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Intro. to Classical Arithmetic. Lesson 10. Rules of Invention: Addition & Subtraction

To complete the objectives of this lesson, complete the following tasks:

  1. Review past lessons to make sure all memory work is fresh in mind.
  2. Read through the lesson below once to become familiar with all the content in the lesson.
  3. Optional: Print Addition/Subtraction Triangle Flashcards from Academy Bookstore.
  4. Study the lesson for mastery.
  5. Complete all memory work.
  6. Complete the lesson assessment.

Lesson

Thus far in Classical Arithmetic we have learned the basics of quantity and number. We have studied definitions, rules, examples and axioms that we will now begin to work with in what are called the “Rules of Invention”.

The First Four Rules of Invention

In Latin, the INVENIRE means “to come upon” or “to find” something. However, it’s most important use doesn’t have to do with the modern idea of inventions. When we hear the word “invention” today, we normally think of Thomas Edison’s “invention” of the light bulb, but this is a narrow idea of what “invention” is.

We have already discussed God’s creation of man with the gift of Reason and how reasons lead men to use what is known to discover what is unknown. If we know that all fire is hot and that the sun is a ball of fire, then we can discover–by reason–that the sun must be hot. If you look outside and see that it is cloudy and the street is wet, you can use that knowledge to discover that it is probably raining outside. When we discover something that was once unknowing by using what is known to us, we call that discovery “invention”.

In Arithmetic, we have already learned about proofs and how reason leads us to connect arguments to make new conclusions. For example, we learned that if 36 in. is equal to 3 ft. and that 1 yd. was equal to 3 ft., then 36 in. must be equal to 1 yd.. In this, what we know leads us–by reason–to the invention or discovery of something we wished to know or something that we simply didn’t know before. This is invention as it applies to Arithmetic.

Now, we don’t go about invention without rules. There are four basic rules that help us to discover new knowledge in Arithmetic using what is already known to us. They are Addition, Subtraction, Multiplication and Division.

Addition

The idea of adding quantities is understood by even the smallest children. After all, adding and counting are one and the same thing. To one we add one and reach two. To two we add one and reach three. If you can count, you can add. If we had no limits on our time, we could add any quantities just by counting. However, we often need to add large quantities and we need to do so quickly. To do this, counting won’t help us. We need faster ways to add quantities.

To add quantities more quickly, we use addition. To begin, let us learn the language of addition. The word comes from the Latin words AD (to) and DARE (to give, put), meaning “giving to” or “putting to”, as we put one quantity to another.

When we wish to discover the unknown quantity that results when two known quantities are combined, we can ask:

  • What is the sum of 2 and 3 ?
  • What is 2 plus 3 ?
  • What is 2 more 3 ?
  • 2 + 3 = __.

In each of these statements, 2 and 3 are the two given quantities. The quantities that are to be added together are called addends (Latin addenda, things to be added). The sum is the quantity to be discovered, which derives its name from the Latin word summa, which means the highest or end amount.

Addition Facts

Again, counting with one’s fingers or with counters has only one problem: it is too slow. Reason and memory can help us to discover quantities by addition with greater ease, speed–and accuracy. By memorizing basic addition facts, we can quickly add from memory in more difficult computations. The table below provides us with the addition facts that should be memorized. The white boxes (adding zero) and red boxes (adding one) do not need to be memorized as they require nothing more than simple counting. The green boxes should be memorized completely. As you memorize this addition table, you’ll find a pleasant surprise later in this lesson.

To use this chart, simply begin with a number in the first column on the bottom left, then select a number of the bottom row. Follow the number on the left across the chart to the right and follow the number on the on bottom up. Where the two lines meet you will find the sum for the selected addends. For example, if you find 2 on the left and 5 on the bottom, they join at 7, which is their sum. Once you are comfortable using the table, you can quickly learn the addition facts and use your lesson exercises to master them.

Addition Flash Cards

If you would like to create printed or hand-written flash cards to help you practice your memory work, you should make them to look like these examples. The addends are on the bottom and the sum at the top. You should have one card for every green box on the table above, for a total of 64 cards.

You read these flash cards in two directions. First, start with the addend on the bottom left and read, “2 plus 5 equals 7”, then start with the added on the bottom right and read “5 plus 2 equals 7”. When you have the facts learned well, have someone quiz you by covering the sum and showing you the addends. Then, have someone show you the sum and one addend and read, “2 plus x equals 7. What is x?” You provide the missing addend in place of x, saying, “2 plus 5 equals 7, therefore x equals 5.”

Addition of Greater Numbers

As you can see the addition facts chart only includes the addends 0-9. You may be wondering, “What do I do if I need to add greater numbers?”. It is very simple.

Rule 102 below states that “Addition is the invention of a number or quantity called the Sum, by collecting together two or more given Homogeneous quantities.”. When we consider the addition of large quantities, we need to focus on the term homogeneous quantities. When we add, we add homogeneous quantities and this helps us to add large numbers.

