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Intro to Classical Arithmetic, Lesson 08. Axioms III

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.

Lesson

In our last lesson, we learned about Ration and Measure. In this lesson, we will learn the next set of axioms in Arithmetic that apply to these. Remember that 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. In this lesson, we will add five more axioms. To view and print a summary of the axioms learned through this lesson, use the link below:

AXIOM XI. If two numbers are increased or decreased by like Parts of themselves, their Ratio remains the same.

As we learned in our last lesson, the goal of Arithmetic is to understand as much as we can about the relationships between numbers. Our reason always wants to understand relationships and wise men have been working since the earliest days of history to understand numbers. Our goal is to learn what they have discovered and continue learning in our lifetime.

We studied Ratio or the relationship between two numbers, but only in the simplest way. We must now note that it is always helpful to state a ratio in the simplest possible way. For example, if we take the numbers 1 and 2, their ratio is 1:2. If we take the numbers 2 and 4, we could write their ratio as 2:4. However, what we want to understand in a ratio is the relationship between the two numbers. We can see that 4 is twice as many as 2 and therefore we can write the ratio between 2 and 4 as 1:2. We can do this because 4 is twice as many as 2, just as 2 is twice as many as 1. This would be the easiest way to write the ratio 2:4 and it would most clearly show us the relationship between the two numbers.

Here are some more examples:

The ratio:

  • 1:1 is equivalent to the ratios: 2:2, 3:3, 4:4, 5:5, 6:6, 7:7, etc..
  • 1:2 is equivalent to 2:4, 3:6, 4:8, 5:10, 6:12, etc..
  • 1:3 is equivalent to 2:6, 3:9, 4:12, 5:15, 6:18, etc.
  • 2:3 is equivalent to 4:6, 6:9, 8:12, 10:15, etc..

You can see that ratios can often be written in simpler forms to better show their relationship and we will almost always want to do so because this is most meaningful way to understand the relationship between two numbers.

Now, what happens when the numbers in a ratio change? Does the relationship between them stay the same if we increase or diminish them? Axiom XI answers this question for us. Read it carefully and commit it to memory.

It is important to note what Axiom XII is not saying. The axiom is not saying that if we add the same number to two numbers their ratio remains the same. Let’s look at an example to see this.

If we begin with the the numbers 2 and 4, we can write their ratio as 2:4 or more simply as 1:2. If we add 2 more to both numbers, we will have a ratio of 4:6, but this can no longer be written as 1:2. The ratio 4:6 is no longer written simply 1:2, but you will see that it is 2:3. Therefore, adding the same number to both numbers in a ratio will not allow the ratio or relationship to remain the same.

Axiom XI teaches us that the ratio between two numbers only remains the same when we add or take away like parts from each number. Here is an example:

If we begin with the numbers 2 and 4 and a half-part of each number, we will add 1 to 2 and 2 to 4, which will give us the numbers 3 and 6. The ratio 3:6 can be written more simply as 1:2. Using Axiom III we can prove that the ratios 2:4 and 3:6 are the same–or we can simply use Axiom XI. That is all you need to understand now.

AXIOM XII. If a number measures all the parts of another number, it will also measure the whole of that other number.

In our last lesson, besides learning about ratios, we also learned about measure. We learned that a smaller number measures a greater number when it is an aliquot part of that greater number. Therefore, the number 2 measures 4, 6 and 8 while 3 measures 6, 9 and 12, and so on.

Knowing this basic definition is very good, but it doesn’t help us determine whether one number measures another when we are dealing with greater numbers that are not so easy to work with. For example, does 3 measure the number 24? 54? 104? This is more difficult.

Axiom XII tells us that if the number 3 can measure each of the parts of some greater number, it will also measure the whole of that number. This is very helpful! Let’s begin with the number 24. If we divide it into two parts, each will contain 12. We know that 3 measures 12, so 3 must also measure 24! If we look at the number 54, we could divide it into two parts and each would contain 27. If we broke 27 into parts, we could get 12 and 15. We can easily see that 3 measures both 12 and 15, so 3 must also measure 54!

If this seems a little bit too difficult for you, don’t worry. We will understand this much better later when we study division. For now, simply make sure that you know your axiom and what it means.

