# Reasonable Basic Algebra by A. Schremmer - HTML preview

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Copyright ©2006, 2007 A. Schremmer. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Section 1 , no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".

1 Educologists who deem the footnotes "inflammatory" need only turn them off by uncommenting "renewcommand-footnote" in the ADJUSTMENTS TO DOCUMENT.

To Franchise, Bruno and Serge.

Contents

Preface xi

I Elements of Arithmetic 1

1 Counting Number-Phrases 3

1.1 What Arithmetic and Algebra are About 3

1.2 Specialized Languages 4

1.3 Real-World 5

1.4 Number-Phrases 5

1.5 Representing Large Collections 7

1.6 Graphic Illustrations 12

1.7 Combinations 14

1.9 Decimal Number-Phrases 17

2 Equalities and Inequalities 19

2.1 Counting From A Counting Number-Phrase To Another ... 19

2.2 Comparing Collections 21

2.3 Language For Comparisons 25

2.4 Procedures For Comparing Number-Phrases 29

2.5 Truth Versus Falsehood 30

2.6 Duality Versus Symmetry 31

3.1 Attaching A Collection To Another 33

3.3 Procedure For Adding A Number-Phrase 36

vn

4 Subtraction 39

4.1 Detaching A Collection From Another 39

4.2 Language For Subtraction 40

4.3 Procedure For Subtracting A Number-Phrase 42

4.4 Subtraction As Correction 43

5 Signed Number-Phrases 45

5.1 Actions and States 45

5.2 Signed Number-Phrases 47

5.3 Size And Sign 49

5.4 Graphic Illustrations 50

5.5 Comparing Signed Number-Phrases 51

5.6 Adding a Signed Number-Phrase 54

5.7 Subtracting a Signed Number-Phrase 57

5.8 Effect Of An Action On A State 59

5.9 From Plain To Positive 61

6 Co-Multiplication and Values 65

6.1 Co-Multiplication 65

6.2 Signed-Co-multiplication 67

II Inequations & Equations Problems 71

7 Basic Problems 1 Counting Numerators 73

7.1 Forms, Data Sets And Solution Subsets 73

7.2 Collections Meeting A Requirement 75

7.3 Basic Formulas 78

7.4 Basic Problems 83

8 Basic Problems 2 (Decimal Numerators) 89

8.1 Basic Equation Problems 90

8.2 Basic Inequation Problems 90

8.3 The Four Basic Inequation Problems 94

9 Translation & Dilation Problems 101

9.1 Translation Problems 102

9.2 Solving Translation Problems 104

9.3 Dilation Problems 108

9.4 Solving Dilation Problems Ill

10 Affine Problems 115

10.1 Introduction 115

10.2 Solving Affine Problems 116

11 Double Basic Problems 119

11.1 Double Basic Equation Problems 119

11.2 Problems of Type BETWEEN 120

11.3 Problems of Type BEYOND 130

11.4 Other Double Basic Problems 139

12 Double Affine Problems 145

III Laurent Polynomial Algebra 149

13 Repeated Multiplications and Divisions 151

13.1 A Problem With English 151

13.2 Templates 152

13.3 The Order of Operations 156

13.4 The Way to Powers 159

13.5 Power Language 162

14 Laurent Monomials 165

14.1 Multiplying Monomial Specifying-Phrases 165

14.2 Dividing Monomial Specifying-Phrases 168

14.3 Terms 172

14.4 Monomials 174

15 Polynomials 1: Addition, Subtraction 181

15.2 Laurent Polynomials 183

15.3 Plain Polynomials 186

15.5 Subtraction 190

16 Polynomials 2: Multiplication 191

16.1 Multiplication in Arithmetic 191

16.2 Multiplication of Polynomials 193

17 Polynomials 3: Powers of x 0 + h 197

17.1 The Second Power: (x 0 + h) 2 198

17.2 The Third Power: (x 0 + h) 3 202

17.3 Higher Powers: (xo + h) n when n > 3 205

17.4 Approximations 208

18 Polynomials 4: Division 209

18.1 Division In Arithmetic 209

18.2 Elementary School Procedure 211

18.3 Efficient Division Procedure 212

18.4 Division of Polynomials 221

18.5 Default Rules for Division 225

18.6 Division in Ascending Powers 229

Epilogue 231

1. Functions 231

2. Local Problems 233

3. Global Problems 237

4. Conclusion 238

1. Applicability And Definitions 241

2. Verbatim Copying 243

3. Copying In Quantity 243

4. Modificatons 244

5. Combining Documents 246

6. Collections Of Documents 246

7. Aggregation With Independent Works 246

8. Translation 247

9. Termination 247

10. Future Revisions Of This License 247

Preface

The prospect facing students still in need of Basic Algebra as they enter two-year colleges 2 is a discouraging one inasmuch as it usually takes at the very least two semesters before they can arrive at the course(s) that they are interested in—or required to take, not to dwell on the fact that their chances of overall success tend to be extremely low 3 .

Reasonable Basic Algebra (RBA) is a standalone version of part of From Arithmetic To Differential Calculus (A2DC), a course of study developed to allow a significantly higher percentage of students to complete Differential Calculus in three semesters. As it is intended for a one-semester course, though, RBA may serve in a similar manner students with different goals.

The general intention is to get the students to change from being "answer oriented", the inevitable result of "show and tell, drill and test", to being "question oriented 4 " and thus, rather than try to "remember" things, be able to "reconstruct" them as needed. The specific means by which RBA hopes to accomplish this goal are presented at some length below but, briefly, they include:

• An expositional approach, based on what is known in mathematics as MODEL THEORY, which carefully distinguishes "real-world" situations from their "paper-world" representations 5 . A bit more precisely, we start with processes involving "real-world" collections that yield either a relationship between these collections or some new collection and the students then have to develop a paper procedure that will yield the sentence representing the relationship or the number-phrase representing the new

2 Otherwise known, these days, as "developmental" students.

3 For instance, students who wish eventually to learn Differential Calculus, the "mathematics of change", face five or six semesters with chances of overall success of no more than one percent.

4 See John Holt's classic How Chidren Fail, Delacorte Press,1982. See Zoltan P. Dienes, for instance Building Up Mathematics.

collection.

EXAMPLE 1. Given that, in the real-world, when we attach to a collection of three apples to a collection of two apples we get a collection of five apples, the question for the students is to develop a paper procedure that, from 3 Apples and 2 Apples, the number-phrases representing on paper these real-world collections, will yield the number-phrase 5 Apples.

In other words, the students are meant to abstract the necessary concepts from a familiar "real-world" since, indeed, "We are usually more easily convinced by reasons we have found ourselves than by those which have occurred to others." (Blaise Pascal).

• A very carefully structured contents architecture —in total contrast to the usual more or less haphazard string of "topics"—to create systematic reinforcement and foster an exponential learning curve based on a Coherent View of Mathematics and thus help students acquire a Profound Understanding of Fundamental Mathematics 6 .

• A systematic attention to linguistic issues that often prevent students from being able to focus on the mathematical concepts themselves.

• An insistence on convincing the students that the reason things mathematical are the way they are is not because "experts say so" but because common sense says they cannot be otherwise.

The contents architecture was designed in terms of three major requirements.

1. From the students' viewpoint, each and every mathematical issue should:

• flow "naturally" from what just precedes it,

• be developed only as far as needed for what will follow "naturally",

• be dealt with in sufficient "natural" generality to support further developments without having first to be recast.

EXAMPLE 2. After counting dollars sitting on a counter, it is "natural" to count dollars changing hands over the counter and thus to develop signed numbers. In contrast, multiplication, division or fractions all involve a complete change of venue.

2. Only a very few very simple but very powerful ideas should be used to underpin all the presentations and discussions even if this may be at the cost of some additional length. After they have familiarized themselves with such an idea, in its simplest possible embodiment, later, in more complicated situations, the students can then focus on the technical aspects of getting

'See Liping Ma's Knowing and Teaching Elementary School Mathematics.

Xlll

the idea to work in the situation at hand. In this manner, the students eventually get to feel that they can cope with "anything".

EXAMPLE 3. The concept of combination-phrase is introduced with 3 Quarters + 7 Dimes in which Quarters and Dimes are denominators and where + does not denote addition as it does in 3 Quarters + 7 Quarters but stands for "and". (In fact, for a while, we write 3 Quarters & 7 Dimes.) The concept then comes up again and again: with 3 hundreds + 7 tens, with f + ^j, with 3x 2 + 7a: 5 , with 3x + 7y, etc, culminating, if much later, with 3i + 7j.

EXAMPLE 4. If we can change, say, 1 Quarter for 5 Nickels and 1 Dime for 2 Nickels, we can then change the combination-phrase 3 Quarters + 7 Dimes for 3 jJwarteTs x ^^ + 7 -Dimes x ^|!5 that is for the specifying-phrase 15 Nickels + 14 Nickels which we identify as 29 Nickels. (Note by the way that here x is a very particular type

7 Cents Xtetfcff

of multiplication, as also found in 3XteWaTs x 7 E ^1 S = 21 Cents.) Later, when having to "add" | + jq, the students will then need only to concentrate on the technical issue of developing a procedure to find the denominators that Fourth and Tenth can both be changed for, e.g. Twentieths, Hundredths, etc.

3. The issue of "undoing" whatever has been done should always be, if not always resolved, at least always discussed.

EXAMPLE 5. Counting backward is introduced by the need to undo counting forward and both subtracting and signed numbers are introduced by the need to undo adding, that is by the need to solve the equation a + x = b.

As a result of these requirements, the contents had to be stripped of the various "kitchen sinks" to be found in current BASIC ALGEBRA courses and the two essential themes RBA focuses on are affine inequations & equations and Laurent polynomials. This focus empowers the students in that, once they have mastered these subjects, they will be able both: i. to investigate the Calculus of Functions as in A2DC and ii. to acquire in a similar manner whatever other algebraic tools they may need for other purposes.

However, a problem arose in that the background necessary for a treatment that would make solid sense to the students was not likely to have been acquired in any course the students might have taken previously while, for lack of time, a full treatment of arithmetic, such as can be found in A2DC, was out of the question here.

Following is the "three PARTS compromise" that was eventually reached. Part I consists of a treatment of arithmetic, taken from A2DC but minimal in two respects: i. It is limited to what is strictly necessary to make sense of inequations & equations in Part II and Laurent polynomials in Part III, that is to the ways in which number-phrases are compared and operated

with. ii. It is developed only in the case of counting number-phrases with the extension to decimal number-phrases to be taken for granted even though the latter are really of primary importance—and fully dealt with in A2DC.

• Chapter 1 introduces and discusses the general model theoretic concepts that are at the very core of RBA: real-world collections versus paper-world number-phrases, combinations, graphic representations.

• Chapter 2 discusses comparisons, with real-world collections compared cardinally, that is by way of one-to-one matching, while paper-world number-phrases are compared ordinally, that is by way of counting. The six verbs, <, >, ^, ^, =, /, together with their interrelationships, are carefully discussed in the context of sentences, namely inequalities and equalities that can be TRUE or FALSE.

