The Algorithms of Arithmetic

7/30/2019 The Algorithms of Arithmetic

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The Algorithms of Arithmetic by Nathan D. Lane

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The Algorithms ofArithmetic

By Nathan D. Lane


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Table of Contents

1 Introduction to Algorithms of Arithmetic

1.1 What is an Algorithm?1.2 What is Arithmetic?1.3 What to Expect1.4 Who is this Document For?

2 Numbers2.1 What Are Numbers?2.2 Number Systems2.3 Whole Numbers2.4 Sequences2.5 Conclusion2.6 Exercises for Chapter 2

3 Addition3.1 What is Addition?3.2 Beginner Algorithms of Addition3.3 Skip-Counting Review3.4 Properties of Addition3.5 Conclusion3.6 Exercises for Chapter 3

4 Subtraction

4.1 What is Subtraction?5 Multiplication6 Division7 Fractions

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1 Introduction to Algorithms of Arithmetic

1.1 What is an Algorithm?An algorithm is an effective, repeatable method of doing something. In the case of this document it isan effective, repeatable method of solving a mathematical problem. Because algorithms are repeatable,that is that by repeating an algorithm similar conclusions may be obtained, their value is great inmathematics. Mathematics is a field of fact and algorithms may be used effectively to producemathematical facts.

In this document arithmetic is used loosely to identify mathematical problems of a particular class. Inparticular arithmetic in this document refers to mathematical problems requiring the operations ofaddition, subtraction, multiplication, division, abstract concepts such as real numbers, negative

numbers, fractions, and variables, and concepts that are easily applied to real-world problem spacesincluding money, time, and patterns.

My primary goal in creating this document is to provide some context for mathematical algorithms anddescribe some mathematical algorithms that have been proven over and over again, both of which maybe used to formulate solutions quickly for simple and complex arithmetic problems. The algorithms Iintroduce are simple and easily repeatable. Over time practice makes perfect is a principle that will berealized in mathematics. I will also attempt to point to and provide resources for such practice.

1.2 What is Arithmetic?

Arithmetic is the oldest most elementary branch of mathematics, [combining] ...the traditionaloperations of addition, subtraction, multiplication and division with smaller values of numbers.[1] Inthis document I will attempt to discuss as many of the facets of elementary arithmetic in as much detailas is necessary in order to fully understand the algorithms I am presenting. I will begin with the simpleand then move toward more complex details, yet keep the algorithms as simple as possible.

Since arithmetic includes the operations of addition, subtraction, multiplication and division, I willwork with these operations discretely and avoid the algebraic operations as much as possible, exceptwhere necessary to describe a concept. I will also used some of the ideas identified in number theory,and I will maintain their simplicity, as number theory as a whole is outside of the scope of thisdocument.

1.3 What to Expect

Throughout the following chapters I will introduce concepts that build upon each other. At the end ofeach chapter I will supply a set of exercises that will help you to better understand the conceptsidentified throughout the chapter. The exercises are meant to give you practical practice and aid in

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teaching each concept.

Each chapter is divided into sections similar to this chapter. Each section will identify a single sub-concept that fits into the primary concept of the chapter. Later chapters will build on concepts fromprevious chapters. I will try and link chapters and sections together where it applies, so that if previousconcepts are forgotten, then they can easily be referred back to.

1.4 Who is this Document For?

I have written this document primarily for anybody struggling to understand the basic algorithms inarithmetic that have proven, at least to me, that anybody should be able to understand and be able to dosimple math. This document should be mostly suitable for children, teenagers and adults. I use somestrong mathematical language, and I try to maintain contextual clues, so that understanding thelanguage isn't too difficult for anybody with a reasonable vocabulary. I will provide a glossary at theend of each chapter in order to define more complicated terms.

This document should also be suitable for anybody wanting to review arithmetic or to learn newalgorithms for solving arithmetic problems, as I have learned them over my many years of educationfrom elementary school through college. This document will not introduce every algorithm available,only those that I feel have benefited me. I am taking a student-teacher approach to teaching in thisdocument, which means that as a student of arithmetic, I have identified some concepts that have beenimportant to me, and I am teaching them. I am generally not following any outline of any alreadypublished document, manual, or book.

