To complete this lesson, complete the following tasks:
1. Study your Lesson.
2. Complete your Memory Work.
3. Complete the Lesson Examination
We have spent some time learning about different units used by men to measure quantities. The last few lessons simply provided some helpful examples of how known quantities are used to measure unknown quantities. As we discussed the different units, we learned about the relationships between them. For example, we learned that twelve inches are equal to one foot and that four quarts equal one gallon. In your lesson quizzes, you were asked to arrange different quantities in order from least to greatest. Those exercises were preparing you for the study of Comparison, which is the study of how things are similar or different. In mathematics, it is very important that we know how to rightly compare quantities.
Equality and Inequality
Let’s begin by clarifying a number of important definitions. When we use the term “Equality”, we mean that two or more things agree in their quantities. If these things did not agree in their quantities, we would call this “Inequality”. If we compare two multitudes, say five apples and seven apples, we will see that the quantities are not referred to Unity (one) in the same way. This means that they are not equal in multitude. However, if we compare five apples with five apples, we will see that both quantities are referred to Unity in the same way and are therefore equal in multitude. If we compare two magnitudes, say a line that is five inches long and a line that is eight inches long, again, we see that the two quantities are not referred to the unit in the same way and are therefore not equal in magnitude.
The ability to judge whether two quantities are equal or unequal is extremely important because when we know that one quantity is equal to another, this tells us that whatever we can say about one of the quantities can also be said of the other. In our next lesson, we will begin working with Axioms, which are the truths from which all of our mathematical reasoning will begin and you will see how important equality is. You’ll have to wait until then to find out how.
More and Fewer
When we are comparing two multitudes, we may wish to know more than whether they are equal or unequal. Once we know that two multitudes are not equal, we may wish to know whether one multitude has more or fewer parts than another. For example, if we are offered two baskets of apples and we know that one basket has seven apples and the other basket has nine apples, we may wish to know which basket has more, or which has fewer.
Everyone has a general idea of more and fewer, but we cannot pursue truth with general ideas. We must understand exactly what is meant when we say that one multitude is more or fewer than another so that we can reason from this idea to others and use these reasonings to solve complicated problems in the future. We will begin to learn how this works in our next lesson.
When we say that one multitude is more than another, we are saying that a part of it is equal to the whole of the other. This is what “more” really means in mathematics.
Alternatively, when we say that one multitude is fewer than another, we are saying that the whole of that multitude is equal to a part of the other. This is what fewer really means in mathematics.
Greater and Lesser
When we are comparing two magnitudes, we may wish to know more than whether they are equal or unequal. Once we know that two magnitudes are not equal, we may wish to know whether one magnitude is greater or lesser than another. For example, if we are told of two containers, one which measures seven gallons and another which measures ten gallons, we may wish to know which container is greater and which lesser. As with more and fewer, everyone has a general idea of greater and lesser, but we need to be exact for the sake of reasoning.
When we say that one magnitude is greater than another, we are saying that a part of it is equal to the whole of the other. This is what greater really means in mathematics.
And when we say that one magnitude is lesser than another, we are saying that the whole of the one is equal to a part of the other. This is what lesser really means in mathematics.
Whole and Part
In speaking of more, fewer, greater and lesser, we came across the terms whole and part, which we should clarify for the sake of our mathematical studies. We learned that Unity or a unit is a known quantity referred to as “one”. It is clear, then, that any whole must be a unity. However, when we use the term whole, we are not focusing on a quantity as the unit, but on the quantity as a collection of parts. We do this when we wish to study the relationships between the parts of that whole and the whole itself.
As we will see in the next lesson, these relationships are very important in reasoning because anything that may be said of a whole may also be said of the collection of all of its parts.