• Chapter 3 discusses the effect of an action on a state and introduces addition as a unary operator representing the real-world action of attaching a collection to a collection.

• Chapter 4 introduces subtraction as a unary operator meant to "undo" addition, that is as representing the real-world action of detaching a collection from a collection.

• Chapter 5 considers collections of "two-way" items which we represent by signed number-phrases.

EXAMPLE 6. Collections of steps forward versus collections of steps backward, Collections of steps up versus collections of steps down, Collections of dollars gained versus collections of dollars lost, etc

In order to deal with signed number-phrases, the verbs, <, >, etc, are extended to ©, ©, etc and the operators + and — to © and ©.

• Chapter 6 introduces co-multiplication between number-phrases and unit-value number-phrases as a way to find the value that represents the worth of a collection.

EXAMPLE 7. 3 Apples x 2J=|2j§ = q Cents as well as 3 Dollars x 7{^j| = 21 Cents We continue to distinguish between plain number-phrases and signed number-phrases with x and ©. Part II then deals with number-phrases specified as solution of problems.

• Chapter 7 introduces the idea of real-world collections selected from a set of selectable collections by a requirement and, in the paper-world, of nouns specified from a data set by a form. Letting the data set then consist of counting numerators, we discuss locating and representing the solution subset (of the data set) specified by a basic formula, i.e. of type x = xq, x < xq, etc where xq is a given gauge.

• Chapter 8 extends the previous ideas to the case of decimal numerators by introducing a general procedure, to be systematically used henceforth, in

which we locate separately the boundary and the interior of the solution subset. Particular attention is given to the representation of the solution subset, both by graph and by name.

• Chapter 9 begins the focus on the computations necessary to locate the boundary in the particular case of "special affine" problems, namely translation problems and dilation problems, which are solved by reducing them to basic problems.

• Chapter 10 then solves affine problems by reducing them to dilation problems and hence to basic problems. It concludes with the consideration of some affine-reducible problems.

• Chapter 11 discusses the connectors and, and/or, either/or, in the context of double basic problems, that is problems involving two basic inequations/equations (in the same unknown). Here again, particular attention is given to the representation of the solution subset, both by graph and by name.

• Chapter 12 wraps up the discussion of how to select collections with the investigation of double affine problems, that is problems involving two affine inequations/equations (in the same unknown).

Part III investigates plain polynomials as a particular case of Laurent polynomials.

• Chapter 13 discusses what is involved in repeated multiplications and repeated divisions of a number-phrase by a numerator and introduces the notion of signed power.

• Chapter 14 extends this notion to Laurent monomials, namely signed powers of x. Multiplication and division or Laurent monomials are carefully discussed.

• Chapter 15 extends the fact that decimal numerators are combinations of signed powers of ten to the introduction of Laurent polynomials as combinations of signed powers of x. Addition and subtraction of polynomials are then defined in the obvious manner.

• Chapter 16 continues the investigation of Laurent polynomials with the investigation of multiplication.

• Chapter 17 discusses a particular case of multiplication, namely the successive powers of xq + u.

• Chapter 18 closes the book with a discussion of the division of polynomials both in descending and ascending powers

This is probably the place where it should be disclosed that, as the development of this text was coming to an end, the author came across

a 1905 text 7 that gave him the impression that, in his many deviations from the current praxis, he had often reinvented the wheel. While rather reassuring, this was also, if perhaps surprisingly, somewhat disheartening.

Some of the linguistic issues affecting the students's progress are very specific and are directly addressed as such. The concept of duality, for instance, is a very powerful one and occurs in very many guises.

• When it occurs as "passive voice", duality is almost invariably confused with symmetry, a more familiar concept 8 . But, in particular, while duality preserves truth, symmetry may or may not.

EXAMPLE 8. "Jack is a child of Sue" is the dual of "Sue is a parent of Jack" and, since both refer to the same real-world relationship, they are either both true or both FALSE.

On the other hand, "Jack is a child of Sue" is the symmetrical of "Sue is a child of Jack" and, here, the truth of one forces the falsehood of the other. But compare with what would happen with "brother" or "sibling" instead of "child".

• When it occurs as indirect definition, duality is quite foreign to most students but absolutely indispensable in certain situations.

EXAMPLE 9. While Dollar can be defined directly in terms of Quarters by saying that 1 Dollar is equal to 4 Quarters, the definition of Quarter in terms of Dollar is an indirect one in that we must say that a Quarter is that kind of coin of which we need 4 to change for 1 Dollar and students first need to be reconciled with this syntactic form. The same stumbling block occurs in dealing with roots since y/9 is to be understood as "that number the square of which is 9" 9 . Other linguistic issues, even though more diffuse, are nevertheless systematically taken into account. For instance:

• While mathematicians are used to all sorts of things "going without saying", students feel more comfortable when everything is made explicit as, for instance, when & is distinguished from +. Hence, in particular, the explicit use in this text of default rules.

• The meaning of mathematical symbols usually depends on the context while students generally feel more comfortable with context-free termi-

7 H. B. Fine, College Algebra, reprinted by American Mathematical Society Chelsea, 2005.

The inability to use the "passive voice" is a most important linguistic stumbling block for students and one that Educologists have yet to acknowledge.

9 Educologists will surely agree that, for instance, these particular "reverse" problems would in fact be better dealt with in an algebraic context, i.e. as the investigation of 4x — 1 and x — 9. Iincidentally, this is the point of view adopted in A2DC where arithmetic and algebra are systematically "integrated".

nology, that is in the case of a one-to-one correspondence between terms

and concepts. • Even small linguistic variations in parallel cases disturb the students who

take these variations as having to be significant and therefore as implying

in fact an unsaid but actual lack of parallelism. In general, being aware of what needs to be said versus what can go without saying is part of what makes one a mathematician and, as such, requires learning and getting used to. Thus, although being pedantic is not the goal here, RBA tries very hard to be as pedestrian as possible and, if only for the purpose of "discussing matters", to make sure that everything is named and that every term is "explained" even if usually not formally defined.

The standard way of establishing truth in mathematics is by way of proof but the capacity of being convinced by a proof is another part of what makes one a mathematician. And indeed, since the students for whom RBA was written are used only to drill based on "template examples", they tend to behave as in the joke about Socrates' slave who, when led through the proof of the Pythagorean Theorem, answers "Yes" when asked if he agrees with the current step and "No" when asked at the end if he agrees with the truth of the Theorem. So, to try to be convincing, we use a mode of arguing somewhat like that used by lawyers in front of a court 10 .

Another reason for using a mode of reasoning more akin to everyday argumentation is that even people unlikely to become prospective mathematicians ought to realize the similarities between having to establish the truth in mathematics and having to establish the truth in real-life. Yet, as Philip Ross wrote recently, "American psychologist Edward Thorndike first noted this lack of transference over a century ago, when he showed that [...] geometric proofs do not teach the use of logic in daily life." 11 .

Finally, it is perhaps worth mentioning that this text came out of the author's conviction that it is not good for a society to have a huge majority of its citizens saying they were "never good in math". To quote Colin McGinn at some length:

" Democratic States are constitutively committed to ensuring and furthering the intellectual health of the citizens who compose them: indeed, they are

See Stephen E. Toulmin, The Uses of Argument Cambridge University Press, 1958 11 Philip E. Ross, The Expert Mind. Scientific American, August 2006.

only possible at all if people reach a certain cognitive level [■■■]■ Democracy and education (in the widest sense) are thus as conceptually inseparable as individual rational action and knowledge of the world. [... ] Plainly, [education] involves the transmission of knowledge from teacher to taught. But [knowledge] is true justified belief that has been arrived at by rational means. [... ] Thus the norms governing political action incorporate or embed norms appropriate to rational belief formation. [...]"

"A basic requirement is to cultivate in the populace a respect for intellectual values, an intolerance of intellectual vices or shortcomings. [... ] The forces of cretinisation are, and have always been, the biggest threat to the success of democracy as a way of allocating political power: this is the fundamental conceptual truth, as well as a lamentable fact of history."

" [However] people do not really like the truth; they feel coerced by reason, bullied by fact. In a certain sense, this is not irrational, since a commitment to believe only what is true implies a willingness to detach your beliefs from your desires. [... ] Truth limits your freedom, in a way, because it reduces your belief-options; it is quite capable of forcing your mind to go against its natural inclination. [... ] One of the central aims of education, as a preparation for political democracy, should be to enable people to get on better terms with reason — to learn to live with the truth." 12

2 Colin McGinn, Homage to Education, London Review of Books, August 16, 1990

Part I

Elements of Arithmetic

(Communicating By Way Of Number-Phrases)

procedure process

Chapter 1

Counting Number-Phrases

This chapter takes a brief look back at arithmetic to present it in a way that will be a better basis for looking at algebra because we will then be able to look at algebra as just a continuation of arithmetic.

1.1 What Arithmetic and Algebra are About

To put it as briefly as possible, Arithmetic and Algebra are both about developing procedures to figure out on paper the result of real-world processes without having to go through the real-world processes themselves. To make this a bit clearer, here are two examples from Arithmetic the Algebra counterpart of which we will deal with in Part III of this book.

EXAMPLE 1. In the real world, we may want to hand-out six one-dollar bills to each of four people. To find out ahead of time how many one-dollar bills this would amount to, we would put on the table six one-dollar bills for the first person, then six one-dollar bills for the second person, etc. The result of this real-world process is that this amounts to twenty-four one-dollar bills.

But with, say, hundreds of one-dollar bills to each of thousands of people, this process would be impractical and what we do instead is to represent on paper both the one-dollar bills and the people and then develop the procedure called multiplication, that is a procedure for figuring-out on paper how many one-dollar bills we will need as a result of the real-world process.

EXAMPLE 2. In the real world, we may want to split fourteen one-dollar bills among three people. To find out ahead of time how many one-dollar each person should get, we would put on the table one one-dollar bill for the first person, one one-dollar bill for the second person, one one-dollar bill for the third person, and then, in a second round, another one-dollar bill for the first person, another one-dollar bill for the second person, and so on until we cannot do a full round. The result of this real-world process is that each person would get four one-dollar bills with two one-dollar bills remaining un-split.

CHAPTER 1. COUNTING NUMBER-PHRASES

mathematical language

boldfaced

margin

index

But with thousands of one-dollar bills to be split among hundreds of people, this process would be impractical and what we do instead is to represent on paper both the one-dollar bills and the people and then develop the procedure called division, that is a procedure for figuring-out on paper how many one-dollar bills to give to each person and how many one-dollar bills will remain un-split as a result of the real-world process.

The difference between these two examples illustrate is not obvious but, as we shall see, it is a significant one which, in fact, is at the root of the distinction between Arithmetic and Algebra.

1.2 Specialized Languages

People working in any trade need to use words with a special meaning. Sometimes, these are special words but often they are common words used with a meaning special to the trade. For instance, what electricians call a "pancake" is a junction box that is just the thickness of drywall.