AXIOM XIII. If a number measures another number, it will also measure all the numbers which that other number measures.

Once again, it is important to remember that our goal in Arithmetic is to understand the relationships between numbers so that we are able to use them in many different and creative ways. If we become masters of Arithmetic, we will be able to find solutions to problems that we have never seen before–because we will know the rules and relationships between numbers.

In Axiom XII, we learned that if a number measures the parts of another number, it will also measure the whole of that number. Axiom XIII adds much more. We learn that if one number measures another, it will also measure all of the numbers that number measures!

This can be very helpful anytime we already know that a greater number is measured by some smaller number. For example, suppose we were told that 10 measured 40, but we needed to know whether 5 also measured 40? We would not have to work to see if 5 measured 40 at all, since we already know that 10 does. All we would need to show was that 5 measured 10. Since 5 measures 10, it also measures all the number which 10 measures. Wow! That’s useful to know–and we’ll find many uses for this axiom down the road.

AXIOM XIV. If a number measuring any other number also measures a Part of that other number, it will also measure the remaining Part.

Modern students are taught to solve difficult problems with calculators, and it feels good to get a quick answer to a problem. However, if a student presses a wrong button, their answer will be wrong! Worse, what happens when we need to solve a problem and don’t have a calculator? It is obviously not a good idea for students to depend on calculators to solve problems. With the gift of reason, we should need one anyway. The axioms and definitions will help us to solve many amazing problems in the future. Let’s look at an example using Axiom XIV.

Imagine that someone asked us whether 4 could measure the number 392. Wow–that’s a tough one! Well, it’s only tough for someone who doesn’t know his axioms. Axiom XIV leads us to the answer to this challenge in just a few seconds–if you think creatively. The axiom tells us that if a number measuring any other number also measures a part of that number, it will also measure the remaining part. Knowing that, watch how easy it is to determine if 4 measures 392.

We can see that the number 392 is close to 400–and we can easily see that 4 measures 400–it’s simply 4 taken 100 times. We can also count and see that if we took 392 as one part of 400, the other part would be 8. Clearly, 4 measures 8. Since 4 measures both 400 and a part of it, it must also measure the remaining part–392. See how easy that was?

How about 395? Does 4 measure 395? Well, we know that 4 measures 400 and if 395 is taken as a part of 400, the other part would be 5. It is clear that 4 does not measure 5, so 4 will also not measure 395.

If you’re scratching your head at that one, it’s OK. Memorize your axiom carefully and read the examples over and over until you understand.

AXIOM XV. If two numbers are commensurate with a third number, they are commensurate to one another.

When you read Axiom XV, it may sound familiar. It sounds, in some ways, like Axiom III:

Quantities which are both equal to one and the same third are equal to one another.

We see that one of the ways in which Reason searches out relationships between numbers is by seeing each relates to a third number. Here, we find a guide to determining whether two numbers are commensurate–that is, whether they are all measured by a common number. The axiom teaches us when we begin with two numbers, if we find a third number with which each of the two numbers is commensurate, then they are commensurate with one another. How does this help us?

Remember that the axioms provide us with self-evident statements that we can use to prove theorems (statements that need to be proven). If we know that the number 20 is commensurate with 40 because they are each measured by 4 and if we know that 8 is commensurate with 40 because they are both measured by 4, we will have a way to prove that 20 and 8 are commensurate with each other–because they are both commensurate with one and the same third.

As always, don’t think you will thoroughly master these axioms now. It will not be until later that you really begin to see the many different ways they can be used and how helpful they are. For now, memorize them carefully and try your best.

Summary

In this lesson, we have simply added five 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.

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 tutor quiz you to test your mastery before taking your lesson exam.

  1. AXIOM XI. If two numbers are increased or decreased by like Parts of themselves, their Ratio remains the same.
  2. AXIOM XII. If a number measures all the parts of another number, it will also measure the whole of that other number.
  3. AXIOM XIII. If a number measures another number, it will also measure all the numbers which that other number measures.
  4. AXIOM XIV. If a number measuring any other number also measures a Part of that other number, it will also measure the remaining Part.
  5. AXIOM XV. If two numbers are commensurate to a third number, or are commensurate to commensurate numbers, they are commensurate to one another.
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