If we consider adding the numbers 24 and 32, we could treat these numbers as homogeneous quantities by thinking of them as 24 units and 32 units–and this is how most people think of numbers. However, if we think of them differently, we can add them more easily. If we think of 24 as 2 tens and 4 units and add to that 32 as 3 tens and 2 units, we will see that they make a sum of 5 tens and 6 units, which is equivalent to 56. By adding like this, we need only to know our basic addition facts from 0-9. Consider these examples:

  1. What is the sum of 35 plus 43?
    • 3 tens and 4 tens = 7 tens;
    • 5 units and 3 units = 8 units.
    • 7 tens and 8 units = 78.
  1. What is the sum of 51 plus 28?
    • 5 tens and 2 tens = 7 tens;
    • 1 unit and 8 units = 9 units.
    • 7 tens and 9 units = 79.
  1. What is the sum of 26 plus 14?
    • 2 tens and 1 ten = 3 tens;
    • 6 units and 4 units = 10 units or 1 ten.
    • 3 tens and 1 ten = four tens or 40.
  1. What is the sum of 37 and 88?
    • 3 tens and 8 tens = 11 tens or 1 hundred and 1 ten.
    • 7 units and 8 units = 15 units or 1 ten and 5 units.
    • 1 hundred and 1 ten and 1 ten and 5 units = 125

Thus, if we know our basic addition facts and the orders of numbers (which we will learn later in detail), we can add any large number using only the basic addition facts.

Subtraction

Like addition, the idea of subtracting quantities is understood by even the smallest children. From three we take away one and have a remainder of two. From two we take away one and have a remainder of one. If you can count backwards, you can subtract. Once again, if we had no limits on our time, we could subtract any quantities just by counting down from a one quantity to another. However, we often need to subtract large quantities and we need to do so quickly. To do this, counting won’t help us. We need faster ways to subtract quantities.

To subrtact quantities more quickly, we use subtraction. To begin, let us learn the language of subtraction. The word subtraction comes from the Latin word SUBTRACTUS, which is formed by the word SUB (under) and TRAHERE (to carry away). Therefore when we subtract, we carry some number away from another. The number we begin with is called the minuend , which means a quantity to be diminished. The number we take away from under the minuend is called the subtrahend, which means “thing to be taken away from under”. The quantity that is left behind after we take the subtrahend away from the minuend is called the remainder or difference. It is this quantity that we seek to discover through subtraction.

When we wish to discover the unknown quantity that results when one quantity is taken away from another, we can ask:

  • What is the difference between 5 and 3 ?
  • What is the remainder of 5 minus 3 ?
  • What is 5 minus 3 ?
  • What is 5 less 3 ?
  • 5 – 3 = __.

Subtraction Facts

Rule 110 below explains that “Subtraction is the converse of Addition.”. What that really means is that there is a relationship between subtraction and addition and if you know one, you will know the other as well! Since you know your addition facts, you already have the basic knowledge necessary to answer questions of subtraction as well. How does this work?

If you look at the flash cards below, you will see that minus signs have been added. For subtraction, we use the same flashcards we originally made for addition, but now read them differently. First, start with the minuend on the top and read, “7 minus 2 equals 5”, then start again with the minuend on the top and read in the opposite direction: “7 minus 5 equals 2”. When you have the facts learned well, have someone quiz you by covering the minuend and showing you the subtrahend and remainder while reading, “x minus 2 equals 5. What is x?”. Then, have someone show you the minuend and the subtrahend and read, “7 minus 2 equals x. What is x?” Lastly, another can show you the minuend and remainder and cover the subtrahend and ask you, “7 minus x equals 5. What is x?”.

Again since you only need one set of cards for both addition and subtraction facts, you will need only 64 total cards.

Subtraction of Greater Numbers

As you can see the subtraction facts chart only includes the subtrahends 0-9. You may be wondering, “What do I do if I need to subtract larger numbers?”. It is very simple–and just like addition.

Like the rule for addition discussed above, subtraction is only possible for homogeneous quantities. When we subtract, we subtract homogeneous quantities and this helps us to subtract large numbers.

If we consider 34 minus 12, we could treat these numbers as homogeneous quantities by thinking of them as 34 units and 12 units–but this would make for a difficult subtraction problem. If we think of them differently, we can subtract them more easily. If we think of 34 as 3 tens and 4 units and subtract from that 12 as 1 ten and 2 units, we will see that they leave a remainder of 2 tens and 2 units, which is equivalent to 22. By subtracting like this, we need only to know our basic subtraction facts from 0-9. Consider these examples:

  1. What is 29 minus 8?
    • 2 tens minus 0 tens = 2 tens;
    • 9 units minus 8 units = 1 unit.
    • 2 tens and 1 unit = 21.
  1. What is 57 minus 36?
    • 5 tens minus 3 tens = 2 tens;
    • 7 units minus 6 units = 1 unit.
    • 2 tens and 1 unit = 21.
  1. What is 26 minus 19?
    • 2 tens minus 1 ten = 1 ten;
    • 6 units minus 9 units = impossible!

What do we do in a situation like example #3 above? We use exchanges.