In the same manner, in order to develop and discuss the procedures of Arithmetic and Algebra, we will have to use a mathematical language, that is words that will sometimes be special words but will most of the time be just common words with a meaning special to MATHEMATICS.

EXAMPLE 3. While the words "process" and "procedure" usually mean more or less the same thing, in this book we shall reserve the word "process" for when we talk about what we do in the real world and we shall reserve the word "procedure" for when we talk about what we do on paper.

In this book, we will encounter a great many such words with special meaning, likely more than usual. The idea, though, is certainly not that the students should memorize the special meaning of all these words. These words are used as focusing devices to help the students see exactly what they are intended to see whenever we discuss an issue. Thus, quite often, these words with special meaning will not reappear once the discussion has been completed as they will have served their purpose.

However, in order to help students find where the special meaning of these words is explained, these words with special meaning will always be:

• boldfaced the first time they appear—which is where they are explained,

• printed in the margin of the page where they first appear and are explained,

• listed in the index at the end of the book with the number of the page where they first appear and are explained.

1.3 Real-World item

represent While in the real-world it is often possible to exhibit the items that are to P ict ure be dealt with this is not possible in a book. So, to start with, we need a way to make it clear when we are talking about real-world items as opposed to when we are talking about what we will use to represent these items on paper. string

In this book, when we will want to talk about real-world items, we will slash use pictures of these items. /

EXAMPLE 4. When we will be talking about real-world one-dollar bills, we will use the following picture

denominator collect collection numerator

1.4 Number-Phrases

Our first task in arithmetic is to find a way to represent real-world items

on paper. The underlying idea is quite simple.

1. Given real-world items, in order to represent them on paper, we need

to convey two pieces of information: • We must write a denominator to say what kind of items we are dealing with. Of course, for this to be possible, all the items will have to be of the same kind and this will not work when the items are of different kinds as, for instance, when we are dealing with ten-dollar bills together with one-dollar bills. So, for the time being, we will deal only with items that are all of the same kind and in this case we will say that we can collect the items into a collection. EXAMPLE 5. Given the following real-world items,

since they are all of the same kind (they make up a collection) we can use as a denominator the name of the President whose picture is on them, that is

Washington

We must write a numerator to say how many of these items there are in the collection we are dealing with.

The first approach that comes to mind is just to write a string of slashes, that is to write a slash / for each and every item in the real-world collection.

CHAPTER 1. COUNTING NUMBER-PHRASES

number-phrase

nature (of a collection)

size (of a collection)

number

quantity

quality

EXAMPLE 6. Given the following real-world items,

since they are all of the same kind they make up a collection and to get a numerator we can just write a / for each and every item in the collection that is

/// It is usual first to write the numerator and then to write the denominator and the result then makes up what we shall call a number-phrase. EXAMPLE 7. Given the following real-world items,

since they are all of the same kind they make up a collection which we can represent by the number-phrase

III Washingtons

2. Conversely, given a number-phrase, to get the collection that it represents,

i. The denominator tells us the nature of the collection, that is what kind of items are in the collection,

ii. The numerator tells us the size of the collection, that is the number of items that are in the collection.

EXAMPLE 8. Given the number-phrase /// Washingtons, to get the collection of

real-world items that it represents:

i. The denominator Washingtons tells us that the items in the collection are like

ii. The numerator /// then tells us that there must be a St; tion for each slash in the numerator. iii. Altogether, the slash number-phrase

III Washingtons

represents the collection of real-world items

in the collec-

3. In other words, compared to a photograph of the collection, a number-phrase causes no loss of information as all we did was just to separate quantity —represented by the numerator — from quality —represented by the

denominator 1 . (Keep in mind, though, that this only works for collections.) accounting

As a matter of fact, this is most likely how, several thousands of years coun

counting miniDGr _ i3iircLSG ago, Arithmetic, got started when, one may imagine, Sumerian merchants, _,. .

faced with the problem of accounting for more goods in the warehouse shorthand and/or money in the safe than they could handle directly, decided to have both the goods and the money represented by various scratches on clay tablets so that they could see from these scratches the situation their business was in without the inconvenience of having to go to the warehouse and/or to open the safe.

1.5 Representing Large Collections

With large collections, a problem arises in that it becomes difficult to see, at a glance, how many items a long string of slashes represents. EXAMPLE 9. Given the number-phrase

//////////////////////////////// Washington

it is not immediately clear how many items are in the collection that the number-phrase represents.

What we will do is to count the collection and we will write what we shall therefore call a counting number-phrase. There are three stages to developing the procedure.

1. We must begin by memorizing the following digits as shorthands

for the first nine strings of slashes:

In spite of which this is precisely the point where, in the name of "abstraction", Educologists cut their students away from denominators without noticing, of course, that this is exactly the point where they start losing them.

CHAPTER 1. COUNTING NUMBER-PHRASES

basic succession basic collection basic counting

number-phrase basic counting count end-digit

Moreover, the various procedures that we shall use will also require that we have already memorized the basic succession, that is the digits in the order:

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

"one, two, three, four, five, six, seven, eight, nine" NOTE. There is nothing sacred about TEN: it is simply because of how many fingers we have on our two hands—"digit" is just a fancy word for "finger"—and we could have used just about any number of digits instead

of TEN.

In fact, deep down, computers use only two digits, 0 and f, because any electronic device is either off or on. At intermediate levels, computers may use eight (0, f, 2, 3,4,5,6, 7) or sixteen digits (0, f, 2, 3,4, 5, 6, 7, 8, 9, a, b, c, d, e, /).

The Babylonians used sixty digits, a historical remnant of which can be seen in the fact that there are sixty seconds to a minute and sixty minutes to an hour.

The point is that all that we do with ten digits could easily be done with any number of digits 2 .

2. We can then represent a basic collection, that is a collection with no more items than we have digits, that is no more than nine items, by a basic counting number-phrase.

a. Given a basic collection, to get the numerator of the counting number phrase the procedure, called basic counting, is:

i. We count the collection, that is we point successively at each and every item in the collection while saying the digits in the basic succession that we memorized.

ii. The numerator is the end-digit, that is the last digit we say. EXAMPLE 10. Given the collection

to get the basic counting number-phrase that represents it:

i. We can use for the denominator the name of the President whose picture is on

them, that is Washington.

ii. We count the collection to get the numerator, that is

We point at each and everyone of: while we say:

2 Z. P. Dienes always used to start his workshops with second graders, base-THREE arithmetic blocks and the digits 0, 1, 2.

1.5. REPRESENTING LARGE COLLECTIONS

and the end-digit gives us 3 for the numerator. iii. Altogether, the collection

is represented by the basic counting number-phrase

3 Washingtons

pick

extended counting

extended collections

extended succession

numerals

b. Conversely, given a basic counting number-phrase, to get the basic collection that it represents:

i. We pick one item —of the kind specified by the denominator—each and every time we say a digit in the basic succession

ii. We stop after we have picked the item for the numeral in the numerator EXAMPLE 11. Given the basic counting number-phrase

5 Washingtons

to get the basic collection that it represents:

i. The denominator Washingtons tells us that the items to be picked must be of the

same kind as

ii. The numerator 5 tell us to pick an item each and every time we say a digit in the

succession; we stop after we have picked the item for the end-digit:

We say:

1, 2, 3, 4, 5

We pick each and every one of:

iii. Altogether, the basic counting number-phrase

5 Washingtons represents the basic collection

3. For extended counting, that is for counting extended collections, that is for collections with more items than we have digits, we can continue to proceed essentially as above: we must begin by memorizing the

extended succession, that is the numerals that follow the basic succes-

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

sion

namely

10, 11,

12,

CHAPTER 1. COUNTING NUMBER-PHRASES

endless

that is: numerals I

we say

meaning

to make us think of:

10 11

12 13

19

20 21

ten

eleve - n

elve-tw

thir - teen

nine - teen

twen - ty

twen - ty - one

ten - one

ten - two ten - three

ten - nine

two - tens

two - tens & one

Trtmui I imuui ii imumiii

irtnuiL iiiiiiiii

TitmuiTTtmui i

NOTE. The words we say for the numerals are far from being as systematic as the numerals themselves. This is due in part to the fact that these words slowly evolved over a very long time.

However, and this is possibly the single most important fact about Arithmetic, while there are only so many digits in the basic succession—NINE in our case, the extended succession is endless.

a. Given an extended collection, to get the numerator of the counting number-phrase that represents it:

i. We begin by pointing successively at each and every item in the collection while saying the digits in the basic succession that we memorized, ii. We continue by pointing successively at each and every item in the collection while saying the numerals in the extended succession that we memorized. iii. The numerator is the end-numeral, that is the last numeral we say.

EXAMPLE 12. Given the extended collection,

to get the counting number-phrase:

we point at each and everyone of:

while we say:

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

1.5. REPRESENTING LARGE COLLECTIONS

11

ii. We continue with an extended count, that is: we point at each and everyone of:

=n ...■.■-.'

.--■—-=W Mr

==m !t* r ~~-=m M: —

while we say: 10

11, 12, 13, 14, ...

>

... 29. 30, 31, 32

iii. Altogether, the extended collection

is represented by the counting number-phrase

32 Washingtons

b. Given an extended counting number-phrase, to get the collection:

i. We begin by picking one item each and every time we say a digit in the

basic succession

ii. We continue by picking one item each and every time we say a digit in

the extended succession

iii. We stop after we have picked the item for the end-numeral.

EXAMPLE 13. Given the extended counting number-phrase,

32 Washingtons

to get the collection that it represents:

i. The denominator Washingtons tells us that the items to be picked must be of the

same kind as

ii. The numerator 32 tells us to pick an item each and every time we say a digit in the

basic succession and then one each and every time we say a numeral in the extended

succession; we stop after we have picked the item for the numerator.

iii. Altogether, the extended counting number-phrase

32 Washingtons

represents the extended collection

CHAPTER 1. COUNTING NUMBER-PHRASES

illustrate

graph

graph (to)

ruler

tick-marks

label

graph

NOTE. The sticklers among us will have rightfully observed that, strictly speaking, counting is neither a paper procedure since it involves the real-world items nor a real world process since it involves the digits we write on paper. Indeed, counting is a bridge from the real-world to the paper-world.

1.6 Graphic Illustrations

As pointed-out at the beginning of this book, it is usually easier to work with representations of collections on paper than with the real-world collections themselves. But, once we have represented collections with number-phrases, we will often also want to illustrate the number-phrase with a graph. For short, we shall often say that we graph the number-phrase.

For that purpose, we will use rulers that are straight lines with:

• an arrowhead to indicate the way the succession goes

• tick-marks to be labeled with the numerators

• a label for the denominator.

EXAMPLE 14. To graph collections represented by basic counting number-phrases whose denominator is Washingtons, we use rulers such as

_i 1 1 1 1 1 1 1 1 l^. Washingtons

However, graphing collections represented by number-phrases can raise issues of its own.

1. In the case of basic counting number-phrases, there is no problem and, in fact, as soon as we label the tick-marks with numerators, the arrowhead ceases to be necessary. (But then, there is no point in erasing it either.)

