To complete the objectives of this lesson, complete the following tasks:
- Study the Lesson.
- Complete your Memory Work.
- Complete your Lesson Exercises.
- Complete the Lesson Examination.
Thus far in Arithmetic, we have learned the different branches of Mathematics: Arithmetic, Geometry, Music and Astronomy. We have learned about Unity and units. We have learned about Comparison: equalities, inequalities, majority and minority, and in the last lesson we learned the first axioms of Arithmetic and how Proofs work. If you are uncomfortable with these subjects, you should go back and review.
In this lesson, we will move on to study how quantities are expressed with numbers and symbols. Our goal at this point in Arithmetic is to learn these definitions exactly. We will work with these ideas in future lessons and should have their basic meanings mastered before we do so.
The Language of Arithmetic
In Grammar, we are studying the languages that we will need to study Philosophy and Literature in the future. We are learning how different ideas are spoken with words and how different sounds are written with letters. We need to do the same thing to study Mathematics. We need to learn the language of Arithmetic.
Just as we use letters in Grammar, we use symbols in Arithmetic. In Grammar, we learned that a noun is a name and we use nouns to call things by their names. We may call a boy “Joe” or “David”, or we may call an animal “cat” or “turtle” when we know their true name. At other times, we may need to speak of something and not know its name! In those cases, we use special words like “someone” or “a certain man”, “some animal” or “a certain animal”. Often, it doesn’t matter what the person or thing’s exact name is since we’re not worried so much about who or what they are, but what they did. For example:
A certain man ran up to Jesus and asked him, “What must I do that I may receive everlasting life?”
In this sentence, we really don’t care who the particular man was, we are interested in his question and what Our Lord’s answer would be. The question could have been asked by any man and the answer is universal–it applies to all men. We could also write this sentence as follows, allowing you to fill any man’s name in:
Man X ran up to Jesus and asked him, “What must I do that I may receive eternal life?”.
In the language of Arithmetic, we often do the same thing. We will need to express quantities with words and symbols. Sometimes we will know the quantities exactly, sometimes we will not. Sometimes we will wish to express the exact quantity and other times it will not matter to us because we will be seeking some other quantity–or a solution that applies to any quantity. Let us then begin our study of the language of Arithmetic.
Species and Number
Quantities can be expressed in two different ways, just like names. It’s important to realize that when we don’t know a person’s name, their name doesn’t change, only our expression of it. In the same way, we should not think the quantities themselves change, but only the way we express them, since sometimes we know the quantity exactly and sometimes we don’t. We may also express the same quantity in several different ways. Let’s work to understand this more clearly.
Sometimes we will express a quantity definitely. That means we know the quantity exactly and wish to express it in an exact way. To do this, we use Numbers. If we saw a bushel of apples, we could count the apples and express the quantity in a definite way using a number.
There are seventy apples in the basket.
This would answer a question such as, “How many apples are in the basket?”. Since we want to know the quantity exactly, we will answer with a number.
At other times, we will express a quantity indefinitely. This means that we either do not know the quantity exactly or do not wish to express it in an exact way. To do this we use Species. If we saw a bushel of apples covered with a cloth and didn’t know how many apples there were, but wanted to know how much they weighed, we could express the quantity with a symbol:
How much do x apples weigh?
The point here is that the exact number of apples is not known and really doesn’t matter to us. If a friend weighed them and said they weighed 40 lbs., we would say,
“A bushel of x apples weighs 40 lbs.”
That would be good enough for us. To express a quantity as a Species we use lowercase letters. Don’t let the letters confuse you–they’re only symbols that express some quantity just as a number would.
Integers and Fractions
When we express a quantity with Numbers, we may do so in two different ways: as Integers or as Fractions.
We may express a quantity with an Integer, or whole number. The word Integer is simply the Latin word for “whole” or “unbroken”, so don’t let that confuse you. For example, St. Paul says,
“And may the God of peace himself sanctify you in all things: that your whole spirit and soul and body may be preserved blameless in the coming of our Lord Jesus Christ.”1 Thessalonians 5:23
In that verse the Latin for “whole spirit soul and body” is integer spiritus anima et corpus. Integer simply means “whole”. Therefore numbers like 1, 2, 3, 5, 30, 100 and so on are all integers or whole numbers–they’re not broken into parts.
The definition in your memory work needs some explanation. When we express a quantity with an integer we are expressing it in a way that refers to Unity (one) as though Unity is a part of the quantity. Thus, the integer 3 should be understood as a quantity that contains Unity three times.
We may express a quantity with a Fraction, or broken number. The word Fraction is simply the Latin word for “broken”, so don’t let that confuse you. Again, in the Bible, the disciples speak about “the breaking of bread”, which is fractio panis. Therefore, let it be clear in your mind that integer simply means unbroken or whole and fraction means broken.