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2 Numbers

2.1 What Are Numbers?Numbers are the atomic unit of arithmetic. Their sequence is discrete and infinite both positively(numbers greater than zero) and negatively (numbers less than zero). Numbers may also becomefractionalized, meaning that their wholeness may be divided into parts, called fractions which may berepresented as decimals, or numbers with a whole part, a decimal point, and a fractional part. Thus onehalf of one whole is possible by dividing one into two parts:

1

2

Figure 2.1.1. One half is representative of one whole being divided into two parts. In

this illustration this is represented by a one, which represents the whole, being writtenover a two, which represents the number of parts, divided vertically by a horizontal line.

Numbers are canonically arranged in an ordered positively progressing sequence beginning at thesmallest value and increasing toward positive infinity, as in the following sequence:

0,1,2,3, 4,5,6,7,8,9,,

Figure 2.1.2. Numbers are normally arranged starting at zero and increase indefinitely

to an infinite end-point.

Numbers may also decrease toward negative infinity in an ordered negatively progressing sequence, as

in the following sequence:

0,1,2,3,4,5,6,7,8,9,,

Figure 2.1.3. Numbers may also start at zero and decrease indefinitely to a negatively

infinite end-point.

Numbers are meant to represent quantities, values, or measurements, thus I may have three oranges, ortwenty dollars, or a clock may represent the time as three o'clock. Three oranges is indicative of aquantity. Quantities usually identify a bulk of objects. Twenty dollars (which may also be representedcanonically by $20.00) is indicative of a value. Values may represent a bulk of objects or a single objectdepending on the context of the value. For example thirty-five cents may be represented in a number ofways:

one quarter and one dime (25+10=35) three dimes and one nickel (10+10+10+5=35)

two dimes and three nickels (10+10+5+5+5=35) thirty-five pennies (35=35)

Measurements are representative of a relative value. For example on Earth, our planet, we identify that

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there are twenty-four hours in a day, or in other words it takes twenty-four hours for the Earth to rotate.Thus we can measure the time, three o'clock, as a point in the period of time that the earth takes torotate, three hours after the peak of the night (midnight) or the peak of the day (noon). There areinfinite contexts to which a number may belong.

Context is always important, thus when solving problems that deal with a specific context, such as thecombined weight of multiple objects, it is important to include the context of the problem in thesolution. Since a weight may be indicated by pounds, kilograms, and so on, expressing the solution to aproblem involving weight should always include the context, such as pounds, kilograms, and so on.

2.2 Number Systems

When we talk about number systems we talk about systems of symbols that represent quantities,values, or measurements. Our basic number system in use today is called the base-ten-number-systemor the decimal number system. Base-ten refers to the number of symbols used to construct a number.

There are ten symbols in our standard system of numbers, which we sometimes call digits:

0123456789

Figure 2.2.1. The decimal number system has ten symbols which are used to representnumbers. The symbols are called digits and may be arranged to identify any number in

the system.

Other number systems also exist. When we talk about computing we speak of the binary orbase-twonumber system. Sometimes we also talk about the hexadecimalorbase-sixteen number system. In thefifties and sixties many schools taught math using the octalorbase-eightnumber system. Again thesenumber systems are only differentiated by the number of symbols used to represent a value. For

example a base-eight number system uses eight symbols to represent values. A base-eight numbersystem might include the symbols:

01234567

Figure 2.2.2. The base-eight, or octal number system uses only eight of the ten

numerical symbols of the decimal system. Only eight distinct symbols are used to

identify all numbers in the system.

Thus a numerical value written as 100 using our base-ten number system would be written as:

82 81 80

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Sixty-fours Eights Ones

1 4 4

Table 2.2.1. This table identifies the first three place-values of the octal number system:

sixty-fours, eights, and ones. The number such as 82 are algebraic sentences that

identify a concept called exponents. Exponents will be discussed more later. For now

simply know that they are used to identify special cases of multiplication.

This is read as one-sixty-four-and-four-eights-and-four-ones. This can later be proven by addition,multiplication, and algebra as the arithmetic sentences: (164)+(48)+(41) = 64+32+4=100. Atthe moment this is simply meant to be an example. If it is confusing now then there is no need to tryand understand it. Because the base-ten system is recognized internationally as the standard number

system to use for arithmetic, we will only focus on using it. It is much simpler, and utilizes the tendigits, zero through nine, nicely.

