There are three assignments for this lesson:
- Study your Lesson.
- Complete your Memory Work.
- Complete your Lesson Exercises.
- Complete the Lesson Examination.
In our last lesson, we learned about some of the ways of expressing quantities: by number and species. In this lesson, we will simply learn the next set of axioms in Arithmetic that applied to these. We are learning these for use LATER, so it is important simply to understand their language by reading through them carefully and memorizing them exactly. These are the tools with which we will solve problems later–we want as many tools in out toolbox as we can get.
Review: Reason and Proofs
In Arithmetic, reason allows us to prove whether statements made are true or false. When we wish to demonstrate or show that a statement us true or false, we use Proofs. A proof is a connection of arguments used to prove that a statement is true or false. We must show that what is said follows from truths that have already been proven to us or that need no proof. Statements that need no proof are called Axioms.
The most basic rule of learning is that we use what we know to understand what we don’t know. For example, if we don’t know how long a piece of wood is, we use the units on a measuring tape (whose length we do know) to measure it. We learned that this is the basic idea of measurement: counting how many times a known quantity is contained in an unknown quantity.
In life, we will face many complicated problems that need to be figured out. What we need to solve these problems is a set of axioms, rules and definitions that could be used to prove our answers to be true and not lazily guess. Of these, axioms are the most important because they are self-evident. What that means is that axioms are statements that need no proof–because everyone can see clearly that they are true.
We begin to write a proof by stating what it is that needs to be demonstrated (demonstrandum). This will be a statement known as a theorem. From the theorem, you will simply work backwards to show how the theorem flows logically from the axioms, rules and definitions you have already learned. Once we have completed our proof, we end with the phrase, “which was to be demonstrated”.
Theorem: Demonstrate that 4 cups is equal to 2 pints.
Proof: We know that 4 cups is equal to 1 quart and that 2 pints is also equal to 1 quart. Axiom III states that quantities which are both equal to one and the same third are equal to one another. Since 4 cups and 2 pints are both equal to 1 quart, then 4 cups must be equal to 2 pints, which was to be demonstrated.
This is a proof and, as we will learn, we can write them in a number of different ways. For now, we will continue to work to understand how to use our axioms to reason out our proofs.
Thus far, we have learned the first five axioms–which cannot be forgotten:
- Axiom I. Anything may be assumed as unity.
- Axiom II. Every quantity is equal to itself.
- Axiom III. Quantities which are both equal to one and the same third are equal to each one another.
- Axiom IV. The whole is more or greater than its part.
- Axiom V. The whole is equal to all of its parts taken together.
Now, let us add five more axioms.
Axiom VI. Every lesser homogeneous number is contained in a greater either as an aliquot or aliquant part.
In the last lesson, we learned the definition of aliquot and aliquant parts, and this is an important topic in Arithmetic. The relationships between numbers will be very important as we get further into Arithmetic and we need to master these concepts now so that we will be comfortable working with them later. Let’s take a look at why they matter.
In our study of Arithmetic, if you can count you could solve just about any Arithmetic problem in the world. Unfortunately, many of them would take you a very long time to count out. For example, if I ask you, “How many is 5 and 2 more?”, you can count: five…six…seven! However, if I ask you, “A man had 12 boxes of pencils and every box had 385 pencils. How many pencils did he have in all?”–you’d be counting for a long time. By studying Arithmetic, we simlpy learn faster and easier ways to solve difficult problems. We could count them out, but we are seeking ways to do it more quickly. That’s really all Arithmetic is about.
If you have a great number of objects and you wish to divide it into parts, how do you know how many parts to divide it into to make each group equal? For example, if you had 24 eggs and you wanted to divide them into groups so that each group would have an equal number of eggs, how would you do it?
If you knew what were the aliquot parts of 24, you’d have no problem! However, if you don’t know the aliquot parts of 24, you’ll have to guess how many groups there should be and count the eggs out–the slow way. Fine, but slow.
The aliquot parts of 24 are 1, 2, 3, 4, 6, 8 and 12. That means that if you take any one of these numbers a certain number of times, the sum will equal 24. For example, if the number 3 is taken 8 times, or if 6 is taken 4 times, the sum will equal 24. The numbers 3 and 6 are aliquot parts of 24.
