## 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.

## India & Zero

Lets look at one piece of evidence cited in the history of zero. In order to find it, you will have to go to Gwalior, India, the site of an impressive late 15th c. medieval fort which occupies a plateau in the center of the city. On the eastern side of this plateau is a 9th century Hindu temple, the Chatarbjuj temple, which is carved out of one single chunk of stone. It is dedicated to Vishnu, but it is no longer an active site of worship for the Hindu faithful. Just inside the inner chamber, there is a dedication tablet. By accident, it records the oldest use of “0” in India, for which one can assign a definite date (876 AD).

You will find a more detailed and fascinating description of this site in an essay , “All for Nought,” written by Dr. Bill Casselman (University of British Columbia, Math Department) for the American Mathematical Society. You will see many numerical values on display in the temple inscriptions. These are the numbers as they appeared in the dedication tablet. The numbers 4 and 6 were not written in any of these values.

The essay shows that by 876 A. D. our current place-value system with a base of 10 had become part of popular culture in at least one region of India. including the concept of using zero as a placeholder for “nothing.”  There is a high degree of certainty that the decimal place value notation was invented and developed in India from the 1st to 5th century A.D. There were many number systems simultaneously being used by various cultures throughout Asia and Europe during that time. The knowledge of this system spread from India in a very indirect and complicated way to western Europe via Persian and Arabic mathematicians. Many refer to the decimal place value notation we use today as the Hindu-Arabic number system.

## Real Numbers (R)

Necessity is truly the mother of invention. Throughout the history of mathematics, different types of numbers were invented to deal with a variety of  situations. Lets start our exploration of mathematical numbers by first identifying the different types of numbers we use. I am expressing the five different types of numbers visually in this Venn Diagram.

The first numbers to be invented in many cultures were counting numbers. It evolved out of a universal need to document how many objects we have in a particular situation. The related concept of addition developed for similar reasons. In mathematics, we call these counting numbers the set of natural numbers. This set is usually represented as {N}. The elements of {N} are {1,2,3,4,5,6,7,8,9,10,…}.

When we are counting, we are adding objects together to find its quantity. But what if we were to remove objects from a group? The need to express the removal of objects lead to the creation of negative forms of natural numbers. The related concept of subtraction developed for similar reasons. We call this set negative addends. There is no letter of the alphabet that is commonly used to represent this set. The elements of this set are {-1,-2,-3,-4,-5,-6,-7,-8,-9,-10,….}.

When I have four objects in my hand and one person takes two objects and a second person takes two objects, how do I mathematically express the fact that I no longer have any objects in my hand? A numerical symbol was needed to represent “nothing.” We use the symbol called zero. This is a set that has only one element,  the symbol we use to represent “nothing,” {0}.  It is also an important placeholder used in the decimal notation system as well as expressing any numbers greater than 9.

You may have heard the term whole numbers used in math classes.  This set is usually represented as {W}.   The set {W} is nothing more than the union of the set of natural numbers {N} and the set zero {0}. The set of natural numbers {N} is a subset of {W}.  The set zero {0} is also a subset of {W}.
The elements of {W} are {0,1,2,3,4,5,6,7,8,9,10,…}.

Another term commonly used in math is integers. This set is usually represented as {Z}. The set {Z} is the union of three sets: the set of natural numbers {N}, zero {0}, and the set of negative addends. Therefore, the elements of {Z} are
{…,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,…}.
For the longest time, mathematicians believed that all numbers could be expressed as a fractions: the ratio of two numbers, like ¾. They referred to these numbers as rational numbers. This set is usually represented as {Q}. We will explore the concept of fractions in future blogs.  Even integers can be expressed as fractions (for example, 3=3/1, -5=-5/1). Therefore the set of integers {Z} is a subset of the set {Q}. The elements of {Q} are all numbers that can be written in the form m/n, where n ≠ 0.

Unfortunately, it was discovered that not all quantities could be expressed as a fraction. Due to the puzzling nature of these seemingly quirky numbers, they collectively became known as irrational numbers. The set is usually represented as {I}. The elements of {I} are all numbers that cannot be represented as a fraction. The value of  Pi (π), the ratio of a circle’s circumference to its diameter, is one such irrational number.

So these are the sets and partitions that make up the set of real numbers.  This set is usually represented as {R}.  All the numbers you will come across will be elements of one or more of these sets.  And they are all real numbers… well, almost all.  There are numbers called complex numbers which pop up in certain situations, but we’ll leave that for another blog… maybe.