2.3 Whole Numbers

We call the ten symbols that are in the base-ten number system digits. Digits have place-value whencombined to represent numbers, which is also derived from our number system. In the base-ten numbersystem the place-values are multiples of ten. For example let's dissect the number 1,520:

Thousands Hundreds Tens Ones

1 5 2 0

Table 2.3.1. This table shows how to dissect the number 1,520 into its place-value

containers. In the number 1,520 there is are one thousand, five hundreds, two tens, and

zero ones. 1,520 is the number of feet in a mile.

That is to say that the canonical number 1,520 (one-thousand-five-hundred-and-twenty) isrepresentative of one thousand, five hundreds, two tens and zero ones. Knowing this is important inarithmetic because it is a fundamental of arithmetic algorithms. Places indicate place-value, and place-value is a driving component of the ideas ofborrowingand carrying. We will discuss these terms lateras the need arises.

Place value increases by an order of ten from right to left as in the diagram below:

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109 108 107 106 105 104 103 102 101 100

1,000,000,000 100,000,000 10,000,000 1,000,000 100,000 10,000 1,000 100 10 1

Table 2.3.2. From right-to-left each significant place-value is identified by ones, tens,

hundreds, thousands, ten-thousands, hundred-thousands, millions, ten-millions,

hundred-millions, and billions. Place-values are infinite in number, just like numbers.

2.4 Sequences

In the base-ten, or decimal number system numbers are ordered from zero to nine and then a seconddigit or place-value is added to the left, starting with one and increasing again to nine as the place-valueon the right moves from zero to nine. Digits or place-values are continually added as the need arises. Iwill show you an example of this.

0,1,2,3,4,5,6,7, 8,9,10,11,12,13,14,15,16,17,18,19,20, ...

Figure 2.4.1. This figure shows counting from zero to twenty. Notice that there are

patterns in numbers when counting by one.

In the above sequence we see that, going from left to right, we start with the number zero and thenincrease by one as we move onto one, two, three, and so on up to nine. Then we reach the number ten.Ten starts over again with the number one, now in the tens position, and zero again in the ones position,and then we progress again, from zero to nine, as in: ten, eleven, twelve, and so on, up to nineteen.Then we reach the number twenty, which again is a single increment in the tens place (from one to two)and we start again at zero in the ones place. Continuing on in this pattern we progress through thenumbers like so:

0,1,2,...,9

10,11,12,...,19

20,21,22,...,29

...

100,101,102,...,109

110,111,112,...,109, ...

Figure 2.4.2. Here we show several sequences from zero to nine. Notice the patterns.

This pattern continues through the entire number system as we approach infinity.

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2.5 Conclusion

In this section we have learned about number systems and different bases. We learned that a base isrepresentative of the number of symbols used in a number system. For example, base-ten refers to the

fact that we use ten symbols, zero through nine, in the base-ten or decimal number system. We alsolearned about place-value and that understanding place-value is important in order to understand theconcepts of borrowing and carrying, which will be discussed later. Finally we learned that numbersfollow a uniform pattern in counting both towards positive infinity and negative infinity.

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Exercises For Chapter 2

For exercises 1-10 use the table below:

109 108 107 106 105 104 103 102 101 100

1,000,000,000 100,000,000 10,000,000 1,000,000 100,000 10,000 1,000 100 10 1

Table 2.e.1. From right-to-left each significant place-value is identified by ones, tens,

hundreds, thousands, ten-thousands, hundred-thousands, millions, ten-millions,

hundred-millions, and billions.

Write the number using the symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 in base-ten or decimal format.

1. Write the number two-hundred-and-forty-nine.

2. Write the number nineteen-thousand-three-hundred-and-ninety-one.

3. Write the number one-billion.

4. Write the number thirteen-million.

5. Write the number two-hundred-and-sixty-three-thousand-and-ten.

Describe the place-value for each digit in the following base-ten numbers. For example, given the

number 490 we can derive that there are: four hundreds, nine tens, and zero ones.6. 1,345

7. 16

8. 171

9. 10,900,036

10. 1400

Write the next three numbers in each sequence:

11. 13, 14, 15, , ,

12. 121, 122, 123, 124, , ,

13. 19, 18, , ,

14. -1, 0, 1, 2, 3, , ,

15. 1099, 1100, 1101, , ,

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3 Addition

3.1 What Is Addition?The subject ofaddition or the combination of values, sets, or groups of things, abstract or tangible, isprobably the simplest of all arithmetic operations above the basic understanding of numbers, numbersystems and number sequences. The symbol used to identify the addition operation is the equilateralcross, +. When reading an addition sentence you read the addition symbol asplus.

