CHAPTER 1 History of Numbers

§1.6 The Maya
Lesson 4 - "The Calendrical Mathematics of the Maya”

For calendrical purposes, the Mayan modified their numeration system slightly, making the third place-value from the bottom units 18 × 20 = 360 instead of 20² = 400. The place-value of the fourth position from the bottom became 18 × 20² = 7200; the fifth place-value was 18 × 20³ = 144000; and so on. The symbols remained the same. For the Mayan, it was understood which system of numeration was being used by the problem. You will be instructed which set of place values to use.

The modified place-values are given below.

:		:
18 × 203	=	144,000
18 × 202	=	7,200
18 × 201	=	360
201	=	20
200	=	1

Figure 1.6.2  Place Values for 
the Mayan Calendrical System of Numbering

 

It is this system we will use in what follows, because it was in calendrical calculations that the Maya used numbers most extensively.

The next example we illustrate the use of this modified system.

Example 1.6.3

Translate the Mayan number shown below which is written in calendrical form.

 

Solution

The given Mayan number occupies the three lowest place values of the calendrical system. These place values are from the bottom up 20⁰, 20¹, and 18 × 20¹, respectively. The Mayan numerals shown in each of these place values are 5, 3, and 1, respectively. Thus, this Mayan number is

	1 × 18 × 201	=  360
	3 × 201	=    60
	5 × 200	=      5
		____       425

Note that the symbols used in Examples 1.6.1 and 1.6.3 were the same. However, a different answer is achieved because of the differences in the place values of the two systems.

We will look next at rewriting one of our own Hindu-Arabic numbers as a Mayan number into the calendrical system.

 

Example 1.6.4

Write the Hindu-Arabic number 158,441 as a Mayan number in their calendrical system.

Solution

As before, we need to first list the place values of this system choosing the largest one that will divide into the given number of 158,441.

18 × 20⁴ = 2,880,000

18 × 20³ = 144,000

18 × 20² = 7200

18 × 20¹ = 360

20¹ = 20

20⁰ = 1

The largest place value we need is 18 × 20³ = 144,000. And so we begin by dividing this into 158,441. The entire succession of division with quotients and remainders is included below.

The last remainder of 1 goes into the ones position (20⁰). Now, using the quotients and the last remainder of 1, this Mayan number is written as shown below. The other two columns are simply checking and verifying our results.

	1 × 18 × 203	=  144,000
	2 × 18 × 202	=    14,400
	1 × 18 × 201	=            0
	2 × 201	=          40
	1 × 200	=            1
		_______             158,441

 
Professorial note: For an excellent exposé of Mayan mathematics and their calendar see Maya Civilization (pp. 34 - 37) by T. Patrick Culbert, published by the Smithsonian Institution.
 

The exercises at the end of this section will present problems similar to those found in the examples. They will be divided into two groups; those which use the base system of general mathematics, and those which use the base system of calendrical mathematics.

 

All materials © Leo Lusk 2003