Exchanges

We can see in the example above that it is impossible to subtract a greater number from a lesser. However, in many subtraction problems we can exchange a ten for ten ones or a hundred for ten tens and then complete our subtraction. Let’s look again at the example above:

Since it is impossible to subtract 9 from 6, we can exchange 1 ten for 10 units and add them to the 6 units, making 16. Now we can complete our subtraction.

  1. What is 26 minus 19?
    • 2 tens minus 1 ten = 1 ten;
    • 6 units minus 9 units = impossible, so we exchange 1 ten for 10 units, giving 16 units
    • 16 units minus 9 units = 7.

You can see that this is still pretty difficult. Therefore, we can memorize the exchanges to make this process faster. Remember that this is only possible when numbers of a higher order are available for exchange.

Exchanges of Ten

  • “minus 1” = minus 1 ten, plus 9 units
  • “minus 2” = minus 1 ten, plus 8 units
  • “minus 3” = minus 1 ten, plus 7 units
  • “minus 4” = minus 1 ten, plus 6 units
  • “minus 5” = minus 1 ten, plus 5 units
  • “minus 6” = minus 1 ten, plus 4 units
  • “minus 7” = minus 1 ten, plus 3 units
  • “minus 8” = minus 1 ten, plus 2 units
  • “minus 9” = minus 1 ten, plus 1 units

At first, this will look complicated, but by the end of this lesson it will be very easy. The key is to focus on the minuend, not the subtrahend. If you focus on the minuend and use exchanges, you will quickly find answers. The more you can learn to ignore the subtrahend as soon as you see the exchange that is needed, the faster you will get. You should note that exchanges may be made in similar ways for hundreds, thousands and so on, just like tens. You can also make exchanges for fives with some practice, but that’s not required. Let’s use a few examples to show how exchanges work.

  1. What is 36 minus 9?
    • 3 tens minus 0 tens equals 3 tens.
    • 6 units minus 9 units equals 6 minus 1 ten and plus 1 unit.
    • 3 tens minus 1 ten equals 2 tens.
    • 6 units plus 1 unit equals 7 units.
    • 2 tens plus 7 units = 27.
  1. What is 44 minus 27?
    • 4 tens minus 2 tens equals 2 tens.
    • 4 units minus 7 units equals 4 minus 1 ten and plus 3 units.
    • 2 tens minus 1 ten equals 1 ten.
    • 4 units plus 3 units equals 7 units.
    • 1 ten plus 7 units = 17.
  1. What is 118 minus 39?
    • 1 hundred minus 0 hundreds equals 1 hundred.
    • 1 ten minus 3 tens equals 1 ten minus 1 hundred, plus 7 tens.
    • 1 hundred minus 1 hundred equals nothing. (Axiom 18)
    • 1 ten plus 7 tens equals 8 tens.
    • 8 units minus 9 units equals 8 minus 1 ten and plus 1 unit.
    • 8 tens minus 1 ten equals 7 tens.
    • 8 units plus 1 unit equals 9 units.
    • 0 hundreds plus 7 tens plus 9 units = 79.

If you’re struggling with this, don’t be discouraged. It takes time for you to memorize the exchanges and we’re going to practice this constantly. Once these skills are mastered by study and practice, your ability in arithmetical invention will be incredibly fast. Be zealous for these skills and work patiently.

Summary

In this lesson, we have learned the first two rules of invention in Arithmetic: Addition and Subtraction. We have learned the language of addition and subtraction and memorized their basic facts. We’ve also learned about exchanges in subtraction. In your exercises and exams below we will concentrate first on accuracy and then on speed.

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.

101. What are the first four rules of Arithmetical Invention?
The first four rules of Arithmetical Invention are: Addition, Subtraction, Multiplication and Division.

102. What is Addition?
Addition is the invention of a number or quantity called the Sum, by collecting together two or more given Homogeneous quantities.

103. What is Subtraction?
Subtraction is the invention of a number or quantity called the Remainder or Difference by taking a lesser given quantity, called the Subtrahend from a greater homogeneous quantity, called the Minuend.

104. What is the Sign of Addition?
The Sign of Addition is +, which is read “plus” or “more”, as 6 + 2 denotes the Sum of 6 and 2, or 6 more 2 or 6 plus 2. Indefinitely, b + d denotes the sum of two given quantities signified by the species b and d.

105. What is the Sign of Subtraction?
The Sign of Subtraction is – , which is read “minus” or “less”, as 6 – 2 denotes the difference of 6 and 2, or 6 minus 2 or 6 less 2. Indefinitely, b – d denotes the difference of two given quantities signified by the species b and d.

106. What is Axiom XVI?
AXIOM XVI: If equal quantities are added to equal quantities, their sums will be equal.

107. What is Axiom XVII?
AXIOM XVII: If equal quantities are subtracted from equal quantities, their remainders or differences will be equal.

108. What is Axiom XVIII?
AXIOM XVIII: If one equal quantity be subtracted from another, the remainder will be nothing.

109. What is Corollary I?
COROLLARY I: Addition and Subtraction are only of homogeneous terms.

110. What is Corollary II?
COROLLARY II: Addition is the converse of Subtraction and Subtraction is the converse of Addition.

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