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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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.

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.