## Mayan Numbers

In the previous blog, we looked at how the numbers we use in mathematics were formed using the decimal number system. In this system, we use multipliers of ten to define the positions for the number’s digits. There are other number systems that used multipliers of twenties instead of tens. This vigesimal number system was used by the Pre-Columbian Mayan civilization as their way of representing quantities.

The Mayans used the following three unique symbols to represent numbers: A seashell symbol  which is used to represent zero, a dot  used to represent one unit, and a stroke  used to represent five units. They used these symbols to represent the numbers 0-19 as follows:

The Mayans wrote their numbers vertically. The bottom slot in any number has a multiplier of 1. The next slots above them increase by multiples of 20, as shown in the diagram below.  They Mayans defined numbers by placing one or more of the twenty digits from above in the appropriate slots.  Note that the Mayans used zero as a placeholder just like the Indians did in the Hindu-Arabic Number System (See “India & Zero” blog).

Lets use this information to decipher a mayan number (A).

We translate the symbol into a hindu-arabic number and we multiply it by the multiplier associated with its vertical position (B).  We add the values of each position and come up with the value of the Mayan number (C).  In this example, the Mayan number (A) has a value of 96,410.

There are only three undisputably authentic Maya codices in existence today. The only one of these three to contain mathematical symbols is the one referred to as the Dresden Codex. It is currently at the Saxon State Library (Sächsische Landesbibliothek) of Dresden, Germany. A high-quality high-resolution image of the codex is available at this website for you to explore. See examples of how the Mayans wrote down numbers and see if you can decipher them using what you learned in this blog.

## Subsets (⊂) and Partitions

Lets look at two more fundamental relationships of sets that are closely related: subsets and partitions.

In the Venn Diagram on the left, I am visually representing three fundamental insights.  (1)  All the elements of set A are also elements of set C. (2) All the elements of set B are also elements of set C. (3)  There are elements in set C that are not elements of A or B.  Thus, set A and B are subsets of set C.  The symbol for subset is ⊂.

Now, in the Venn Diagram on the right, we have a slightly different relationship.  Statements (1) and (2) are still accurate.  However, there are no elements in set C that are not elements of A or B.  In other words, all elements of C are either elements of A or B (but not both of them).  In this case, sets A and B are partitions of set C.  They are unique forms of subsets.

You may not realize this, but the most common use of subsets and partitions can be found in the visual display of organizational charts.  Most of the time, such relationships are represented as flow charts.  Although they can tell you which individual (“element”) is part of which group (“set”), they tend to visually emphasize the hierarchy of authority to the audience.  I personally believe  organizational relationships can be best represented as subsets and partitions.

For example, If Manager John Doe manages office A and B, I can show Office A and B as the subset of Manager John Doe.  This relationship would show that both manager and the workers in Office A and B have certain responsibilities in common with each other.  It would also show that Manager John Doe is responsible for things that are outside the scope of the employees in Office A and B.  Now if Manager John Doe’s only responsibilities is to manage the work of Office A and B, then this relationship should be visually displayed as a partition.

The development of this and other concepts of set theory can be traced back to one man: Georg Cantor (1845-1918), a German mathematician.  Cantor published a six part treatise on set theory from the years 1879 to 1884.  These publications defined most of the concepts of set theory we teach in secondary school and college.  He accomplished much during his career, despite much opposition to his ideas from prominent colleagues.  Explore his interesting life.

In the entrance lounge of the institute for mathematics of the Martin-Luther University at Halle-Wittenberg in Germany. There, you will find a display of a bust representing Georg Cantor.  He was a professor of mathematics at this institution from 1879 to 1913. His work on set theory was among his many contributions to math he made during his time there.

Although Georg Cantor is considered a German mathematician, he was actually born in Russia.  A plaque also marks the place of his birth on Vasilievsky Island in Saint Petersburg. “In this building was born and lived from 1845 till 1854 the great mathematician and creator of set theory Georg Cantor.”