When we express a quantity with a fraction we are expressing it in a way that refers to Unity (one) as though Unity is the whole and the quantity is a part of it. Thus, the fraction one-half should be understood as a quantity that is contained two times in Unity.
This all becomes very clear if we remember our old memory work and look at an example. Remember that our first axiom stated that anything may be taken as Unity. Therefore, if we wished to express a quantity, we would first need to determine what Unity would be–and we could do this in more than one way. If we had a length of rope and wished to express the length we might set unity as one foot. Then, we might say that the length of the rope was 2 feet and express the quantity as an integer. If we set Unity as one yard, we would express the length as a fraction–as two-thirds of a yard. In the first expression, we referred to the quantity (rope length) as a whole to a part with an integer. In the second expression, we referred the quantity (rope length) as a part to a whole with a fraction.
Aliquot and Aliquant Parts
As we discuss fractions, it is a good time to discuss the two different kinds of parts: Aliquot and Aliquant. The concept is really very simple to understand and will be very helpful in the future. Earlier in this course, we learned the definition of a Part as follows:
What are Parts?
Parts are those things collected in a whole.
When we consider a part of a whole we may find that the part is contained in the whole a certain number of times. For example, if we take 6 as the whole and 2 as a part, we will find that 2 is contained exactly 3 times in 6. Again, if we take 8 as the whole and 4 as the part, we will find that 4 is contained exactly two times in 8. These are called Aliquot Parts.
Again, there are other parts which when taken a number of times can never equal the whole. For example, if we took 6 again as a whole and 4 as a part, we find that 4 taken one time is less than the whole and 4 taken two times is greater than the whole. If we took 8 as the whole and 3 as a part, we would find that 3 taken two times is less than 8 and 3 taken three times is greater than 8. These are examples of Aliquant Parts.
In this lesson, we begin to understand how quantities are expressed in speech and in writing. This, however, is only the beginning of this study. In our next lesson, we will learn another set of axioms and continue our study of Classical Arithmetic.
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.
- REVIEW: What is a Quantity?
Quantity is any thing that may be increased or diminished.
- REVIEW 19. What is a Homogeneous Multitude?
A Homogeneous Multitude is a multitude that is made up of things of the same kind, such as a pile of pennies.
- REVIEW 20. What is a Heterogeneous Multitude?
A Heterogeneous Multitude is a multitude that is made up of things of different kinds, such as a pile of mixed coins.
- By what two terms is Quantity expressed?
Quantity is expressed by the terms Species and Number.
- What is Species?
Species is a term that expresses quantity indefinitely and universally, such as “a certain number”, “some” etc…
- Into what two classes are Species distinguished?
Species are distinguished into Known (Given) and Unknown (Sought).
- How are Species of Quantities signified?
The Species of Quantities are signified by lowercase letters: Unknown quantities by the first letters of the alphabet (a, b, c, d, etc..); Known quantities by the last letters (u, x, y, etc.)
- What is Number?
Number is a term that expresses quantity definitely and particularly, such as one, five, seven, and so on.
- Into what two classes are Numbers distinguished?
Numbers are distinguished into Integers and Fractions.
- What is an Integer?
An Integer, or Whole Number, is that which is referred to Unity as a Whole to a Part as, 1, 2, 3, 4, etc..
- What is a Fraction?
A Fraction, or Broken Number, is that which is referred to Unity as a Part to a Whole as, 1 half, 2 thirds, 1 third, 3 fourths, etc..
- What are the two kinds of parts?
The two kinds of parts are Aliquot and Aliquant Parts.
- What is an Aliquot Part?
An Aliquot Part is a part which, being repeated a number of times, becomes equal to the whole; as 4 is of the numbers 8 and 12.
- What is an Aliquant Part?
An Aliquant Part is a part which, being repeated a number of times, always exceeds or falls short of the whole, as 5 is of the numbers 8 and 12.
Note: Species and Number may be Homogeneous or Heterogeneous, just like the Quantities they express.
Mr. William C. Michael, O.P. is the founding headmaster of the Classical Liberal Arts Academy. Mr. Michael is a Lay Dominican in the Catholic Church and is a homeschooling father to ten children, all of whom have studied in the Academy. He graduated from Rutgers University with an honors degree in Classics & Ancient History, is a member of Phi Beta Kappa and is currently studying at Harvard University. Mr. Michael has worked in private education as a Classics teacher and administrator for over 20 years. Mr. Michael is known for his talks on the Academy YouTube channel and his sponsorship of Classical Catholic Radio.