EXAMPLE 15. To graph collections represented by basic counting number-phrases whose denominator is Washingtons, we use the ruler

—i 1 1 1 1 1 1 1 1 l^. Washingtons

0123456789

Then, given a basic counting number-phrase, one usually places a dot on the corresponding tick-mark.

EXAMPLE 16. The graph that represents the collection represented by the counting number-phrase 3 Washingtons is

—' ' ' * ' ' ' ' ' >-^- Washington

0123456789

2. In the case of extended counting number-phrases, one problem is that we may not be able to draw a long enough ruler.

EXAMPLE 17. We can barely graph 15 Washingtons (by extending the ruler into the margin):

i i i i i i i i i i i i i i

-•—>• Washingtons

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

but we cannot extend the ruler enough to represent 37 Washingtons

A work-around could be to draw the tick-marks closer together. But then

we may not be able to label all the tick-marks.

EXAMPLE 18. On the following ruler

0123456789 * wasmngions

we don't have enough room to write two-digit numerators.

One workaround to that is to label the tick-marks only every so often. However it is usually better to do so regularly, that is every so many. To make it easier to read the ruler, it is usual in this case to make the tick-marks that are labeled longer and, if these are far apart, to make the middle tick-marks a bit longer too.

EXAMPLE 19. In the following ruler, only every eighth tick-mark, that is 8, 16, 24, 32, etc, is labeled:

I ■ ■ ■ ' ■ ■ ■ I ■ ■ ■ I ■ ■ ■ I ■ ■ ■ I ■ ■ ■ I ■ ■ ■ I ■ ■ ■ I ■ ■ ■ I ■ ■ ■ I ■ ■ ■ I ■ ■ ■ I ■ ■ ■ I ■ ■ ■ I ^. lllachinntnnc

0 8 16 24 32 40 48 56 > washingtons

and the middle tick-marks, 4, 12, 20, etc, are made easier to see by being made a bit larger.

EXAMPLE 20. The graphic that represents the collection represented by the extended counting number-phrase 37 Washingtons is

1 ■ ■ ■ I ■ ■ ■ I ■ ■ ■ I ■ ■ ■ I ■ ■ ■ I ■ ■ ■ I ■ ■ ■ I ■ ■ ■ I ■ ■ ■ I ■#■ I ■ ■ ■ I ■ ■ ■ I ■ ■ ■ I ■ ■ ■ I ^. lllaehinntnnc

0 8 16 24 32 40 48 56^ «U0SHIIiyiUIIS

3. But, to graph collections represented by really large counting number-phrases, we will not even be able to draw all the tick-marks—and even so we will often have to write the labels at an degree angle for them to fit.

EXAMPLE 21. In the following ruler, only every thousandth tick-mark, that is 1000, 2000, 3000, etc, is drawn and labeled.

o L 4 % % 4 k >o 4 V Washingtons

% % % % % % % % %

And another workaround may be not to start at 0.

EXAMPLE 22. Suppose we are not involved with any numerator less than 4000 and more than 13000. Then we would use rulers such as

—' ' ' ' ' ' ' ' ' *-> Washingtons

%, •% °0 0 \ •%>„ % ^o„ % <?„ <fr °o °o °o °o °o °o \ % °o 0 %

CHAPTER 1. COUNTING NUMBER-PHRASES

sort

set

combination-phrase

&

1.7 Combinations

When there is more than one kind of items, they do not make up a collection and we cannot represent them by number-phrases. EXAMPLE 23. Given the following real-world items,

since they are not all of the same kind (they do not make up a collection) there is no one President whose name we can use as a denominator. We then proceed as follows:

1. We sort the items by kind into collections so that we now have a set of collections.

EXAMPLE 24. In the above example, we can sort the real-world items into a set of collections:

2. We represent the set of collections by a combination-phrase by writing the number-phrases that represent each one of the collections separated by the symbol & to be read as "and".

EXAMPLE 25. In the above example, we can represent the set of collections by the combination-phrase:

4 Washington & 2 Hamiltons & 1 Franklin

3. The graphic representation of a combination-phrase requires as many rulers as there are kinds of collections in the set of collections that the combination-phrase represents.

EXAMPLE 26. In the above example, since there are three kinds of bills, we need three rulers:

1 >-^ Washington's

0 12 3 4 5 6 7

0123456789

0 12 3 4 5 6 7

-> Hamiltons

Franklins

We end this chapter with a few remarks about why we are using the term number-phrases as opposed to just the term numbers as is usual in most current Arithmetic textbooks 3 .

1. A numerator by itself, that is without a denominator, represents a number which is not something in the real-world that we can see and touch. EXAMPLE 27. When asked "Can you show what 3 represents?", we usually respond

by showing three real world items, for instance

but this is what the number-phrase 3 Washingtons represents and not what 3 by itself

represents. In fact, there is no way we can show what 3 by itself represents.

In contrast, number-phrases represent collections which are things in the real-world that we can see and touch. This is exactly the reason why we use number-phrases even if they make things more cumbersome.

2. Aside from anything else, we should realize that when textbooks use the word number they are talking—usually without saying it—about the concepts represented by the numerators that are actually printed. EXAMPLE 28. When a textbook says "3 is the number of one-dollar bills on the desk", what is meant is "3 is the numerator that represents the number of one-dollar bills on the desk". Indeed, 3 is only a mark on paper that tells us how many

li s there are in the real-world collection that is on the desk. So, when textbooks use the term number instead of the term numerator, they are not just using one term instead of another, they are, at best, blurring the distinction between the real-world and the paper-world we use to discuss the real-world 4 .

3. Number phrases allow us to be very precise as to what we are dealing with. In particular, the use of number phrases allows us to distinguish:

• matters of quality, that is questions about the kind of the items under consideration

from

• matters of quantity, that is questions about the number of the items under consideration.

3 Educologists will be glad to measure the progress accomplished since Chrystal's Textbook of Algebra infamous opening: iL The student is already familiar with the distinction between abstract and concrete arithmetic. The former is concerned with those laws of, and operations with, numbers that are independent of the things numbered; the latter is taken up with applications of the former to the numeration of various classes of things." At worst, one can wonder if educologists are not just confusing the two worlds.

EXAMPLE 29. Given the collection m^zjmm&iiim sitting on a desk, we can ask three very different questions:

■ " What is on the desk?" which we answer in Arithmetic by writing the counting-number-phrase

5 Washingtons

■ " What kind of items are on the desk?" which we answer in arithmetic by writing the denominator

Washingtons

■ "How many items are on the desk?" which we answer in Arithmetic by writing the numerator

5

4. The distinction we make in Arithmetic between denominators and number-phrases with the numerator 1 is very similar to the informal distinction we make in English between "a" and "one".

EXAMPLE 30. In Arithmetic, we distinguish the denominator Washington from the number-phrase 1 Washington the same way as in English we distinguish between

■ "This looks like a five-dollar bill"

which, just like "This looks like a ten-dollar bill" or "This looks like a twenty-dollar bill" is a qualitative statement because they all are statements about what kind of bills they look like.

■ "This looks like one five-dollar bill"

which, just like "This looks like two five-dollar bills" or "This looks like three five-dollar bills", is a quantitative statement because they all are statements about how many bills it looks there are.

Quite often, though and as we will see in many different situations, the numerator 1 "goes without saying".

Example 31.

3 Washingtons + Washingtons

is understood to mean

3 Washingtons + 1 Washington

and, to take an example from things to come, in the same manner

3x + x

is understood to mean

3x + Ix

So, even though we shall avoid letting the numerator 1 "go without saying", just in case and to be on the safe side, we set the

1.9. DECIMAL NUMBER-PHRASES

17

DEFAULT RULE # 1. When there is no numerator in front of a denom- amount inator and it is otherwise clear that we are dealing with a counting number- '' ll "

phrase, it then goes without saying that the numerator is understood to be 1.

NOTE. Unfortunately, this default rule is often abbreviated as "when there is no numerator, the numerator is 1" which is dangerous because when we say that there is no numerator it is tempting to think that the numerator is 0!

number-phrase, decimal numerator, decimal denominator

1.9 Decimal Number-Phrases

We will work not only with collections of items but also with amounts of stuff and, just as we use counting number-phrases to represent collections of items, in order to represent amounts of stuff we will use decimal number-phrases that consist of a decimal numerator and a denominator:

EXAMPLE 32. We can represent twenty-four apples by the counting number-phrase

24 Apples

but in order to represent an amount of gold, we need a decimal number-phrase such as

31.72 Grams of gold

Unfortunately, this being a text on Basic Algebra, there was space only for the smallest possible investigation of Arithmetic, that is one limited to the introduction, illustration and discussion of the concepts strictly necessary to the understanding of Basic Algebra. So, for lack of space, this was done using only counting number-phrases even though, as just noted above, many real-world situations require decimal number-phrases instead.

More precisely, even though the investigation of decimal number-phrases is intimately related to the representation of large collections, in the above

section and for lack of space we had to take a short cut, namely use extended counting rather than only basic counting together with combinations. Had we had the space to develop the latter approach for representing large collections, it would then have immediately and effortlessly led to decimal number-phrases.

So, here we will have to rely on the reader's own knowledge of decimal numbers. However, the interested reader will find a full investigation in Self-Contained Arithmetic as well as in From Arithmetic To Differential Calculus.

Chapter 2

Comparisons:

Equalities and Inequalities

We investigate the first of the three fundamental processes involving two collections. We will introduce the procedure in the case of basic collections using basic counting number-phrases.

2.1 Counting From A Counting Number-Phrase To Another

Before we can develop the procedures for these three fundamental processes, we must make the concept of counting more flexible by allowing a count

• to start with any digit which we will call the start-digit. (So, the start-digit doesn't have anymore to be 1 as it always did in Chapter 1.)

• to end with any digit which we will call the end-digit. (So, the end-digit may be "before" the start digit as well as "after" the start digit.)

Specifically, when we count from the start-digit to the end-digit:

i. We start (just) after the start-digit

ii. We stop (just) after the end-digit. However, given a start-digit and a end-digit, we may have to count in either one of two possible directions:

• We may have to count-up, that is we may have to use the succession

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

count

start-digit

end-digit

count from ... to

direction

count-up

which we read along the arrow, that is from left to right. EXAMPLE 1. To count from the start-digit 3 to the end-digit 7:

CHAPTER 2. EQUALITIES AND INEQUALITIES

count-down precession

i. We must count up, that is we must use the succession

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

ii. We start counting up in the succession after the start-digit 3, so that 4 is the first digit we say,

iii. We stop counting up in the succession after the end-digit 7 so that 7 is the last digit we say

Altogether, the count from the start-digit 3 to the end-digit 7 is

4, 5, 6, 7

y

We may have to count-down, that is we may have to use the precession

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

<

which we read along the arrow, that is from right to left.

NOTE. If we prefer to read from left to right, we may also write the

precession as

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

which we read along the arrow, that is from left to right.