If you take any of the numbers 5, 7, 9, 10, 11 or any number between 12 and 24 a number of times, their sum will never equal 24, because they are aliquant parts. Thus, if the number 9 is taken 2 times, the sum is 18, but if taken 3 times, the sum is 27. Taking 9 a number of times will never equal 24 because 9 is an aliquant part of 24. These relationships between numbers can be very useful for solving problems quickly, and we will study them in more detail down the road.
Our axiom states plainly that any lesser number of the same kind must be either an aliquot or aliquant part of a greater number. Thus, the number 3 must be either an aliquot or aliquant part of 10 (it is an aliquant part). The number 5 must be wither an aliquot or aliquant part of 10 (it is an aliquot part).
Remember: ALIQUOT parts can equal a certain greater whole when taken a certain number of times, while ALIQUANT parts can’t.
Axiom VII. Every number is contained in itself once.
Axiom VIII. Every lesser number is contained in a greater more than once.
The two axioms above assist us in demonstrating a number of basic truths in Mathematics. First, anytime we wish to prove that one number is equal to another, we can do so by showing that the one number contains the other exactly once, using Axiom VII. Second, anytime we need to write the number 1 in another way (which we will down the road) we can write it as a fraction showing that a number is divided by itself (more about this when we study division and fractions). Third, any time we wish to prove that a number is less than another, we simply need to show that it is contained in the greater number more than once.
You should also see some similarities between Axioms VII and VIII and Axioms II and IV learned in lesson 07. Review them and see if you can notice the similarities.
Axiom IX. The greater any number is in comparison to another, the more equal parts will it contain of the other.
In Axiom III, we learned that quantities which are equal to one and the same third are also equal to each other. Here in Axiom IX, we find a related truth. If two numbers are compared with one and the same third and we find that one number contains the third a greater number of times than the other number does, this proves that the first number is greater than the second. Read that again slowly!
So, as an example, how would we prove that the number 12 is greater than the number 8? We would use Axiom IX and reason like this:
Theorem: Demonstrate that 12 is greater than 8.
Proof: We know that the number 4 is contained twice in 8, but three times in 12. Axiom IX states that the greater any number is in comparison to another, the more equal parts will it contain of the other. Since 12 contains 4 three times and 8 contains 4 only two times, 12 must be greater in comparison to 4 than 8 is, and is therefore the greater number, which was to be demonstrated.
Axiom X. The Nearer any lesser number approaches to being equal to a greater number, the less often it will be contained in that greater number.
Like Axiom IX, here we find another relationship between greater and lesser numbers that can be used to demonstrate many different statements. Here is an example:
Theorem: Demonstrate that 6 is greater than 3.
Proof: We know that the number 6 is contained twice in 12, but 3 is contained 4 times in 12. Axiom X states that the nearer any lesser number approaches to being equal to a greater number, the less often it will be contained in that greater number. Since 12 contains 6 two times and 12 contains 3 four times, 6 must be nearer to the greater number than 3 is, and therefore greater than 3, which was to be demonstrated.
In this lesson, we have simply added eight new axioms with a brief description of the possible uses of each. We will use these axioms to demonstrate the truth or falsehood of statements made in Arithmetic in the future. We have reviewed the fact that demonstration is made by proofs which begin with a theorem to be demonstrated and move step-by-step through the reasoning that joins the theorem (which is unknown) to the most sure axioms (which are known). In your exercises below, you will continue in practicing to your own proofs.
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.
Note: It is recommended that students review Axioms I-V and the method for writing proofs in Lesson 07.
- AXIOM VI. Every lesser homogeneous number is contained in a greater either as an Aliquot or an Aliquant part.
- AXIOM VII. Every number is contained in itself once.
- AXIOM VIII. Every lesser number is contained in a greater more than once.
- AXIOM IX. The greater any number is in comparison to another, the more equal parts will it contain of that other.
- AXIOM X. The nearer any lesser number approaches to being equal to a greater number, the less often it will be contained in that greater number.
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