3+4

Figure 3.1.1. This addition sentence is read as three plus four.

Numbers can represent values, measurements or quantities. Numbers may also represent sets or groupsof things. The numbers in an addition sentence are called addends. When we talk about things, we referto them as units. When we talk about units with reference to the above addition sentence we might say,three somethings plus four somethings. The word somethings denotes units. You may substitute anyobject, or noun, for the word somethings, such as apples. Then we say the addition sentence as, threeapples plus four apples.

Figure 3.1.2. We can formulate the addition sentence to indicate units, as in three

apples plus four apples.

3.2 Beginner Algorithms in Addition

Units can make the addition sentence more tangible or real for beginners, thus making the problem of

solving the addition sentence somewhat more real. For example, change the addition sentence threeplus fourto if I have three apples and somebody gives me four more apples, then how many apples do Ihave?

Counting is an important algorithm which enables beginners to solve addition sentences, howevercounting is usually limited to what is available for counting, such as ten fingers, a few marbles, oranything that is similar. As an example a beginner might recognize that the answer to the addition

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sentence, three apples plus four apples, is thesum or count of all of the apples illustrated, and thus theycount each of the apples in the illustration, starting at one and finishing once all of the illustrated appleshave been counted.

Figure 3.2.1. Counting all of the apples illustrated is one algorithm beginners learn in

order to solve addition sentences.

Another algorithm of counting for beginners who can count on from a certain value in a set of numbersmight begin at the value of the first addend of the addition sentence, three, and then count upfourtimesfrom there. He may accomplish this by holding up four fingers and starting with the number three,count up using each of his fingers: [three,] four, five, six, seven. This method, is more aligned with theskills expected in problem solving, and enables more advanced beginners to solve problems requiringaddition more quickly.

Figure 3.2.2. Starting at the distinct value of the first addend in the set of all of the

addends and then counting up is another algorithm beginners discover in order to solve

addition sentences.

Yet another example of counting, that shows more intuition, is to identify which addend in an additionsentence is greater in value, and starting with that number count up the number of units left. In the case

of the example above the intuitive counter would identify that the addend fourisgreaterin value thanthe addend three, thus starting with four and counting up three times: [four,] five, six, seven. Thisalgorithm works with addition sentences because the operation of addition is commutative, whichmeans that the addends in an addition sentences, can be placed in their original presented order or theycan be placed in reverse order, and the same answer results.

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Figure 3.2.3. Recognizing that one quantity is higher than the other, and starting there

for counting is a higher-level algorithm for beginners that can count.

Ideally a reader of the addition sentence, three apples plus four apples, would recognize three applesandfour apples as two discrete groups of apples, for which we know the quantities. Then because theanswer was memorized, the reader would deduce that three apples plus four apples equals seven

apples.

Figure 3.2.4. Recognizing that three and four are discrete groups of apples and addingthem together as a memorized pattern is ideal. Likewise a beginner with such a skill

could recognize the commutativity of addition and memorize that four plus three is the

same addition sentence.

It should be the goal of all beginners to have memorized all combinations of addition for single-digitaddition or to learn skills that enable the beginner to deduce the result of an addition sentence. Thismeans memorizing the answers to all addition problems, 0+0 to 9+9 , this being the fundamentallevel of addition.

This level of intuition may result in a mixture of memorized concepts such as skip-countingand theaddition of well-known values such as zero, one, two,five, and ten. Reasoning about how numbers areformed through the memorization of simpler math problems may result in a finer tuned set of skills insolving addition sentences. For example, given the original addition sentence, three apples plus fourapples, one might deduce that four is actually two plus two, resulting in the new addition sentence,three apples plus two apples plus two apples. Because the numbertwo is a standard value used in skip-counting, the beginner may deduce that skip-counting twice by twos from three would result in thecorrect answer, and thus the beginner begins: [three,] five, seven, coming up with seven apples for hisanswer.