EXAMPLE 2. To count from the start-digit 6 to the end-digit 2: i. We must count down, that is we must use the precession

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

y

ii. We start counting down in the precession after the start-digit 6 so that 5 is the first digit we say

5, ...

y

iii. We stop counting down in the precession after the end-digit 2 so that 2 is the

last digit we say.

... 2

Altogether, the count from the start-digit 6 to the end-digit 2 is

5,4,3,2

„ T , „ A , . . ,, . 9, 8, 7, 6, 5, 4, 3, 2, 1 . , ,.,

NOTE. Memorizing the precession > just like we memo-

. , ,, . 1, 2,3,4,5,6, 7,8,9 nzed the succession > makes hie a lot easier.

2.2. COMPARING COLLECTIONS

21

Finally, the length of a count from a start-digit to an end-digit is how length (of a count)

many digits we say regardless of the direction, that is whether up in the com P are

, ■ , i • match one-to-one

succession or down m the precession. , .

leftover

EXAMPLE 3. When we count from the start-digit 3 to the end-digit 7, the length relationship of the count is 4. hold (to)

EXAMPLE 4. When we count from the start-digit 6 to the end-digit 2, the length simple of the count is 4.

What that does, as in Chapter 1, is again to separate quality —represented by the direction of the count, up or down, from quantity — represented by the length of the count, how many digits we count.

NOTE. As already mentioned, we will only use basic counting, whether up or down, but extended counting would work essentially the same way.

2.2 Comparing Collections

Given two collections, the first thing we usually want to do is to compare the first collection to the second collection but an immediate issue is whether the kinds of items in the two collections are the same or different. • When the two given collections involve different kinds of items, they don't they cannot be compared.

EXAMPLE 5. If Jane's collection is

and Nell's collection is ~*

we don't really want to compare them because that would mean that we are really

looking at the items as and — , that is that we

would be ignoring some of the details in the pictures.

• When the two given collections involve the same kind of items, the real-world process we will use to compare the two collections will be to match one-to-one each item of the first collection with an item of the second collection and to look in which of the two collections the leftover items are in.

When the two given collections involve the same kind of items, there are

six several different relationships that can hold from the first collection

to the second collection.

1. Up front, we have two very simple relationships:

CHAPTER 2. EQUALITIES AND INEQUALITIES

is-the-same-in-size-as is-different-in-size-from

When there are no leftover objects, we will say that the first collection is-the-same-in-size-as the second collection.

EXAMPLE 6. To compare in the real-world Jack's

with Jill's

we match Jack's collection one-to-one with Jill's collection:

Since there is no leftover item in either collection, the relationship between Jack's collection and Jill's collection is that:

Jack's collection is-the-same-in-size-as Jill's collection

When there are leftover objects, regardless of where they are, we will say that the first collection is-different-in-size-from the second collection.

EXAMPLE 7. To compare Jack's

® II®

,.:.;fl vvith Jill's

in the real-world, we match Jack's collection one-to-one with Jill's collection:

Since there are leftover items in one of the two collections, the relationship between Jack's collection and Jill's collection is that:

Jack's collection is-different-in-size-from Jill's collection

EXAMPLE 8. To compare in the real-world Jack's

with Jill's

we match Jack's collection one-to-one with Jill's collection:

Since there are leftover items in one of the two collections, the relationship between Jack's collection and Jill's collection is that:

2.2. COMPARING COLLECTIONS

2::;

Jack's collection is-different-in-size-from Jill's collection strict

is-smaller-in-size-than 2. When two collections are different-in-size, then there are two possible is-larger-in-size-than strict relationships depending on which of the two collections the leftover mutually exclusive item, if any, are in:

• When the leftover items are in the second collection, we will say that the first collection is-smaller-in-size-than the second collection.

EXAMPLE 9. To compare Jack's

P

with Jill's

in the real-world, we match Jack's collection one-to-one with Jill's collection:

Since the leftover items are in Jill's collection, the relationship between Jack's collection and Jill's collection is that:

Jack's collection is-smaller-in-size-than Jill's collection

When the leftover objects are in the first collection, we will say that the first collection is-larger-in-size-than the second collection.

EXAMPLE 10. To compare in the real-world Jack's

with Jill's

we match Jack's collection one-to-one with Jill's collection:

Since the leftover items are in Jack's collection, the relationship between Jack's collection and Jill's collection is that:

Jack's collection is-larger-in-size-than Jill's collection

The relationship is the same as and the two strict relationships, is-smaller-than and is-larger-than, are mutually exclusive in the sense that as soon as we know that one of them holds, we know that neither one of the other two can hold.

3. Quite often, though, instead of the above three relationships, we will need to use another two relationships that we shall call lenient.

CHAPTER 2. EQUALITIES AND INEQUALITIES

is-no-larger-than is-no-smaller

a. Instead of wanting to make sure that a first collection is-smaller-than a second collection, we may just want to make sure that the first collection is-no-larger-than the second collection, that is we may include collections that are-the-same-as.

What this mean is that instead of requiring that, after the one-to-one matching, the leftover items be in the second collection, we only require that the leftover items not be in the first collection and this is of course the case when the leftover items are in the second collection as before ... but also when there are no leftover items in either collection and therefore certainly no leftover in the first collection.

and Jill's collection is

EXAMPLE 11. If Jack's collection is then we have that:

Jack's collection is no-larger-in-size-than Jill's collection since, after one-to-one matching,

IB

there is no leftover item in Jack's collection.

EXAMPLE 12. If Mike's collection is it is also the case that:

*' and Jill's collection is

Mike's collection is no-larger-in-size-than Jill's collection since, after one-to-one matching,

m

there is no leftover item in either collection and therefore certainly no leftover item in Mike's collection.

b. Similarly, instead of wanting to make sure that a first collection is-larger-than a second collection, we may just want to make sure that the first collection is-no-smaller than the second collection, that is we include collections that are-the-same.

What this mean in the real-world is that instead of requiring that, after the one-to-one matching, the leftover items be in the first collection, we only require that the leftover items not be in the second collection and this

2.3. LANGUAGE FOR COMPARISONS

25

is of course the case when the leftover items are in the first collection as before ... but also when there are no leftover items in either collection and therefore certainly no leftover in the second collection.

and Jane's collection is

EXAMPLE 13. If Dick's collection is then we have that:

Dick's collection is no-smaller-in-size-than Jane's collection since, after one-to-one matching,

IF

there is no leftover item in Jane's collection

1

EXAMPLE 14. If Mary's collection is it is also the case that:

and Jane's collection is

Mary's collection is no-smaller-in-size-than Jane's collection since, after one-to-one matching,

there is no leftover item in either collection and therefore certainly no leftover item in Jane's collection.

The two lenient relationships are not mutually exclusive in the sense that, given two collections, even if we know that one lenient relationship is holding from the first collection to the second collection, we cannoi be sure that the other lenient relationship does not hold from the first collection to the second collection because the first collection could be holding because the first collection is-the-same-as the second collection in which case the other lenient relationship would be holding too.

On the other hand, if both lenient relationships hold from a first collection to a second collection, then we know for sure that the first collection is-the-same-as the second collection.

2.3 Language For Comparisons

In order to represent on paper relationships between two collections, we first need to expand our mathematical language beyond number-phrases.

CHAPTER 2. EQUALITIES AND INEQUALITIES

verb

is-equal-to

is-not-equal-to

<

is-less-than

>

is-more-than

is less-than-or-equal-to

is more-than-or-equal-to strict

lenient verbs sentence (comparison)

1. Given a relationship between two collections, we need a verb to represents this relationship. In keeping with our distinguishing between what we do in the real-world and what we write on paper to represent it, as between a real-world process and the paper procedure that represents it, we use different words for a real-world relationships and for the verbs we write on paper to represent it:

• To represent on paper the real-world simple relationships:

— is-the-same-in-size-as, we will use the verb = which we will read as is-equal-to,

— is-different-in-size-from, we will use the verb ^ which we will read as is-not-equal-to,

• To represent on paper the real-world strict relationships:

— is-smaller-in-size-than, we will use the verb <, which we will read as is-less-than.

— is-larger-in-size-than, we will use the verb > which we will read as is-more-than,

• To represent on paper the real-world lenient relationships

— is-no-larger-in-size-than, we will use the verb ^, which we will read as is less-than-or-equal-to.

— is-no-smaller-in-size-than, we will use the verb ^, which we will read as is more-than-or-equal-to.

We will say that

• The verbs > and < are strict verbs because they represent the strict relationships is-smaller-in-size-than and is-larger-in-size-than.

• The verbs ^ and ^ are lenient verbs because they represent the lenient relationships is-no-larger-in-size-than and is-no-smaller-in-size-than.

2. Then, to indicate that a relationship holds from one collection to another, we write a comparison-sentence that consists of the number-phrases that represent the two collections with the verb that represents the relationship in-between the two number-phrases.

EXAMPLE 15. Given Jack's the relationship

m and Jill's

we represent

Jack's collection is the same as Jill's collection

by writing the comparison-sentence

3 Dollars = 3 Dollars

three dollars is-equal-to three dollars.

2.3. LANGUAGE FOR COMPARISONS

27

EXAMPLE 16. Given Jack's 'S.-i=^fe™Ja anc j jjii's resent the relationship

Jack's collection is different from Jill's collection

by writing the comparison-sentence

3 Dollars ^ 7 Dollars which we read as

three dollars is-not-equal-to seven dollars.

EXAMPLE 17. Given Jack's sent the relationship

and Jill's

Jack's collection is different from Jill's collection by writing the comparison-sentence

5 Dollars ^ 3 Dollars

five dollars is-not-equal-to three dollars.

EXAMPLE 18. Given Jack's resent the relationship

f-0 I

and Jill's

Ifipip

Jack's collection is smaller than Jill's collection

by writing the comparison-sentence

3 Dollars < 7 Dollars

three dollars is less than seven dollars.

EXAMPLE 19. Given Jack's sent the relationship

sh, we rep-

we repre-

we rep-

and Jill's

we repre-

Jack's collection is larger than Jill's collection

by writing the comparison-sentence

5 Dollars > 3 Dollars

five dollars is more than three dollars.

EXAMPLE 20. Given Jack's

and Jill's

we re pre-

CHAPTER 2. EQUALITIES AND INEQUALITIES

equality inequality (plain)

sent the relationship

Jack's collection is no-larger than Jill's collection by writing the comparison-sentence

3 Dollars g 5 Dollars, which we read as

three dollars is less-than-or-equal-to five dollars.

IP IP

EXAMPLE 21. Given Mike's S=i and Jill's

resent the relationship

Mike's collection is no-larger than Jill's collection by writing the comparison-sentence

5 Dollars ^ 5 Dollars, which we read as

five dollars is less-than-or-equal-to five dollars.

we rep-

EXAMPLE 22. Given Dick's 1 and Jane's

sent the relationship

Dick's collection is no-smaller than Jane's collection by writing the comparison-sentence

5 Dollars ^ 2 Dollars, which we read as

three dollars is more-than-or-equal-to five dollars.

we repre-

EXAMPLE 23. Given Mary's 1 a nd Jane's " 9, we repre-

sent the relationship

Mary's collection is no-smaller than Jane's collection which we represent by writing the comparison-sentence

2 Dollars ^ 2 Dollars, which we read as

three dollars is more-than-or-equal-to five dollars. 3. Finally, comparison-sentences are named according to the verb that they involve

• Comparison-sentences involving the verb = are called equalities.

