To complete this lesson, complete the following tasks:
- Review your previous exams in Arithmetic as needed.
- Study your Lesson.
- Complete your Memory Work.
- Complete your Exercises.
- Complete the Lesson Examination.
1. Lesson
In this lesson, we will learn axioms in Arithmetic that apply to the four rules of invention: Addition, Subtraction, Multiplication and Division. 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 seven more axioms. To view and print a summary of the axioms learned through this lesson, use the link below:
AXIOM XVI. If equal quantities are added to equal quantities, their sums will be equal.
AXIOM XVII. If equal quantities are subtracted from equal quantities, the remainders or differences will be equal.
In Axiom III, we learned that quantities which are equal to one and the same third are also equal to each other. Here in Axioms XVI and XVII, we find related truths. Let us use them in an example first. Suppose that we were given that the quantities ( a + 7 ) and ( c + 7 ) are both equal to 12. Then, the quantity ( a + 7 ) must be equal to the quantity ( c + 7 ) by Axiom III. Now, if ( a + 7 ) is equal to ( c + 7 ), then a must be equal to c because “If equal quantities are subtracted from equal quantities, the differences will be equal.” We know that, if we start with ( a + 7 ) and ( c + 7 ), which are equal, and subtract 7 from each of them, we obtain a and c, respectively. Indeed, subtraction is the converse of addition, and therefore ( a + 7 ) – 7 is equal to a. Similarly, ( c + 7 ) – 7 is equal to c. Therefore, a and c must be equal by Axiom XVII. Let us look at a precise proof of the statement above. We will actually prove it in two different ways.
Theorem: Granted that ( a + 7 ) = 12 and ( c + 7 ) = 12, prove that a = c.
First Proof: Since ( a + 7 ) and ( c + 7 ) are each equal to 12, they must be equal to each other by Axiom III. Therefore, by Axiom XVII, if we subtract the quantity 7 from each of them, we will obtain equal quantities. In other words, ( a + 7 ) – 7 is equal to ( c + 7 ) – 7. Since addition is the converse of subtraction, however, ( a + 7 ) – 7 is equal to a, and ( c + 7 ) – 7 is equal to c. Therefore, a is equal to c. Q.E.D.
Second Proof: Since ( a + 7 ) = 12, we know by Axiom XVII that ( a + 7 ) – 7 is equal to 12 – 7. Since 12 – 7 is equal to 5 by subtraction, we know that ( a + 7 ) – 7 is equal to 5 by Axiom III, because they are both equal to 12 – 7. However, since addition is the converse of subtraction, ( a + 7 ) – 7 is equal to a. Since a and 5 are both equal to ( a + 7 ) – 7, they must be equal to each other by Axiom III. Therefore, a = 5. We can now repeat the exact same reasoning starting with ( c + 7 ) = 12, and deduce that c = 5. Indeed, Since ( c + 7 ) = 12, we know by Axiom XVII that ( c + 7 ) – 7 is equal to 12 – 7, which is 5. However, since addition is the converse of subtraction, ( c + 7 ) – 7 is equal to c. Since c and 5 are both equal to ( c + 7 ) – 7, they must be equal to each other by Axiom III. We deduced that a = 5 and c = 5. Since a and c are both equal to 5, they must be equal to each other by Axiom III. Q.E.D.
While the second proof is longer and a little more tedious, it has the advantage of also providing us with the precise value for a and c. At the end of the second proof, we not only know that a and c are equal to each other, but we also know that they are equal to 5. Since we were not required to find their value, the first proof is to be preferred. However, keep in mind for future reference that sometimes the best way to show that two quantities are equal to each other is to actually find out what quantities they are.
Let us now look at a similar proof, this time using Axiom XVI:
Theorem: Granted that ( a – 3 ) = 6 and ( c – 3 ) = 6, prove that a = c.
Proof: Since ( a – 3 ) and ( c – 3 ) are each equal to 6, they must be equal to each other by Axiom III. Therefore, by Axiom XVI, if we add the quantity 3 to each of them, we will obtain equal quantities. In other words, ( a – 3 ) + 3 is equal to ( c – 3 ) + 3. Since addition is the converse of subtraction, however, ( a – 3 ) + 3 is equal to a, and ( c – 3 ) + 3 is equal to c. Therefore, a is equal to c. Q.E.D.
Just like in the example above, we could solve this problem by finding the precise value for a and c, which would be 9. Try to write a proof of it on your own.
With these axioms, we will be able to solve and prove a number of different problems:
- We can prove that quantities are equal when, added to the same quantity, produce the same sums.
- We can prove that two sums must be equal if equal quantities are added to equal quantities.
Let’s look at a fun example of where these axioms can help us out.
One day, David, Jonathan and Elizabeth wanted to build a see-saw. They found an old tree stump and set a log on top, but when they climbed onto the log, Elizabeth and David were stuck on the ground and Jonathan was stuck way up in the air on the other side! Jonathan sat there, high up in the air, while Elizabeth and David sat on the ground laughing at him.