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3.3 Skip-Counting Review

Skip-counting is defined as counting by increments larger than one in either a positive or negativedirection. Common skip-counting patterns for many beginners include skip-counting by twos, skip-

counting by threes, skip-counting byfives, and skip-counting by tens. Often beginners will start skip-counting with zero, whether zero is used as a term in the skip-counting or not. Thus a beginner whoskip-counts by twos will start: [zero,] two, four, six, eight, ten, and so on. Similarly a beginner whoskip-counts by threes will start: [zero,] three, six, nine, twelve, fifteen, and continue on.

Figure 3.3.1. Showing a skip-counting-by-two diagram with the zero that is usually

implied. The even numbers have a gray background.

Skip counting can eventually be mastered to a degree that enables even a beginner to transpose the skip

counting. For example while a beginner will typically start skip-counting by twos with the numberzero, resulting in counting only even numbers (numbers that are a factor of two), transposing the skip-counting by one in the positive direction, resulting in starting with the number one, is still valid skip-counting by twos. The result of this transposition is the sequence one, three, five, seven, nine, eleven,and so on.

Skip-counting is the natural result of the beginning realization of patterns in a number system. Thesepatterns are the result of adding the same number over and over again, starting with zero or anothernumber. For example, skip-counting by twos using even numbers results when we start with zero andthen add two, and then add two again, adding two as many times as is required.

0 + 2 + 2 + 2 + 2 + 2 ...

0 2 4 6 8 10 ...

Figure 3.3.2. Showing that skip-counting by twos is a result of adding two to zero

repeatedly and calculating the result.

If we take a simple graph of the numbers from one to one-hundred laid out in a ten-column and ten-row

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grid, then we can visualize some of the patterns that occur as a result of skip-counting. Following aresome of these visualizations. The gray squares represent the counted numbers, while the white squaresrepresent the numbers skipped over in the various skip-counting sequence.

Figure 3.3.3. Showing pattern created by skip-counting by twos using even numbers.

Figure 3.3.4. Showing pattern created by skip-counting by twos using odd numbers.

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Figure 3.3.5. Showing pattern created by skip-counting by threes.

Figure 3.3.6. Showing pattern created by skip-counting by fives.

Figure 3.3.7 Showing pattern created by skip-counting by tens.

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The simple skip-counting that has been identified here, while valuable and useful, is only helpful in alimited manner, because it only enables the beginner to realize the next closest value through skip-counting if the current value is one of the gray squares. For example, if an addition sentence revealsthat skip-counting by twos is valuable, as in the sentence three plus four or the derived additionsentence three plus two plus two, then we can use the skip-counting by twos using odd numbers skip-counting diagram by starting on the gray number three and skip-counting by two two times: three, five,seven.

Figure 3.3.8. Showing skip-counting by twos starting with three in order to solve the

addition sentence: three plus two plus two (originally three plus four).

If however we are given an addition sentence such as four plus five, then none of the given skip-counting diagrams are immediately helpful, because there is no skip-counting diagram that shows agray numberfour that also shows skip-counting by fives. This sort of problem can be solved bycombining other operations, through memorization of addition facts, and through other counting

algorithms. For example if a beginner understands that five is the same as two plus three, then he couldrewrite the addition sentence as four plus two plus three, replacing the five with two plus three. Thensince there is a skip-counting by twos diagram that has a gray four(skip-counting by twos using evennumbers), the beginner can use this diagram in order to get to the next element of the additionsentence, (four plus two equals six) plus three . Finally we end up with the new addition sentence sixplus three, so we look at the skip-counting by threes diagram because our final addend is three, andrealize that six is a gray number on that diagram. Thus we can use skip-counting by threes to solve theremaining problem: [six,] nine. Six plus three equals nine.

4+5=?

4

+(2

+3

)=?

Use skip-counting by twos using evens.

6+3=9

Use skip-counting by threes.

4+5=9

Figure 3.3.9. showing the progression from the initial addition sentence of four plus five

to the conclusion of four plus five equals nine, using skip-counting.

Alternatively we could use the commutative property of addition and reorder the addends in theaddition sentence to readfive plus four. We could then similarly break down the addition sentence into

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five plus two plus two (because two plus two equals four). Since two is a skip-counting pattern andthere is a skip-counting diagram for skip-counting by twos that includes a gray five ( skip-counting bytwos using odds), we can use that diagram to skip-count up to our answer: [five,] seven, nine.