Example 24.

3 Dollars = 3 Dollars is an equality

• Comparison-sentences involving the verb ^ are called plain inequalities.

Example 25.

3 Dollars =£ 5 Dollars is a plain inequality

• Comparison-sentences involving the verbs > or < are called strict in- inequality (strict) equalities inequality (lenient)

Example 26.

3 Dollars < 7 Dollars and 8 Dollars > 2 Dollars are strict inequalities

• Comparison-sentences involving the verbs ^ and ^ are called lenient inequalities.

Example 27.

3 Dollars ^ 7 Dollars and 8 Dollars ^ 2 Dollars are lenient inequalities

2.4 Procedures For Comparing Number-Phrases

Given two number-phrases, the procedure for writing the comparison-sentences that are true will depend on whether the number-phrases are basic counting number-phrases or decimal number-phrases.

Given two basic counting number-phrases, we must see whether we must count-up or count- down from the first numerator to the second numerator 1 .

There are three possibilities depending on the direction we have to count when we count from the numerator of the first number-phrase to the numerator of the second number-phrase:

• We may have to count up, in which case the comparison-sentence is: first counting number-phrase < second counting number-phrase (with < read as "is-less-than")

EXAMPLE 28. To compare the given basic counting number-phrases 3 Washingtons

and 7 Washingtons

i. We must count from 3 to 7:

4, 5, 6, 7

that is we must count up.

ii. So, we write the strict inequality:

3 Washingtons < 7 Washingtons

We may have to count down, in which case the comparison-sentence is:

first counting number-phrase > second counting number-phrase

EXAMPLE 29. To compare the given basic counting number-phrases 8 Washingtons

and 2 Washingtons

i. We must count from 8 to 2:

7, 6, 5, 4, 3, 2

1 Educologists will be glad to know that, already in 1905, Fine was using the cardinal aspect for comparison processes in the real world and the ordinal aspect for comparison procedures on paper.

t me that is, we must count down.

false jj. So, we write the strict inequality:

8 Washingtons > 2 Washingtons • We may have neither to count up nor to count down, in which case the comparison-sentence is:

first counting number-phrase = second counting number-phrase (with = read as "is-equal-to")

EXAMPLE 30. To compare the given basic sentences 3 Washingtons and 3 Washingtons. i. We must count from 3 to 3, that is we must count neither up nor down. ii. So, we write the equality:

3 Washingtons = 3 Washingtons

2.5 Truth Versus Falsehood

Inasmuch as the comparison-sentences that we wrote until now represented relationships between real-world collections, they were true.

However, there is nothing to prevent us from writing comparison-sentences regardless of the real-world. In fact, there is nothing to prevent us from writing comparison-sentences that are false in the sense that there is no way that anyone could come up with real-world collections for which one-to-one matching would result in the relationship represented by these comparison-sentences. EXAMPLE 31. The sentence

5 Dollars < 3 Dollars

is false because there is no way that anyone could come up with real-world collections for which one-to-one matching would result in there being leftover items in the second collection.

EXAMPLE 32. The sentence

5 Dollars = 3 Dollars,

is false because there is no way that anyone could come up with real-world collections for which one-to-one matching would result in there being no leftover item.

EXAMPLE 33. The sentence

3 Dollars ^ 3 Dollars,

is true because we can come up with real-world collections for which one-to-one matching would result in there being no leftover item.

EXAMPLE 34. The sentence

5 Dollars £ 3 Dollars,

is false because there is no way that anyone could come up with real-world collections negation

for which one-to-one matching would result in there being leftover items in the second NOT[ J

collection or no leftover item. slash

However, while occasionally useful, it is usually not very convenient to duant y (linguistic; ^

write sentences that are false because then we must not forget to write that y \ & )

i r i n ■ -, i . , . , opposite

they are false when we write them and we may miss that it says somewhere

that they are false when we read them. So, inasmuch as possible, we shall

write only sentences that are true and we will use

DEFAULT RULE # 2. When no indication of truth or falsehood is given, mathematical sentences will be understood to be true and this will go without saying.

When a sentence is false, rather than writing it and say that it is false, what we shall usually do is to write its negation —which is true and therefore "goes without saying". We can do this either in either one of two manners:

• We can place the false sentence within the symbol NOT[ ],

• We can just slash the verb which is what we shall usually do. EXAMPLE 35. Instead of writing that

the sentence 5 Dollars = 3 Dollars is false

we can either write the sentence

NOT [5 Dollars = 3 Dollars]

or the sentence

5 Dollars ^ 3 Dollars

2.6 Duality Versus Symmetry

The linguistic duality that exists between < and > must not be confused with linguistic symmetry, a concept which we tend to be more familiar with 2 .

1. Linguistic symmetry involves pairs of sentences—which may be true or false —that represent opposite relationships between the two people/collections because, even though the verbs are the same, the two people/collections are mentioned in opposite order.

Example 36.

This confusion is a most important linguistic stumbling block for students and one that Educologists utterly fail to take into consideration.

dual . j ac k i s a c hj|d 0 f jj|| versus Jill is a child of Jack

■ Jill beats Jack at poker versus Jack beats Jill at poker

■ Jack loves Jill versus Jill loves Jack

■ 9 Dimes > 2 Dimes versus 2 Dimes > 9 Dimes

Observe that just because one of the two sentences is true (or false) does not, by itself, automatically force the other to be either true or false and that whether or not it does depends on the nature of the relationship.

2. Linguistic duality involves pairs of sentences—which may be true or false —that represent the same relationship between the two people/collections because, even though the people/collections are mentioned in opposite order, the two verbs are dual of each other which "undoes" the effect of the order so that only the emphasis is different.

Example 37.

■ Jack is a child of Jill versus Jill is a parent of Jack

■ Jill beats Jack at poker versus Jack is beaten by Jill at poker

■ Jack loves Jill versus Jill is loved by Jack

■ 9 Dimes > 2 Dimes versus 2 Dimes < 9 Dimes

Observe that here, as a result, if one of the two sentences is true(or false) this automatically forces the other to be true (or false) and this regardless of the nature of the relationship.

3. When the verbs are the same and the order does not matter for these verbs, the sentences are at the same time (linguistically) symmetric and (linguistically) dual .

Example 38.

■ Jack is a sibling of Jill versus Jill is a sibling of Jack

■ 2 Nickels = 1 Dime versus 1 Dime = 2 Nickels

Observe that, here again, as soon as one sentence is true (or false), by itself this automatically forces the other to be true (or false) and that it does not depend on the nature of the relationship.

operation

attach

result

resulting collection

Chapter 3

We investigate the second of the three fundamental processes involving two collections. We will introduce the procedure in the case of basic collections using basic counting number-phrases and we will then extend the procedure to extended collections using decimal-number phrases.

3.1 Attaching A Collection To Another

Given two collection, the second fundamental issue is the first instance of what is called an operation —as opposed to a relationship: it is to attach the second collection to the first collection.

To get the collection that is the result of the real-world process: i. We set the second collection alongside the first collection ii. We move the second collection along the first collection iii. The resulting collection is made of all the items in the first collection as well as the moved items. .

KIP

• ■■I I 'Ilk.'—"jra,-: —M . II' i fc.'S g

Jill s pi u , * smsBii*~m to Jack s K —■*

Example 1. To attach

i. We set Jill's collection to the right of Jack's collection:

i. We move Jill's collection along Jack's collection

operator

+

specifying-phrase

bar

iii. The items in the first collection together with the moved items make up the resulting collection.

In order to represent on paper the result of an operation, such as attaching a second collection to a first collection, we need to expand again our mathematical language.

1. The first thing we need is a symbol, called operator, to represent the operation. In the case of attaching a second collection to a first collection, we will of course use the operator +, read as "plus".

NOTE. It should be stated right away, though, that this use of the symbol + is only one among very many different uses of the symbol + and that this will create in turn many difficulties. We shall deal with these difficulties one at a time, as we encounter each new use of the symbol +.

2. Given two collections represented by number-phrases, we will represent f attaching the second collection to the first by a specifying-phrase that we write as follows:

i. We write the first number phrase:

first number phrase ii. We write the symbol for adding:

first number phrase —

iii. We write the second number-phrase over the bar:

+ second number phrase

first number phrase

Altogether then, the specifying-phrase that corresponds to attaching to a first collection a second collection is:

first number phrase + second number phrase

EXAMPLE 2. In order to say that we want to add to the first number-phrase 5 Washingtons the second number-phrase 3 Washingtons we write the specifying phrase:

5 Washingtons + 3 Washingtons

3. This language gives us a lot of flexibility:

35

• Before we count the result of attaching a second collection to a first collection, we can already represent the result by using a specifying-phrase.

• After we have found the result of attaching a second collection to a first collection, we can represent the result by a number-phrase.

• Altogether, to summarize the whole process, we can identify the specifying phrase with an identification-sentence which we write as follows i. We write the specifying phrase

ii. We lengthen the bar with an arrowhead iii. We write the number-phrase that represents the result. Example 3.

identify

identification-sentence

Before we attach to Jack's

Jill's

~™, we ca n a I-

ready represent the result by the specifying-phrase

6 Washington + 3 Washington

ii. After we have found that the result of attaching to Jack's

Jill's

™ we can represent the result by 9 Washingtons iii. Altogether, to summarize the whole process with an identification-sentence we lengthen the bar with an arrowhead and we write the number-phrase that represents the result of the attachment.

+ 3 Washingtons

6 Washingtons

9 Washingtons

4. Usually, though, we will not write things this way and we only did it above to show how the mathematical language represented the reality. As usual, some of it "goes without saying":

• In the specifying phrase, the bar goes without saying

• In the identification sentence, the arrowhead is replaced by the symbol

EXAMPLE 4. Instead of writing the specifying phrase

6 Washingtons + 3 Washingtons

we shall write

6 Washingtons + 3 Washingtons and instead of writing the identification sentence

addition 6 Washington + 3 Washingtons , 9 Washington

we shall write

6 Washingtons + 3 Washingtons = 9 Washingtons

3.3 Procedure For Adding A Number-Phrase

Given two collections, the paper procedure that gives (the numerator of) the number-phrase that represents the result of attaching the second collection to the first collection is called addition and depends on whether the two number-phrases are basic counting number-phrases or decimal number-phrases.

In order to add a second basic collection to a first basic collection, we count up from the numerator of the first collection by a length equal to the numerator of the second collection.

There are then two cases depending on whether, when we count up from the numerator of the first number-phrase by a length equal to the second numerator, we need to end up past 9 or not. • If we do not need to end up past 9, the result of the addition is just the

end-digit.

