Category Archives: History

Babylonian Numbers

In an October blog “Decimals,” I showed you how we use the decimal (base-10) number system to generate the natural numbers we use today in math. In the blog “Mayan Numbers,” I showed you how the ancient Mayan civilization created numbers using a vigesimal (base-20) number system. In today’s blog, we’ll visit another ancient culture and learn how the Babylonian civilization used a sexegesimal (base-60) number system to do math.

Babylonia existed from about 18th c. BC until the 6th c. BC. This nation existed in the southern region of Mesopotamia between the Tigris and Euphrates River. Today, this area of the world is now part of Iraq.

The Babylonians used one of the earliest forms of written expression: the cuneiform script. They wrote in soft clay tablets using a wedge-shaped reed stylus. When the clay was hardened in the sun, the writing permanently became a part of the tablet.
They used just two symbols to define the 59 digits as follows:


The Babylonians wrote their numbers from right to left. The rightmost slot in any number has a multiple of 1. The multiplier for the next slots to the left of this column increased by a multiple of 60. The Babylonians defined their numbers by using one or more of the 59 digits from above. Although the Babylonians understood the concept of nothingness, they did not have a symbol for the digit or value of zero. They would leave a space or use a non-numeric placeholder to indicate that no digit was defined for a position within a number.

Now let’s use this knowledge to translate a Babylonian number (A).


We first translate the Babylonian digits into a Hindu-Arabic number and we multiply it by the multiplier associated with its position (B). We add the values of each position and come up with the value for the Babylonian number (C).

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Can Alice do Addition?

tenniel-portraitJohn Tenniel (1820-1914) was a British illustrator, graphic humorist and political cartoonist. He achieved considerable fame as the illustrator of Alice. Tenniel drew ninety-two drawings for Lewis Carroll’s “Alice’s Adventures in Wonderland” (London: Macmillan, 1865) and “Through the Looking Glass” (London: Macmillan, 1871). There is an illustration to the ninth chapter of Through the Looking Glass and excerpt that touches upon the topic of recent blogs.


Illustration to the ninth chapter of “Through the Looking Glass” by John Tenniel. Wood-engraving by the Dalziels.

“Manners are not taught in lessons,” said Alice. “Lessons teach you to do sums, and things of that sort.”
“Can you do Addition?” the White Queen asked. “What’s one and one and one and one and one and one and one and one and one and one?”

“I don’t know,” said Alice. “I lost count.”
“She can’t do Addition,” the Red Queen interrupted..

The .pdf copy of this book was made available to the public by Lenny de Rooy at her website, Lenny’s Alice in Wonderland Site.

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The Mechanical Arithmetic of the Pascaline

Blaise_pascalBlaise Pascal (1623-1662) is known today as a brilliant French mathematician, physicist, inventor, writer and Christian philosopher. In 1642, he attempted to help his father, a tax collector, deal with the repetitive arithmetic calculations that were part of the task of reorganizing the tax revenues of the French province of Upper Normandy. He was thus motivated to develop and invent the only functional calculator of the 17th century, known as the Pascaline, in 1645. It could add and (indirectly) subtract two numbers. It could also multiply and divide by repetition. In 1649 a royal privilege, by Louis XIV of France, gave him the exclusivity of the design and manufacturing of calculating machines in France. He designed the only functional calculator of the 17th century.


As far as I can tell, nine Pascalines still exist today. Four of them are on display at the National Conservatory of Arts and Crafts (CNAM) museum in Paris, France. There is also an interesting video that explains how this fascinating mechanical calculator worked. I encourage you to watch, learn and explore this intriguing bit of math history and technology.

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Addition and the Ishango Bone

In today’s blog, we are going to start a steady and cumulative process of exploring the concepts and techniques of mathematics in an evolutionary way. You will soon realize that even the most basic concepts can be seen in different and interesting ways. As always, and whenever possible, I’ll point you to some interesting places, events and people in the history of mathematics.  Lets start with the concept of addition. Addition evolved out of a very fundamental desire to count and know the quantity of similar objects we have.

Suppose we begin with two sets of objects (1). Set A consists of six elements (blue discs) and Set B has three elements (green discs). These two sets have no elements in common. The intersection of sets A and B is the empty set. Thus, they are referred to as disjoint sets. The union of sets A and B consists of both the six blue discs and the three green discs. In the Hindu/Arabic number system, there are numbers that are used to represent the quantities of discs in each set (2). In this case, it is the numbers “6” and “3.” The union of two disjointed sets is a visual way of representing the sum of two numbers. The addition symbol (+) is used to represent the operation of addition in a mathematical expression or equation. The sum of these two numbers is equivalent to counting the number of blue and green discs in the union of two sets (3).  In this case, six plus three is equal to nine.  One should note that counting is nothing more than the repeated addition of 1 to generate a set of natural numbers.