4+5=?

5+4=?

5+(2+2)=?

5+4=9

4+5=9

Figure 3.3.10. Showing another algorithm for solving the simple addition sentence: fourplus five using skip-counting.

Using skip-counting in order to solve addition sentences is a valuable algorithm for beginners andadvanced mathematicians alike. Skip-counting is a valuable tool for the beginner, and it should not beoverlooked or under-utilized.

3.4 Properties of Addition

The operation of addition has several properties, one of which has been briefly revealed. That is thecommutativeproperty. The addition operation is commutative because the addends may appear either in

their originally presented order or in reverse order in the addition sentence to result in the sameoutcome. In figure 3.1.5 above we revealed this briefly by showing that a beginner, who understandswhen one value is greater than another value, you can start counting at that value and count up fewertimes in order to arrive at the answer to an addition sentence. In that example the beginner swapped theorder of the addends, three and four, and the meaning of the addition sentence did not change. That is tosay that 3+4 and 4+3 have the same meaning, because of the commutative property.

In fact we can prove that addition has the commutative property by answering both addition sentences:

3+4=74+3=7

so3+4=4+3

Figure 3.4.1. Showing that addition has the commutative property.

Here we see both variants of the addition sentence, 3+4 and 4+3 . We also see that both variantsof the addition sentence equalseven, and thus conclude that both addition sentences are equal in value,

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and contain the same symbols only in a different order, resulting in addition having the commutativeproperty.

The next property of addition that I would like to discuss is the associative property. When we say thataddition is associative we mean that when we add more that two addends together, their order does notmatter. So far we have talked about adding two operands together as in the addition sentence, 3+4 ,but arithmetic does not only allow for two addends in an addition sentence, which we have already seena little bit. In fact there may be an infinite number of addends in an addition sentence, every twoaddends are usually separated by an addition symbol. For example, let us say that we start with threeapples and then add four more, and then add three more apples:

Figure 3.4.2. An addition sentence with more than two addends: three plus four plus

three.

Now we have an addition sentence that contains three addends: three, four, and three. All of these needto be added together in order to solve the addition sentence.

Now let's note that this addition sentence can be figuratively broken down into three different additionsentences that illustrate only part of the addition sentence, as shown in the following diagram:

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Figure 3.4.3. Three addition sentences derived from the addition sentence in figure

3.4.2: three plus four, four plus three, and three plus three.

Infigure 3.3.3, we identified three different addition sentences that could be derived from the addition

sentence in figure 3.3.2, using the commutative property and taking each operand from the originaladdition sentence. But you should note that there are not actually three addition sentences in figure3.3.2, rather there are only two, as identified by the fact that there are only two addition symbols in theaddition sentence.

Why did we identify three different possible addition sentences by the commutative property then? Thereason we identified three new addition sentences is because there are three addends, which means thatthere are three ways to start an addition sentence that has two addends. If we think about combinationswith order, meaning if we were to re-combine the three addends based on the order they fall in theoriginal addition sentence, then actually there would be six addition sentences. We won't go into muchdetail about this right now, but I will show you an example of this using the figure below.

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Figure 3.4.4. Combinatorics, which we will discuss later. Showing that when order

matters, we can re-combine the addends from a three-addend addition sentence into six

two-addend addition sentences.

Let's think of these possible re-combinations from the three-addend addition sentence into two-addendaddition sentences as possible starting points for solving the three-addend addition sentence:3+4+3 . Now let's take the first addition sub-sentence and start there. We already know that3+4=7 from the earlierfigure 3.2.4. If we remove one of the threes and four from the addition

sentence and replace it with seven, the answer to the addition sentence 3+4 , now we have a newaddition sentence:

Figure 3.4.5. We know that 3+4=7, so we can replace the first part of the addition

sentence, 3+4 with our answer, 7.

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Now we have the new addition sentence: seven plus three. We solve the new addition sentence in thesame way that we have done so far. Since this is an addition sentence, I want to impress upon you againthat it carries the two properties that we have already discussed, namely it is both commutative andassociative, meaning that the order of the addends, in this case seven and three, may be changed andthe meaning and solution of the addition sentence will remain the same. For example we could rewritethe addition sentence 7+3 as 3+7 and the meaning and solution of the addition sentence willremain the same.