EXAMPLE 5. To add Jill's 5 Washingtons to Jack's 3 Washingtons, that is, to identify the specifying-phrase

3 Washingtons + 5 Washingtons i. Starting from 3, we count up by a length equal to 5:

4, 5, 6, 7, 8

ii. The end-digit is 8.

iii. We write the identification-sentence:

3 Washingtons + 5 Washingtons = 8 Washingtons

If we need to end up past 9, then we must bundle and change ten of the items.

EXAMPLE 6. To add Jill's 8 Washingtons to Jack's 5 Washingtons, that is to identify the specifying-phrase

5 Washingtons + 8 Washingtons i. Starting from 5, we count up by a length equal to 8 but stop after ten:

4, 5, 6, 7, 8, 9, ten

ii. We bundle ten Washingtons and change for a 1 DEKAWashingtons and count the

rest

1, 2, 3

iii. We write the identification-sentence:

5 Washingtons + 8 Washingtons = 1 DEKAWashingtons & 3 Washingtons

which of course we could also write

5 Washingtons + 8 Washingtons =1.3 DEKAWashingtons

or

5 Washingtons + 8 Washingtons = 13. Washingtons

or . . .

Actually, we usually do the latter a bit differently, that is, instead of basic counting up just past 9, interrupt ourselves to bundle and change, and then start basic counting again, it is easier to use some extended counting and count all the way and then bundle and change what we must and count the rest.

EXAMPLE 7. To add Jill's 8 Washingtons to Jack's 5 Washingtons, that is to identify the specifying-phrase

5 Washingtons + 8 Washingtons

i. We count up from 5 by a length equal to 8 using extended-counting:

4, 5, 6, 7, 8, 9, TEN, ELEVEN, TWELVE, THIRTEEN >

ii. Then we say that we can't write thirteen Washingtons since we only have digits up to 9 so that we should bundle ten Washingtons and change for a 1 DEKAWashingtons with 3 Washingtons left

iii. We write the identification-sentence:

5 Washingtons + 8 Washingtons = 1 DEKAWashingtons & 3 Washingtons

that is, using a decimal number-phrase,

5 Washingtons + 8 Washingtons = 1.3 DEKAWashingtons

or, if we prefer,

5 Washingtons + 8 Washingtons = 13. Washingtons

or

The difference is of course not a great one. It is only that we said that we would deal with extended collections using only basic counting and indeed, in the second example, we fudged a bit when, after having counted to THIRTEEN, we said that after bundling and changing we had 3 left: officially, we cannot do so since we have not yet introduced subtraction.

However, if the first example illustrates the fact that, when needed, we can indeed do things "cleanly", the second example illustrates the fact that, while we are usually not willing to count very far, a bit of (extended) counting beyond 9 makes life easier.

detach

resulting collection

Chapter 4

Subtraction

We investigate the third of the three fundamental processes involving two collections. We will introduce the procedure in the case of basic collections using basic counting number-phrases and we will then extend the procedure to extended collections using decimal-number phrases.

4.1 Detaching A Collection From Another

Given two collections, the third fundamental issue is to detach the second collection from the first collection. This is the second instance of an operation.

The real-world process is to mark off the items of the first collection that are also in the second collection and to look at all the unmarked items as making up a single collection that we shall also call the resulting collection.

from Jack's

EXAMPLE 1. To detach Jill's

i. We set Jill's collection to the right of Jack's collection

P

We mark off the items in Jack's collection that are also in Jill's collection

««EftL

'i

minus bar

40 CHAPTER 4. SUBTRACTION

iii. The unmarked items in the first collection make up the resulting collection

4.2 Language For Subtraction

In order to represent on paper the result of an operation, such as detaching a second collection from a first collection, we need to expand again our mathematical language but we will proceed in essentially the same manner as we did with the language for addition.

1. The first thing we need is a symbol, called operator, to represent the operation. In the case of detaching a second collection from a first collection, we will of course use the operator —, read as "minus".

To represent on paper the result of detaching a second collection from a first collection, we will of course use the operator — read minus. Here again, just as with the symbol +, this use of the symbol — is only one among very many different uses of the symbol — and that this will create in turn many difficulties. We shall deal with these difficulties one at a time, as we encounter each new use of the symbol —.

NOTE. It should be stated right away, though, that this use of the symbol — is only one among very many different uses of the symbol — and that this will create in turn many difficulties. We shall deal with these difficulties one at a time, as we encounter each new use of the symbol —.

2. Given two collections represented by number-phrases, we will represent detaching the second collection from the first by a specifying-phrase that we write as follows:

i. We write the first number phrase:

first number phrase ii. We write the symbol for subtracting:

first number phrase —=

iii. We write the second number-phrase over the bar:

— second number phrase

first number phrase

Altogether then, the specifying phrase that corresponds to detaching from a first collection a second collection is:

n , i _i — second number phrase

first number phrase

EXAMPLE 2. In order to say that we want to subtract from the first number-phrase 5 Washingtons the second number-phrase 3 Washingtons we write the specifying phrase:

4.2. LANGUAGE FOR SUBTRACTION

41

5 Washingtons

3 Washingtons

3. This language gives us a lot of flexibility:

• Before we count the result of attaching a second collection to a first collection, we can already represent the result by using a specifying-phrase.

• After we have found the result of attaching a second collection to a first collection, we can represent the result by a number-phrase.

• Altogether, to summarize the whole process, we can identify the specifying phrase with an identification-sentence which we write as follows i. We write the specifying phrase

ii. We lengthen the bar with an arrowhead iii. We write the number-phrase that represents the result. Example 3.

identify

identification-sentence

Before we detach from Jack's

Jill's

already represent the result by the specifying-phrase

6 Washingtons ~ 4 Washingtons

ii. After we have found that the result of detaching from

Jill's

we can represent the result by 4 Washingtons iii. Altogether, to summarize the whole process with an identification-sentence we lengthen the bar with an arrowhead and we write the number-phrase that represents the result of the detachment.

~ ... . . — 2 Washingtons . ... . .

6 Washingtons > 4 Washingtons

4. Usually, though, we will not write things this way and we only did it above to show how the mathematical language represented the reality. As usual, some of it "goes without saying":

• In the specifying phrase, the bar goes without saying

• In the identification sentence, the arrowhead is replaced by the symbol

EXAMPLE 4. Instead of writing the specifying phrase

6 Washingtons ~ g Washingtons

we shall write

subtraction 6 Washington - 2 Washingtons

and instead of writing the identification sentence

„..,.. — 2 Washingtons , ,„, . .

6 Washingtons > 4 Washingtons

we shall write

6 Washingtons — 2 Washingtons = 4 Washingtons

4.3 Procedure For Subtracting A Number-Phrase

Given two collections, the paper procedure that gives (the numerator of) the number-phrase that represents the result of detaching the second collection from the first collection is called subtraction and depends on whether the two number-phrases are basic counting number-phrases or decimal number-phrases.

In order to subtract a second basic collection from a first basic collection, we count down from the numerator of the first collection by a length equal to the numerator of the second collection.

There are then two cases depending on whether, when we count down from the numerator of the first number-phrase by a length equal to the second numerator, we can complete the count or not. • If we can complete the count, then the result of the subtraction is just

the end-digit.

EXAMPLE 5. To subtract Jill's 3 Washingtons from Jack's 7 Washingtons, that is to identify the specifying-phrase

7 Washingtons — 3 Washingtons

i. Starting from 7, we count down by a length equal to 3:

6, 5, 4

ii. We can complete the count and the end-digit is 4 iii. We write the identification-sentence:

7 Washingtons — 3 Washingtons = 4 Washingtons

In particular, the end-digit can be 0.

EXAMPLE 6. To subtract Jill's 5 Washingtons from Jack's 5 Washingtons, that is to identify the specifying-phrase

5 Washingtons — 5 Washingtons i. Starting from 5, we count down by a length equal to 5:

4, 3, 2, 1, 0

ii. We can complete the count and the end-digit is 0

iii. We write the identification-sentence: outcast

5 Washingtons - 5 Washingtons = 0 Washington incorrect

• If we cannot complete the count, then the subtraction just cannot be stnke out done. (At least in this type of situation. We shall see in the next Chapter other situations in which we can end down past 0.)

EXAMPLE 7. To subtract Jill's 5 Washingtons from Jack's 3 Washingtons, that is to identify the specifying-phrase

3 Washingtons — 5 Washingtons

But, to identify the specifying-phrase, we would have to start from 3 and count down by a length of 5 but, by the time we got to 0, we would have counted only by a length of 3 and so we cannot complete the count which is as it should be.

4.4 Subtraction As Correction

Subtraction often comes up after we have done a long string of additions and realized that there is an outcast, that is a number-phrase that we shouldn't have added (for whatever reason), so that, as a consequence, the total is incorrect.

EXAMPLE 8. Suppose we had an ice-cream stand and that we had added sales as the day went which gave us the following specifying-phrase:

6 Washingtons + 3 Washingtons + 7 Washingtons + 9 Washingtons

and that at the end of day we identified the specifying-phrase which gave us

25 Washingtons

but that we then realized that 3 Washingtons was an outcast (it was not a sale but money given for some other purpose) with the consequence that 25 Washingtons is incorrect in that it is not the sum total of the sales for the day.

To get the correct total, we have the following two choices for the procedure: • Procedure A would be to strike out the outcast and redo the entire

EXAMPLE 9. In the above example, we strike out the outcast 3 Washingtons

6 Washingtons + tyjffflfftffpfyffl + 7 Washingtons + 9 Washingtons

which gives us

22 Washingtons

Of course, since Procedure A is going to involve a lot of unnecessary work redoing all that had been done correctly, it is very inefficient.

cancel out . Procedure B would be to cancel out the effect of the outcast in the

adjustment incorrect total by subtracting the outcast from the incorrect total. (Ac-

countants call this "entering an adjustment".)

EXAMPLE 10. In the above example, we subtract 3 Washingtons (the outcast) from 25 Washingtons (the incorrect total):

25 Washingtons — 3 Washingtons

which gives us:

22 Washingtons

We now want to see that the two procedures must give us the same result either way. For that, we place the specifying-phrases in the two procedures side by side and we see that that the remaining number-phrases are the same either way.

EXAMPLE 11. In the above example, we have: 6 Washingtons + yftffffffffffflffft + 7 Washingtons + 9 Washingtons and

6 Washingtons + j5_Waskmgtons + 7 Washingtons + 9 Washingtons — 3JflZashmgtons We see that, either way, the remaining number-phrases are: 6 Washingtons + 7 Washingtons + 9 Washingtons

plain number-phrases

cancel

two-way collections

Chapter 5

Signed Number-Phrases

We have seen in Chapter 1 that we can use plain number-phrases, that is either counting number-phrases or decimal number-phrases, only in situations where the items are all of the same one kind. We shall now introduce and discuss a new type of number-phrase that we shall use in a type of situations that occurs frequently in which the items are all of either one of two kinds.