Again, knowledge of the relationship between the parts and the whole also leads us to the study of complements. Don’t get this word confused with compliments (with an “i” in the middle), which are nice things said of a person. Complement (with an “e” in the middle) comes from the Latin word complere, which means to complete or fill out. The complement of a part is the rest of the parts that complete the whole. This idea is very important in reasoning because whatever can be said about a whole can be said about the collection of a part and its complement. This simple idea will help you to understand many important problems in mathematics. For example:
If Mary had a gallon of water in a bucket and she poured out a part of it into a bottle that held one quart of water. You can figure out the complement of that part. The complement is three quarts.
If Joseph had a piece of wood that was 10 feet long and he cut a part of the wood to measure 4 feet, you can figure out the complement to that part. It’s 6 feet.
The reason why this is important is because what can be said of the whole can also be said of a part and its complement. If Joseph’s 10 foot board weight 10 lbs., how much would the collection of the part and its complement weigh? The same!
However, we must always watch out for a trap we can fall into when we reason about a whole and its parts. Note that the lesson teaches that whatever can be said of a whole can be said of the collection of its parts. It does not say that whatever can be said of a whole can be said of its parts by themselves, or that what can be said of each part can be said of the whole when they are collected together. You will see how important this is in the future.
It would be perfectly fine for you to write “Twelve inches is equal to one foot.” However, just as men have established different units of measurement, we have also established symbols that can help us to write things in a shorter and easier way that everyone can understand–even if they speak different languages!
Equality can be written with the symbol = . Therefore instead of writing out the sentence above, you could more easily write, “12 in. = 1 ft.”. You can see that the lines of the symbol show that the two quantities are in agreement.
Inequality can be written with the symbol ≠ . If you wished to write “One mile is not equal to one yard.”, you could more easily write, “1 mi. ≠ 1 yd.”. You can see that the lines of the symbol show that the two quantities are not in agreement since the equal sign is crossed out.
However, if we wish to express the details of an inequality, we use the symbols for Majority and Minority. Inequalities are always written in order from left to right.
Majority is symbolized with the symbol >. By majority, we mean either more or greater. So if we wanted to write “One yard is greater than two feet.”, we could write, “1 yd. > 2 ft.”. You can see that the lines of the symbol show that the quantity on the left is greater than the one on the right.
Minority is symbolized with the symbol <. By minority, we mean fewer or lesser. So if we wanted to write “One pound is less than one ton.”, we could write, “1 lb. < 1 ton.”. You can see that the lines of the symbol show that the quantity on the left is lesser than the one on the right, and that the symbol for minority is simply the symbol for majority turned around.
In this course, we have studied the important concepts of Quantity, Multitude, Magnitude, Unity and Units. In this lesson, we begin our move into mathematical reasoning, which is necessary for problem solving. We have studied the concepts of Equality and Inequality, More and Fewer, Greater and Lesser and have seen the symbols with which these concepts are written. In our next lesson, we will begin the study of Axioms–the truths from which all of our reasoning will begin.
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.
50. What is Equality?
Equality is the agreement of things in Quantity.
51. What is Inequality?
Inequality is the disagreement of things in Quantity.
52. When are things Equal in Multitude?
Things are Equal in Multitude when they are referred to Unity in the same way.
53. When are things Equal in Magnitude?
Things are Equal in Magnitude when they are referred to the same Unit.
54. What is meant by the terms “More” and “Fewer”?
Of two Unequal Multitudes, one is said to be “More” that has a part equal in Multitude with the Whole of the other Multitude; one is said to be “Fewer” whose Whole is equal in Multitude to a Part of the other.
55. What is meant by the terms “Greater” and “Lesser”?
Of two Unequal Magnitudes, one is said to be “Greater” that has a part Equal in Magnitude with the Whole of the other Magnitude; one is said to be “Lesser” whose Whole is Equal in Magnitude to a Part of the other.
56. What is a Whole?
A Whole is a collection of things taken as a Unity. A bushel of wheat is a whole.
57. What are Parts?
Parts are those things collected in a whole.
58. What is a Complement?
If any one Part of a Whole is assumed, then the rest of the parts are called the Complement of that part to the whole.