“This is no fun!” Jonathan groaned as he jumped down to the ground. “We need to find another kid to sit with me so we can ride.” Elizabeth asked, “How can we do that?” “It’s pretty easy, I think…we just have to use our Arithmetic.” said Jonathan. “We need the kids on each side of the see-saw to weigh the same. Then the log will balance. Mom said that you and I weigh the same, Elizabeth, so we just need to find someone–or something–that weighs the same as David. Then it’ll work!”. “How do you know that will work?” asked David. “Because of the axiom that says, ‘If equal quantities are added to equal quantities, their sums will be equal.’ Since Elizabeth and I are equal quantities, we just need another quantity equal to you. Then the sums of our weight will be the same. David–you stay here on the see-saw and I’ll find something.”
Jonathan found Samuel playing in the backyard and pulled him over to the see-saw. He sat him up on the see-saw and the log moved down a little, but not enough. Then, Jonathan grabbed Beckett (the dog!) and told Samuel, “Here, hold the dog!”. When Samuel held Beckett, the log balanced!
“That’s it!” shouted Jonathan. “Sammy and the dog weigh the same as David. Now, when we climb on, Liz, our see-saw will work!”. Jonathan and Elizabeth climbed up and the see-saw worked perfectly. “How did you figure that out, Jonathan?” Elizabeth asked. “Well, like I said, you and I are equal quantities, you know, so we needed to find a quantity equal to David that would make both sides equal…then we could all ride the see-saw together. Sammy and the dog are equal to David. When we added them to our weight, the sides were equal to one another–just like my Arithmetic lesson said. I just used an axiom and figured it out.”
AXIOM XVIII. If one equal quantity be subtracted from another, the remainder will be nothing.
Axiom XVIII provides us with another useful starting point for reasoning in Arithmetic. This axiom can help us to prove that if one quantity is subtracted from another equal quantity, there will be no remainder. Let’s consider a proof:
Theorem: Prove that if Michael has 10 apples and then gives 7 to Mary and 3 to Anna, he will have none left.
Proof: We know from by addition that 7 + 3 = 10. Therefore, the number of apples Michael had to begin with is equal to the sum of the number of apples he gave to Mary and Anna. Axiom XVIII states that if one equal quantity is subtracted from another, the remainder will be nothing. Since the quantity Michael began with (the minuend) and the number of apples given to Mary and Anna (subtrahends) are the same, then when the apples given are subtracted, Michael will have none left. Q.E.D.
This axiom also helps us to prove that if a quantity is subtracted from another and the remainder is zero, then those quantities are equal to one another.
Theorem: Prove that 15 = ( 10 + 5 ).
Proof: We know from our subtraction facts that if we take 10 away from 15, we are left with 5 and that if we take 5 more away we are left with 0. Therefore, 15 minus 10 and 5 equals 0. This can be written as 15 – ( 10 + 5 ) = 0. Axiom XVIII states that if one equal quantity be subtracted from another, the remainder will be nothing. When ( 10 + 5 ) is subtracted from 15, the remainder is 0. Therefore, 15 must be equal to ( 10 + 5 ). Q.E.D.
AXIOM XIX. If equal quantities are multiplied by equal quantities, the products will be equal.
AXIOM XX. If equal quantities are divided by equal quantities, the quotients will be equal.
We learned in lesson 15 that multiplication is the converse of division and that division is the converse of multiplication. Therefore, whatever can be said about multiplication often can be said about division as well. In Axiom XIX, we learn that if equal quantities are multiplied by equal quantities, then the products will be equal. Axiom XX is its converse, and it allows us to conclude that, if two equal quantities are multiplied by some unknown quantities, and their products are the same, then those unknown quantities are equal to one another. Let’s check it out with a proof.
Theorem: Granted that a × 3 = m and b × ( 2 + 1 ) = m , prove that a = b.
Proof: From addition we know that 2 + 1 = 3. Therefore, we can rewrite the second expression above to say that b × 3 = m. Since a × 3 and b × 3 are both equal to m, they are equal to each other by Axiom III. Axiom XX states that, if equal quantities are divided by equal quantities, the quotients will be equal. Therefore, ( a × 3 ) / 3 is equal to ( b × 3 ) / 3. Since division is the converse of multiplication, ( a × 3 ) / 3 is equal to a, and ( b × 3 ) / 3 is equal to b. Therefore, a and b must be equal quantities. Q.E.D.
Similarly, we can use the same idea with division, using Axiom XIX.
Theorem: Granted that a / 3 = z and b / ( 2 + 1 ) = z, prove that a = b.
Proof: From addition we know that 2 + 1 = 3. Therefore, we can rewrite the expression above to say that b / 3 = z. Since a / 3 and b / 3 are both equal to z, they must be equal to each other by Axiom III. Axiom XIX states that, if equal quantities are multiplied by equal quantities, the products will be equal. Therefore, ( a / 3 ) × 3 is equal to ( b / 3 ) × 3. Since multiplication is the converse of division, ( a / 3 ) × 3 is equal to a, and ( b / 3 ) × 3 is equal to b. Therefore, a and b must be equal quantities. Q.E.D.
