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.
We'll try to give you an answer to the following question :
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.gif (470 bytes)
0
1.gif (187 bytes)
1
2.gif (225 bytes)
2
3.gif (261 bytes)
3
4.gif (288 bytes)
4
5.gif (195 bytes)
5
6.gif (255 bytes)
6
7.gif (303 bytes)
7
8.gif (314 bytes)
8
9.gif (363 bytes)
9
10.gif (266 bytes)
10
11.gif (323 bytes)
11
12.gif (362 bytes)
12
13.gif (395 bytes)
13
14.gif (410 bytes)
14
15.gif (304 bytes)
15
16.gif (374 bytes)
16
17.gif (399 bytes)
17
18.gif (437 bytes)
18
19.gif (460 bytes)
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.gif (187 bytes)(1)
1's 12.gif (362 bytes)(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 1.gif (187 bytes)   1.gif (187 bytes)   2.gif (225 bytes)
400's 3.gif (261 bytes)   6.gif (255 bytes)   9.gif (363 bytes)
20's 12.gif (362 bytes) + 1.gif (187 bytes) = 13.gif (395 bytes)
1's 9.gif (363 bytes)   5.gif (195 bytes)   14.gif (410 bytes)
  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*202; the fourth 18*203, 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)

CONCLUSION

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

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