Intro to Classical Arithmetic, Lesson 09.

There are three assignments for this lesson:

  1. Re-take all of your previous exams in Arithmetic for review.
  2. Study your Lesson.
  3. Complete your Memory Work.
  4. Complete the Lesson Examination.


In this course, we have studied Unity, but only in the simplest way. We learned that Unity is a known quantity used to measure unknown quantities. In this lesson, we will learn the definitions of Even, Odd, Prime, Composite and Perfect numbers, but all of that would be meaningless unless we knew why they are important in Arithmetic. In this lesson, we will take a deeper look at Unity and numbers and to begin we need to begin at the beginning.

As Christians, we are very familiar with the story of Creation in the book of Genesis:

“In the beginning, when God created the heavens and the earth, the earth was a formless wasteland, and darkness covered the abyss, while a mighty wind swept over the waters. Then God said, “Let there be light,” and there was light. God saw how good the light was. God then separated the light from the darkness.”

For Christians today, these words are well known and the ideas seem very simple. However, these words were written almost 3,500 years ago and when they were written they were very mysterious. The truth about the beginning of the world was made known to Moses by God Himself, but for thousands of years different wise men from many nations worked to understand when the world was made, by whom it was made and for what purpose it was made. Some of these men were very famous and had many followers who believed their explanations An explanation of the creation of the world is called a Cosmogony.

Some philosophers believed that God created the world, some that many gods created the world, some that the world was God, some that God was in everything in the world. Some philosophers believed that the world was made out of water. Some believed it was made out of fire. Some believed it was made out of air, earth, water and fire. One of the most famous ancient philosophers was Pythagoras and those who followed him were called Pythagoreans. Pythagoreans believed that the world was created out of numbers.

The World of Numbers

How in the world did Pythagoras come up with this idea that the world was created out of numbers? First, we should know that his teaching was very popular and be slow to laugh it off as silly. Many excellent men were Pythagoreans and most later philosophers made adjustments to his ideas rather than come up with ideas of their own. Once understood, his ideas are not very strange at all. In fact, they may be seen to have a lot in common with the creation story in the book of Genesis. It is believed that Pythagoras himself learned some of his ideas from the Egyptians and from the writings of Moses.

Pythagoreans believed that everything in the world was either a multitude or a magnitude. All of the world was created from God’s knowledge of numbers and Wisdom was the knowledge of these. To get Wisdom, the Pythagoreans taught that man had to master the four mathematical arts: Arithmetic, Geometry, Music and Astronomy–which were necessary to study multitude and magnitude. Of these, Arithmetic was the first and most important because it was by the knowledge of numbers that God created all things.

Now, if the world was created by numbers, where would that Creation begin? It would begin with the beginning of all numbers: Unity. In Greek, “Unity” was called the Monad, which comes from the Greek word for one: monos. This idea of the Monad became the Pythagorean idea for God, who was the beginning of all things–the One.

When the Pythagoreans thought about God, they thought of Him as the One (Monad). When He created the world, they referred to this as the Two, or Dyad. Since God was believed to be the greatest Good, One was seen as good. Since the created world was seen to be filled with evils, Two was seen as evil. From this, the Pythagoreans noted many different opposites in the world, such as Monad and Dyad, Even and Odd, Male and Female, Hot and Cold, Light and Dark, etc. These opposites guided them as they developed a way of life that was good in many ways because they believed that the goal of life was to maintain a balance or harmony between good and evil.

Even and Odd Numbers

When we look back at the passage from Genesis 1, we see that the Earth was, “a formless wasteland”, covered with darkness and in this it appears that God begins with a world of chaos and disorder and then acts to bring it under control. The created world, when first made, was wild and the forces of nature had no limits. The seas raged, the winds blew, the elements of the world were disordered and out of harmony. All was chaos and could have been very evil. However, God was not finished with creation, and continued to create those elements which limited the chaos and brought order to the world. He created light to limit the darkness. He formed the dry lands to limit the seas, He created the stars, moon and sun to give limits to time. He commended the animals to reproduce, but by making different kinds of animals set limits on how much each species could grow. Finally, God made man to rule over the earth and bring it under control. When creation was completed, all was brought into harmony and the order of the world (called the Cosmos in Greek) was completed.

