacred than others like 20 as it represented the
number of fingers and toes a human being could count on. Another
special number was five, as this represented the number of digits
on a hand or foot. Thirteen was sacred as the number of original
Maya gods. Another sacred number was 52, representing a number of
years in a "bundle", a unit similar in concept to our
century.**

**In the following
table, you can see how the system of dots and bars works to
create Maya numerals compared to our equivalent present notation
for the numbers from 0 to19. **

0 |
1 |
2 |
3 |
4 |

5 |
6 |
7 |
8 |
9 |

10 |
11 |
12 |
13 |
14 |

15 |
16 |
17 |
18 |
19 |

**Because the base of
the number system was 20, larger numbers were written down in
powers of 20. We do that in our decimal system too: for example
32 is 3*10+2. In the Maya system, this would be 1*20+12, because
they used 20 as base. **

**Below you can see
how the number 32 was written from bottom to top : **

20's |
(1) |

1's |
(12) |

**It was very easy to
add and subtract using this number system, but they did not use
fractions. Here's an example of a simple addition: **

8000's |
|||||

400's |
|||||

20's |
+ |
= |
|||

1's |
|||||

9449 |
+ |
10425 |
= |
19874 |

**As you can see,
adding is just a matter of adding up dots and bars! Maya
merchants often used cocoa beans, which they layed out on the
ground, to do these calculations.**

**The following table
compares our notation to their vertical and horizontal form for
numbers 0 to19.**

A very
interesting fact is that the Maya introduced a variation
in the third order. In a perfect vigesimal system of
numeration, the third term should be 400 but the Maya
took 18*20 because 360 was a closer approximation to the
length of the solar calendar. In higher orders they
continued multiplying by 20. Therefore the first place
value is 1; the second, 20, the third 18*20, the fourth
is 18*20^{2};
the fourth 18*20^{3},
and so forth. |

**Maya head numerals**

**These are the head
numerals for 0 through 19. **

**They also used a
head symbol representing the moon for 20. This head symbol is
used in the following representation of the 31st day in their
calendar. You will notice that the head symbol is combined with
the "normal" numerical system.**

**(Closs, Michael P.
"Mathematical Notation of the Maya" **__Native
American Mathematics__**, edited by Michael P.
Closs, University of Texas Press, Austin, 1986 p. 345, figure
11.20a)**

**We discovered an
ancient civilization that had a very accurate, sophisticated, and
complex vigesimal numerical and calendrical system. The Maya
representation of the numbers using bars and dots was most of the
time combined with a set of beautiful head numerals : the Maya
glyphs. In the jungles of southern Mexico and the neighbouring
countries you can find carved stelaes, altars, ceramics and
stucco all showing that the Maya civilization had not only a
unique system of writing but also a precious artistical
represention which beauty is a real inspiration for many artists.**

**Last but not least
a funny thing to do. There are three Maya numbers. See if you and
your classmates can translate it into our numerical system! **

**Sources**

http://www.saxahali.com/historymam2.htm

http://www.vpds.wsu.edu/fair_95/gym/UM001.html

http://www.astro.uva.nl/~michielb/maya/calendar.html

http://www.civilization.ca/membrs/civiliz/maya/mmc05eng.html

The Ancient Maya, G. Morley, Stanford University Press