Finally we solve the last addition sentence 7+3 as we did the previous addition sentence 3+4 ,which results in ten.

Figure 3.46. The addition sentence, seven plus three, solved as ten.

3.5 Multi-Digit Addition

Now that we have discussed some basic algorithms of addition using single-digit numbers it is onlynatural to begin talking about multi-digit addition algorithms. Multi-digit addition builds on all of theprinciples that we have discussed thus far. To review, multi-digit addition still exhibits the twoproperties ofcommutativity and associativity, multi-digit addition sentences may contain more than two

addends, multi-digit addition works with skip-counting, and multi-digit addition may be easier tounderstand in the real world when units are applied. Multi-digit addition also builds onto the principleofplace-value. Remember that place-value is important when reading numbers and identifying theirvalue.

When we talk about multi-digit addition, we are referring to addition sentences that contain at least onemulti-digit number. Thus the following addition sentence is considered to be the minimum multi-digitaddition sentence:

10+0=10

Figure 3.5.1. Showing that ten plus zero equals ten is the minimum multi-digit

addition sentence, because ten is a multi-digit number. The minimum requirement is that

at least one addend in the addition sentence has more than one digit.

With multi-digit addition, it is often simpler to understand addition sentences when they are written inanother format. Up until this point we have written all of our addition sentences using infix notation.Infix notation is when we have a number, then an operator symbol such as the plus symbol, thenanother number, and so on. Numbers with symbols in between each number. Multi-digit addition is

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usually more understandable when addition sentences are written in columnar form. An additionsentence is written in columnar form by stacking the addends of the addition sentence so that the place-values of the digits appear in the same column. For example take the addition sentence: ten plus zero,which we previously wrote as 10+0=10 . We can now write this addition sentence in columnarform:

10

+ 0

10

Figure 3.5.2. Showing columnar format for writing an addition sentence.

We can see in this example that the addends, ten andzero are aligned to the right side. This is becauseplace-value is the smallest on the right side and increases in value from right-to-left. Here we have twoplace-values for each addend because the largest addend has two place-values. The place-values areones and tens. Zero technically has a place-value for tens even though it is not shown. The value of the

tens column is zero when it is not shown.

Once we have rewritten the addition sentence in columnar form, and properly aligned the addends, weare now able to solve the problem using a columnar algorithm, which means that we will solve theaddition sentence one column at a time. Beginners may wish to physically divide the columns by a linein order to see the correct line-up of addend place-values as in the diagram below.

1 0

+ 0

Figure 3.5.3. Showing a line to split the place-values of the columnar form of theaddition sentence: ten plus zero.

When we split the place-values of the addends in the addition sentence in columnar form, weeffectively break the addition sentence into two single-digit addition sentences: 0+0 and 1+[0] . Ihave placed brackets around the zero in the second addition sentence because the zero is not actuallypresent and we don't write it either. In reality you could assume that the second addition sentence,

1+[0] is actually just the number sentence 1 .

We begin solving addition sentences from right-to-left and then we write the result in the same columnas the solved addition sentence below the horizontal line below the addends.

1 0

+ 0

0+0=0

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1 0

+ 0

0

1+[0]=1

1 0

+ 0

1 0

Figure 3.5.4. Showing the algorithm of solving the addition sentence: ten plus zero, in

columnar form.

Now that we have solved the simplest multi-digit addition sentence, let's take a look at a morecomplicated addition sentence: thirteen plus six.

1 3

+ 6

Figure 3.5.6. Showing the addition sentence: thirteen plus six.

Remember that the first step is to split the addition sentence into columns that represent the digits of theaddends place-values:

1 3

+ 6

Figure 3.5.7. Showing splitting the addends into columns by the place-values of their

digits.

Next we treat each column as its own addition sentence so that now we have two addition sentences:3+6 and 1+[0] . We solve each addition sentence, starting with the addition sentence on the right

and then moving from right-to-left, placing the result of each addition sentence into the additionsentence's column:

1 3

+ 6

3+6=9

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1 3

+ 6

9

1+[0]=1

1 3

+ 6

1 9

Figure 3.5.8. Showing the algorithm of solving the addition sentence: thirteen plus six,

in columnar form.

