Expressing Mathematically – Part 2

In one of my earliest blogs (“Expressing Mathematically”), I introduced the idea that mathematical concepts and skills can be expressed in different ways. Our senses identify natural and artificial objects {world model} that convey math concepts. We use language to identify and communicate our understanding of the sensed math concept written and spoken ways {language model}. We also use geometric curves, shapes and volumes to represent these objects {visual model} in a 2D or 3D mathematical world. We can also plot our visual models on a coördinate plane or volume {graphical model} to generate numerical data about the geometry. Finally, we can express mathematical concepts and skills using symbols, numbers, variables and equations {numerical model}. Keep in mind that all these models are different aspects of what we generally refer to as mathematics. I will use these terms often throughout future blog to explore math concepts and skills through these unique perspectives.

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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.

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.

Decimals

Today, we are going to take a closer look at the elements that make up the set of natural numbers. If you remember from a previous blog, this set represents all positive integers within the set of real numbers.

All real numbers (which include natural numbers) are made up of one or more digits. The digits we use in our modern numbering system are 0,1,2,3,4,5,6,7,8,9. Note that there are ten numbers that are made up of one digit (0-9). There are ninety two-digit numbers (10-99), and there are one thousand three-digit numbers (100-999). All real numbers can be expressed in this decimal notation, even fractions. This will be looked into in a future blog.

The position of the digits within a numeral is as important as the digits themselves. The numeral “476” is not the same as the numeral “674” even though both numerals use the same three digits: 4, 7, and 6. In order to understand why this is the case, we need to understand the decimal number system. I’ll show the structure of this system by illustrating the number “six hundred and seventy-four” in three different ways.

In this system, each position represents a multiplier used to determine the magnitude of the digit in its slot (A). The rightmost slot in any number has a multiplier of 1. The slot to its left represents a multiplier of 10. The third slot from the right represents a multiplier of 100. Theoretically, this can continue indefinitely.  Very large (or very small) multipliers can also be expressed using scientific notation.Each slot represents a different multiplier that is ten times more than the slot to the right.  Since all the multipliers in this system are multiples of ten, the decimal notation system is also referred to as a base-10 system. The arithmetic expression (B) is a way of expressing the relationship between the multipliers and its digits. The number (C) is a shorthand way of expressing this arithmetic expression.  There are other numerical systems that use a multiplier other than ten.  We’ll explore these systems and their practical applications in a future 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.

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.”

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).

Word Problems

One of the most common forms of problems in mathematics are word problems.  They are essentially mathematical exercises presented in the form of a hypothetical situation that requires an equation to solve.  They are usually expressed in a narrative form.   I consider this a verbal (literary) form of expressing math (See the blog “Expressing Mathematically”).  There are two insights I want to share with you about word problems:

Hypothetical Situations
First, when I solve word problems, I am actually executing at most five steps, not necessarily in the order given: (1) I translate the hypothetical situation into a visual expression, like a diagram using geometric concepts, or straight into an equation or expression. (2) I find the values that we are given in the  word problem, and (3) I find the variable(s) we need to solve for (the “unknown”) to find the solution to the hypothetical situation. (4) I then decide which concepts and/or equations we need to apply to the hypothetical situation. (5) Finally, I apply the proper skills from our math toolbox to find the solution to the word problem.

Actual Situations
Second, a traditional word problem explicitly supplies the students all the necessary information to do steps (1),(2),and (3).  When I am presented with  a problem in a real-world situation, I usually have to define the situation in a way that I can solve it mathematically and express it as a word problem, or a labeled diagram.  Many times, I have to conduct measurements in order to establish the “given” values.  Usually, the “unknown(s)” end up being those values I need to know that I cannot measure in a practical or direct way.  It usually takes some work to properly define the situation that is usually given in a classroom-type word problem.

Word problems are one of many ways we can test the competency of a math student.  They are also effectively used to help the math student achieve mastery in applying certain skills.  I understand that it is impractical to consistently present students with real-world situations to solve.  However, it is just as important for students to know how to properly generate a problem as it is to solve it.

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.