How does his help us? Let’s consider another fun example.
Jonathan and David were each given a deck of cards for Christmas. One day, David found the babies playing with his cards and they had thrown them all over the room! David quickly gathered all of his cards together, but he had no way of knowing if any were lost.
Jonathan came in and found David upset and asked, “What’s wrong, David?”. David replied, “The babies threw my cards all over the room and I don’t know if I’ve lost any. How can I play any card games if I am missing cards? I can only count to 20 so there’s no way I can count them all!”
“Hmmm…let me think about this.” Jonathan said. “I’ve got it! I know an axiom from Arithmetic that can help us figure this out. It’s a little tricky, but it will work. The axiom states that if equal quantities are divided by equal quantities, their quotients will be the same. You don’t know what all that means, but don’t worry–just do what I say. There are four different kinds of cards, right? There are hearts, diamonds, clubs and spades. So, let’s each divide our deck into four groups.” So they did.
“There!”, Jonathan said, “Each of our decks has been divided by four. Now, we can count how many cards are in each group. If they are all the same, then you have all of your cards!” “Really?!”, asked David, “OK…let’s count them!”
Jonathan counted each group of cards and David counted his. When David finished, Jonathan asked, “How many hearts did you have? Did you have 13?” “Yes!” said David. Jonathan continued, “How many diamonds did you have?” “I think thirteen.”, answered David. “Me too!” shouted Jonathan, “What about the rest?”. “I have 13 clubs and 12 spades.”, said David. “Uh oh!”, Jonathan said, looking around to see if any were missing, “You are missing one spade. Ah!! look! Baby Joshua has it! He’s going to eat it!” Jonathan and David ran across the room and saved the card just before it was devoured.
David was happy to have all of his cards back and said, “Thanks, buddy. How did you figure that out, again?”. “Easy”, Jonathan answered, “I just used my axioms. Since we should have started with the same number of cards, and since we divided them by the same number, I knew that we should end up with the same number in each group–that’s called the “quotient”.” David shook his head and said, “Well, I don’t know about all that fancy quotient stuff, but I know it works!”.
AXIOM XXI. Unity neither multiplies nor divides.
This axiom is what every axiom should be: obvious.
We learned that to “multiply” means to add a number to itself a number of times. To multiply 3 times 2, we add 3 to itself, as 3 + 3. It would be meaningless to ask, “What is 3 times 1?”. The number taken once is simply itself–and that is not multiplication. Unity does not multiply.
We also learned that to “divide” means to discover how many times a number called the dividend contains a number called the divisor. When we name a number, we are stating how many times a quantity contains unity–that’s what a number is. Therefore, asking “How many times does 9 contain 1?” is really pretty dumb. You already know the answer…it’s 9!
To “divide” also means to break a quantity into a number of groups to learn how many units will be found in each group. Again, it makes no sense to ask, “How many units will be in each group if 9 is divided into 1 group?” Uh….we already know that! The number 9 is one group of 9 units. Thus, it is clear that unity does not divide.
AXIOM XXII. To multiply by an Integer increases the value of the Multiplicand and to divide by an Integer decreases the value of the Dividend.
In Lesson 08, we learned that an Integer is a number which is referred to Unity as a Whole to a Part as, 1, 2, 3, 4, etc.. This was compared in that lesson with a Fraction, which is a number referred to unity as a Part to a Whole, as half, quarter, and so on. We also call Integers “Whole Numbers” because they are not broken into parts like Fractions.
This axiom states that any time a number is multiplied by a whole number, the product will be a greater number. Conversely, anytime a number is divided by a whole number, the quotient will be a lesser number. Let’s face it, this axiom is pretty obvious—but axioms are supposed to be obvious! That’s why we begin our reasoning in Arithmetic from them.
Summary
In this lesson, we have added seven new axioms with brief descriptions 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, so you must memorize them and think on them anytime you face a problem in Arithmetic. Axioms are tools–your diligence and thoughtfulness will determine how many things they can discover and prove. Lastly, 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 praeceptor quiz you to test your mastery before taking your lesson exam.
- AXIOM XVI: If equal quantities are added to equal quantities, their sums will be equal.
- AXIOM XVII: If equal quantities are subtracted from equal quantities, their remainders or differences will be equal.
- AXIOM XVIII: If one equal quantity be subtracted from another, the remainder will be nothing.
- AXIOM XIX: If equal Quantities are multiplied by equal Quantities, the Products will be equal.
- AXIOM XX: If equal Quantities are divided by equal Quantities, the Quotients will be equal.
- AXIOM XXI: Unity neither multiplies nor divides.
- AXIOM XXII: To multiply by an Integer increases the value of the Multiplicand, and to divide by an Integer decreases the value of the Dividend.
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.