The result of the addition sentence: thirteen plus six, is therefore nineteen. Now let us recall thataddition has the property of commutativity, which means that the order of the addends in an addition

sentence may be reversed and the meaning and the answer of the addition sentence remains the same.Having stated the property of commutativity again we can realize the answer to the multi-digit additionsentence: two plus seventeen, more easily:

2+17=?

2

+ 1 7

1 7

+ 2

Figure 3.5.9. Showing how the property of commutativity applies to and helps in solving

multi-digit addition sentences in columnar form. All three of these addition sentences

mean exactly the same thing.

In both of the above columnar forms of the addition sentence there is an imaginary zero in the columnto the left of the two. Placing the two under the seventeen makes the columnar form of the additionsentence slightly easier for many beginners to understand.

Multi-digit addition in columnar form simplifies addition sentences significantly when there aremultiple addends containing multiple digits. For example let's take the addition sentence: one-hundred-and-thirty-one plus three-hundred:

1 3 1

+ 3 0 0

Figure 3.5.10. Showing the addition sentence: one-hundred-and-thirty-one plus three-

hundred, in columnar form.

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3.7 Conclusion

In this chapter we have learned about the operation of addition in arithmetic. We learned that additionis both commutative, which means that addends may be in the original order or in the reverse order and

the meaning and solution of the addition sentence does not change its meaning or solution, andassociative, which means that if there are more than two addends, then it doesn't matter which part ofthe addition sentence is solved first. We also solved an addition sentence that contained more than twoaddends and showed you an algorithm on how to solve such problems, namely solving part of theproblem first, and then solving the next put with the solution to the first part. Finally we showed youcolumnar form and how to solve multi-digit addition sentences using columnar form in order tosimplify the work that needs to be done.

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Exercises for Chapter 3

Write the addition sentences below at least one other way that doesn't change the meaning or solution

of the addition sentence.

1. 4 + 7

2. 9 + 3

3. 3 + 1

4. 0 + 5

5. 6 + 7

Write at least two two-operand addition sentences that can be formed from the addition sentence.

6. 5 + 4 + 3

7. 7 + 12 + 9

8. 8 + 1 + 1

9. 15 + 6 + 2

10. 213 + 71 + 309

Solve the addition sentence.

11. 6 + 8

12. 7 + 4

13. 5 + 1 + 3

14. 7 + 2 + 9

15. 8 + 8 + 3 + 3

Solve the addition sentence using columnar form.

16. 163 + 22

17. 3758 + 1221

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18. 23 + 5

19. 70 + 10

20. 791 + 205

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4 Subtraction

4.1 What is Subtraction?Subtraction involves removing or separating objects, sets, values, or groups of objects from a usuallylarger group of objects.

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7 Fractions

7.1 What are Fractions?When we talk about fractions, there are place values. Fractions may be represented by whole numbers,as in:

1

8

or as decimals, or using a decimal point, as in:

0.125

Both of these representations are for the value one eighth. The decimal point, usually represented by asingle period, identifies a place value separator. On the left of the decimal place-values increase by amultiple of ten, and on the right of the decimal place-values decrease, or go down, by a multiple of ten.This is represented by the followind diagram:

102 101 100 Decimal 10-1 10-2 10-3

1 0 7 . 1 2 5

The value in the table is read as one-hundred-and-seven-point-one-two-five. This number may also berepresented by the whole numbers:

1071

8

and:

8578

The numbers are read as one-hundred-and-seven-and-one-eighth and eight-hundred-and-fifty-seven-eighths. These numbers are equal through the rule of distribution, which will be discussed later.

One important thing that you should notice from the table above is that place-values on the right of the

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decimal point begin with 10-1, which is different from the place-values to the left of the decimal point,which begin with 100. This is because generally ones are the smallest atomic number that can berepresented wholly, or on the left side of the decimal point. Thus on the right side of the decimal pointwe begin counting in tenths. Moving from left-to-right on the right side of the decimal point place-values become successively smaller. One tenth representing a single part out of ten parts of one whole,thus we identify one tenth as:

1

10

one

divided-by

ten

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Bibliography

1. Wikipedia, 2011, Arithmetic, introductory paragraph, http://en.wikipedia.org/wiki/Arithmetic

2.
http://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Arithmetic

Documents

The Algorithms of Arithmetic