In 1960, a Belgian geologist, Jean de Heinzelin de Braucourt (1920-1998) was exploring an area of Africa near the headwaters of the Nile River at Lake Edwards, called Ishango. At the time, the area was part of the Belgian Congo. He discovered a large number of tools, artifacts, and human remains.  In the world of archeology, there have been numerous discoveries of prehistoric animal bones that have notches carved into them. These artifacts are generally known as tallies or tally sticks. Some of these notched bones have proven to be more interesting to the mathematical community than others. One of these interesting bones was discovered at Ishango by Jean de Heinzelin de Braucourt . Today, this artifact is referred to as the Ishango bone.  It is considered the second oldest mathematical object. However, its exact purpose has yet to be determined.

The Ishango bone is currently on display at the Royal Institute of Natural Sciences in Brussels, Belgium.  There, you will find a Flash-based website on the Ishango archeological site, which will give you a detailed explanation of the bone’s markings and its possible uses. Explore this site and you will find out how far back the history of mathematics stretches.

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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 Mayan seashell symbol for zero which is used to represent zero, a dot Mayan symbol for 1 unit, a dot used to represent one unit, and a stroke Mayan stroke symbol for 5 units used to represent five units. They used these symbols to represent the numbers 0-19 as follows:

The digits of the Mayan Number System

The vertical positions of the Mayan numbersThe 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.

A page from the Mayan Dresden CodexThere 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.

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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.

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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.”

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Unions (∪) and Intersections (∩)

Lets start exploring the various relationships between sets. Lets look at the first two type: the union and intersection of two sets. The union of two sets A and B is the collection of elements which are in A or in B or in both A and B (1). The symbol ∪ (2) is used to represent this concept in mathematical statements.  Such a relationship can also be expressed using a Venn diagram (3).   The orange areas represent the union of both sets. Given two sets, there is sometimes the need to know what elements the sets have in common. The intersection of two sets A and B is s a math term used to describe the collection of elements which are in A and B (1).  In mathematical statements, the symbol ∩ (2) is used to represent this concept. This concept of intersection can also be expressed in a Venn Diagram (3).  The orange area represents the intersection of Set A and B.

Lets visually consider two random lines, A and B, that are not parallel to each other. Set A is the collection of all points that make up line A. Set B is the collection of all points that make up line B. In Euclidian geometry, two line that are not parallel will always intersect each other. In this case, the intersection of Sets A and B consists of one point and only one point. Such visual representation of intersections  are found on all street maps. If you are out and about, you will notice that the intersection of any two streets represents the section of road that both streets have in common.

Most of the basic symbols of logic and set theory in use today were introduced between 1880 and 1920. The symbols ∩ and ∪ were introduced by Giuseppe Peano (1858-1932), an Italian mathematician, for intersection and union in Calcolo geometrico (“Geometrical Calculus”) secondo l’Ausdehnungslehre di H. Grassmann (1888).

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Sets and Elements

Human beings love to categorize things. Just look at all the nouns we have to describe groups of things in many ways. Our ability to categorize objects evolves out of our mental ability to notice the differences and similarities between two objects. The somewhat unconscious application of this skill gives us a chance to mentally focus on and communicate about many things at once. There is a visually mathematical way of representing the concept. A “collection of well-defined and distinct objects” is referred to as a set (B,C). It’s like the collections of marbles within the white circle (A).  Each of the objects within a given set is  an element. We can express these collections as a rule which describes the attributes common to all elements of the set. Visually, we traditionally use circles or ellipses (I believe any closed shape can be used) to represent the relationships between multiple sets. The resulting diagram is commonly known as a Venn diagram (D). The rectangle enclosing the sets represents the universal set. We’ll explore the most common relationships between sets in a future blog….

Mathematics, like most subjects, is rich in terms and vocabulary. In many cases, there is some fascinating history behind the words we commonly use to express ourselves mathematically. For example, the Venn diagram is named after John Venn (1834-1923), a British logician and philosopher, and a fellow of the Royal Society. Among the many things he did during his life, he published three texts on the subject of logic and he introduced Venn diagrams in his second book Symbolic Logic (1881).

This work was uniquely commemorated in a stained glass window which is on display in the dining hall of his alma mater, Gonville and Caius College, a college in the University of Cambridge, England. Read about his life online and explore his accomplishments. You will find out that he, like most mathematicians, are interested in a variety of things.

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π (Pi)

The most recognizable constant  in mathematics is the ratio of the circumference to the diameter of a circle. Today, we use the Greek symbol, π,  to represent this ratio. The existence of this constant has been known for a very long time.  It actually comes from nature, so it was always there, waiting for us to discover it.  The earliest textual evidence of this ratio dates back to the Babylonian and Egyptian civilizations.  One particular artifact fascinates me.  It consists of a list of eighty-four (84) practical problems encountered in administrative and building works.  It is known as the Rhind Mathematical Papyrus, now in the British Museum. It does not explicitly state the value of the ratio.   Instead, one of the problems calculates the area of a circle as the square of eight-ninths of its diameter.  The resulting solution was 3.16.  Explore this fascinating artifact.   Learn how math was used to solve practical problems in the world of the ancient Egyptian civilization.

By the way,  the symbol, π, has been used in this mathematical sense for only the past 250 years.  But that’s a topic for another blog.

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