Now, we can learn all of that from the book of Genesis, but the Pythagoreans taught that behind all of this was God’s knowledge of numbers and that these numbers provided the reasons behind all things. In the beginning was the Monad (One) and from it came the Dyad (Two) and all other numbers—and all other created things, which are made of numbers. Every line in the world was made of two points. Every triangle of three lines, and so on through all the world. Moreover, among the things made, there were two kinds of elements: unlimited things (bad) and things that limited them (good). Two great forces thus filled the world, all coming from numbers. Of the numbers, monad was good and dyad evil. Every number that could be measured by the Dyad was considered evil and every number that was not measured by the dyad–but by the monad alone–was considered good. These good numbers we call Odd numbers. These evil numbers we call Even numbers. Why did they think this?

We can divide an even number into two equal parts. For example, the number 4 can be divided into 2 and 2. An odd number cannot be divided into two equal parts because its division is limited by the Monad. For example, the number 5 can be divided into 2 and 2 with 1 left standing between them. Thus, if we consider unlimited things as evil and limited things as good, the Pythagoreans would see odd numbers are good since their division is limited and even numbers as evil because they are unlimited.

Prime and Composite Numbers

Above we have learned that Unity–or the Monad–or One was considered by the Pythagoreans to be a sacred number, representing God who is the source of all things. Even and odd numbers were valued because of their relationship to One. Another relationship in numbers that interested the Pythagoreans was whether a number was measured by One alone. For example, the number 4 can be measured by 1 taken 4 times, and it can also be measured by 2 taken 2 times. However, there are many numbers that are composed of Ones alone. For example, 3 cannot be measured by any number other than 1.

The Pythagoreans believed that these numbers, called Prime numbers, were special because they were composed of no other number besides One. The others, called Composite numbers were composed of other numbers and were therefore less “pure” than the Prime numbers. Prime numbers include: 3, 5, 7, 11, 13, 17, 19, 23 and so on.

Perfect Numbers

In addition to even and odd numbers and prime and composite numbers, another relationship was also sought out by the Pythagoreans. This relationship has to do with the aliquot parts of a number-so take a moment and remind yourself of what they are.

The Pythagoreans noticed that there were rare numbers that had a special relationship to their aliquot parts. It was noticed that or some rare numbers, the number itself was equal to the sum of the aliquot parts–not including the number itself. For example, the number 6 has the aliquot parts 1, 2, 3 and 6. When we collect the aliquot parts 1, 2 and 3 they form a total of 6, which is equal to the original number. The number 6, therefore, is called Perfect number.

In ancient times, only four perfect numbers were known: 6, 28, 496 and 8128. Today computers have allowed mathematicians to find many more, but the goal is not to find perfect numbers using a computer. The goal for the Pythagoreans was to find those special numbers that would help us understand the world. The number 6 has been called the perfect number throughout history and it is no surprise to find that God created the world not in any random number of days, but in 6 days–a perfect number.


In this lesson, we have spent some time looking into the ancient world of numbers, but understand that this is a very simple look into the mysteries of the Pythagoreans. The reason why this philosophy is so important is because it is the reason why many of the ideas we have in Arithmetic exist. Most students (or teachers!) cannot tell you why they learn about even or odd numbers, or prime numbers at all–and they usual don’t learn anything at all about perfect numbers. The reason why is because modern schools teach Arithmetic only for its uses in everyday life–not the important religious ideas that it was used to communicate by wise men of the past. We will continue to learn about ancient ideas about numbers in future lessons. Let this be enough for now.

Memory Work

Directions: The following questions help you to memorize the most important points of this lesson. Commit them perfectly to memory and have a parent or praeceptor quiz you to test your mastery before taking your lesson exam.

  1. What is an Even Number?
    An Even Number is a number which is measured by 2, or that can be divided into two equal halves.
  2. What is an Odd Number?
    An Odd Number is a number which is one greater than an even number, or that cannot be divided into two equal halves.
  3. What is a Composite Number?
    A Composite Number is a number which is measured by some one or more numbers beside Unity, as 4, 6, 8, 9, 10, 12, 14, 15, etc..
  4. What is a Prime or Incomposite Number?
    A Prime or Incomposite Number is a number which is measured by no number but Unity, as 3, 5, 7, 11, 13, 17, 19, etc..
  5. What is a Perfect Number?
    A Perfect Number is a number which is equal to all of its aliquot parts taken together, as the number 6, whose aliquot parts are 1, 2 and 3.

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