Just as we did for plain number-phrases in Chapters 2, 3, and 4, we will have to define for this new type of number-phrase what we mean by:

i. To "compare" two number-phrases,

ii. To "add" a second number-phrase to a first number-phrase,

iii. To "subtract" a second number-phrase from a first number-phrase, and in particular to develop the corresponding procedures.

What will complicate matters a little bit, though, is that the procedures for the new type of number-phrases will involve the procedure that we developed for plain number-phrases. So, until we feel completely comfortable with the distinction, we shall use new symbols for "comparison", "addition" and "subtraction" for the new kind of number-phrases 1 .

5.1 Actions and States

Quite often we don't deal with items that are all of the same kind but with items of two different kinds and a special case of this is when two items of different kinds cannot be together as they somehow cancel each other. As a result, we will now consider what we shall call two-way collections, that

One can only wonder as to how Educologists can let their students use, without warning, the same symbols in these rather different situations.

CHAPTER 5. SIGNED NUMBER-PHRASES

action

step

state

degree

benchmark

is collections of items that are all of one kind or all of another kind with items of different kinds canceling each other.

1. In the real-world, two-way collections come up very frequently and in many different types of situations but they generally fall in either one of two types:

• In one type of two-way collections, called actions, the items are steps in either this-direction or that-direction.

EXAMPLE 1. In fact, we already encountered in the previous chapter this kind of items: counting up and counting down. Of course, the situation there was not symmetrical: we could always count steps up but we could not always count steps down. But there would have been no point counting at the same time three steps up and five steps down since steps up would cancel out steps down and this would have just amounted to counting two steps down.

Example 2.

— Actions that a businesswoman may take on a bank account are to deposit three thousand dollars, withdraw two thousand dollars, etc

— Actions that a gambler may take are to win fifty-eight dollars, lose sixty-two dollars, etc

— Actions that a mark may take on a horizontal line include moving two feet leftward, five feet rightward, etc.

— Actions that a mark may take on a vertical line include moving five inches upward, five inches downward, etc.

• In the other type of two-way collections, called states, the items are degrees of one kind or another but they have to be either on this-side or that-side of some benchmark.

EXAMPLE 3.

— States that a business may be in include being three thousand dollars in the red, being seven thousand dollars in the black, etc.

— States that a gambler may be in include being sixty-two dollars ahead of the game, being thirty-seven dollars in the hole, etc.

— States that a mark may be in on a horizontal line with some benchmark include being two feet to the left of the benchmark, being nine feet to the right of the benchmark, etc.

— States that a mark may be in on a vertical line with some benchmark include being five inches above the benchmark, being three inches below the benchmark, etc.

2. Since all the items in a given two-way collection are of the same kind, a two-way collection is essentially a collection with a twist. So, just as we said that, in the real world,

• the nature of a collection is the kind of items in the collection,

• the extent of a collection is the number of items in the collection, we shall now say that:

• the nature of an action is the kind of steps in the action and the nature (of an action) nature of a state is the kind of degrees in which the state can be nature (of a state)

,, j. j. j? u_- ■ .1 z. r i • .Li .l j ii extent (of an action)

• the extent of an action is the number of steps m the action and the . , r x

size (of a state)

size of a state is the number of degrees of the state. ,. ,. , c ,. x

° direction (of an action)

• the direction of an action is the direction of the steps in the action s j^ e ( 0 f a s tate)

and the side of a state is the side of the degrees in the state. signed number-phrase

EXAMPLE 4. When a person climbs up and down a ladder, an action may be record

climbing up seven rungs. Then, standard direction

- the nature of the action is climbing rungs opposite direction

- the size of the action is seven

- the direction is up

5.2 Signed Number-Phrases

Plain number-phrases are not sufficient to represent on paper either actions or states because they do not indicate the direction of the action or the side of the state. Example 5.

- 3000 Dollars does not say if the businesswoman made a deposit or a withdrawal or if the business is in the red or in the black.

- 62 Dollars does not say if the gambler is ahead of the game or in the hole.

- 2 Feet does not say if the mark is to the left or to the right of the benchmark.

- 5 Inches does not say if the mark is moving up or down.

1. Since a two-way collection is just a collection with a direction or a side, we will represent on paper a two-way collection by a signed number-phrase that will consist of:

• a denominator to represent on paper the nature of the action (that is the kind of the steps in the action) or of the state (that is the kind of the degrees in the state).

• a numerator to represent on paper the extent of the action (that is the number of steps in the action) or the extent of the state (that is the number of degrees in the state),

• a sign to represent on paper the direction of the action (that is the direction of the steps in the action) or the side of the state (that is the side of the benchmark that the degrees of the state are on.)

2. However, in order to say what direction the action or what side the state, we must always begin by recording for future reference:

• which direction is to be the standard direction and which direction is therefore to be the opposite direction,

CHAPTER 5. SIGNED NUMBER-PHRASES

standard side

opposite side

sign

+

positive

negative

context

signed-numerator

positive numerators

negative numerators

• which side of the benchmark is going to be the standard side and which side is therefore to be the opposite side,

NOTE. Historically, it has long gone without saying that standard was what was "good" and opposite what was "bad". Example 6.

- To deposit money is usually considered to be "good" as it goes with saving while to withdraw money is usually considered to be "bad" as it goes with spending.

- To win is usually considered to be "good" while to lose is considered to be "bad".

- To go up is usually considered to be "good" while to go down is usually considered to be "bad".

3. Once we have recorded what is standard and therefore what is opposite, we can use a sign to represent on paper the direction of the action (that is the direction of the steps in the action) or the side of the state (that is the side of the benchmark that the degrees of the state are on):

• we will use the sign +, read here as positive, to represent on paper whatever is standard, whether an action or a state.

• we will use the sign —, read here as negative, to represent on paper whatever is opposite, whether an action or a state.

NOTE. This use of the symbols + and — is entirely different from their use in Chapter 1 where they denoted addition and subtraction. This complicates reading the symbol as we need to rely on the context, that is the text that is around the symbol, to decide what the symbol stands for.

4. However, because this will make developing and using procedures a lot easier, we will lump the sign together with the numerator and call the result a signed-numerator. Signed-numerator with a + are said to be positive numerators and signed-numerators with a — are said to be negative numerators.

NOTE. Historically, just as with standard and opposite and perhaps as a result, positive has been identified with "good" and negative with "bad". So, altogether, a signed number-phrase will consist of:

• a signed-numerator

• a denominator

EXAMPLE 7. Say that we have put on record that the standard direction is to win money so that to lose money is the opposite direction. Then,

When a real-world gambler:

• wins forty-seven dollars

• loses sixty-two dollars

We write on paper: +47 Dollars -62 Dollars

EXAMPLE 8. Say we have put on record that the standard side is in-the-black so that in-the-red is the opposite side. Then,

When a real-world business is: We write on paper: sign, of the numerator

• three thousand dollars in-the-black +3000 Dollars size

• seven hundred dollars in-the-red —700 Dollars

5. We are using the same symbol, 0, both for

• the counting numerator that is left of the succession of counting numerators 1, 2, 3,4,...

• the signed numerator which is inbetween the succession of positive numerators + 1,+2,+3,+4,... and the recession of negative numerators -1,-2,-3,-4,....

In this case, we shall have to live with the ambiguiity and decide each time, according to the context, which one the numerator 0 really is.

5.3 Size And Sign

On the other hand, given a signed numerator, we shall say that:

• the sign of the numerator is the sign which was put in front of the plain numerator to make the signed numerator

• the size of the numerator is the plain numerator from which the signed numerator was made.

Example 9.

Signed Numerator = - 5

44

Sign of Signed Numerator =

Size of Signed Numerator =

In other words, —5 is a signed-numerator whose size is 5 and whose sign is

Example 10.

Signed Numerator = + 3

44

Sign of Signed Numerator = Size of Signed Numerator =

In other words, +3 is a signed-numerator whose size is 3 and whose sign is +.

Indeed, signed number-phrases can contain more information than is necessary for a particular purpose and then all we need is either the sign or the size of the signed number-phrase.

CHAPTER 5. SIGNED NUMBER-PHRASES

signed ruler minus infinity —oo plus infinity

+00

1. In many circumstances, what matters is only the size of the signed number-phrases and not the sign. EXAMPLE 11. Say we are told that

■ Jill's balance is +70,000,000 Dollars

■ Jack's balance is -70,000,000 Dollars.

We can safely conclude that neither Jack nor Jill belongs to "the rest of us".

EXAMPLE 12. If we are stopped on the turnpike doing +100 ^, that is while driving from Philadelphia to New York, or doing —100 j^ that is while driving back from New York to Philadelphia , it does not matter which way we were going: regardless of the direction, we are going to get into big trouble.

So, in such cases, it is the size of the given signed numerator that matters.

EXAMPLE 13. The size of Jill's +70,000,000 Dollars is 70,000.000 and the size of Jack's -70,000,000 Dollars is also 70,000,000 Dollars.

So, what makes Jack and Jill different from "the rest of us" is the size of their balance and not its sign.

EXAMPLE 14. The size of our speed when we are going +100 p^ (that is from Philadelphia to New York) is 100 ^^ and the size of our speed when we are going -100 ^§ (that is from New York to Philadelphia) is also 100 jg. So, what gets us into trouble is the size of our speed.

2. In many other circumstances, what matters is only the sign of the signed number-phrase and not the numerator.

EXAMPLE 15. Usually, banks do not accept negative balances, regardless of their size. In other words, all bank care about is the sign of the balance.

EXAMPLE 16. If we are stopped going the wrong way on a one way street, it won't matter if we were well under the speed limit. In other words, what gets us into trouble is the sign of our speed and not its size.

5.4 Graphic Illustrations

To graph a two-way collection represented on paper by a signed number-phrase, we proceed essentially just as with counting number-phrases and/or decimal number-phrases. The only differences are that on a signed ruler:

• we shall have the symbol for minus infinity, — oo, and the symbol for plus infinity, +oo, at the corresponding ends of the ruler

> Dollars

—oo +oo

• the tick-marks, if any, are labeled with signed number-phrases.

As with all rulers and depending on the circumstances, 0 may or may not appear. Example 17.

5.5. COMPARING SIGNED NUMBER-PHRASES

51

Example 18.

—> Dollars

+00

-> Dollars

+00

algebraic viewpoint \$<\$ (signed) \$>\$ (signed) \$\leqq\$ (signed) \$\geqq\$ (signed) algebra-compare

Example 19.

-> Dollars

,_. >— o

Example 20.

—> Dollars

+O0

Example 21.

■> Dollars

+00

5.5 Comparing Signed Number-Phrases

We investigate the first fundamental process involving actions and states: Given two actions or two states we would like to be able to compare the signed number-phrases that represent them.

However, there are actually two viewpoints from which to compare signed number-phrases.

1. From what we shall call the algebraic viewpoint, the comparison depends both on the sign and the size of the two signed number-phrases. In the real-world, the comparison corresponds to the relationship is-smaller-than understood as is-poorer-than extended to the case when being in debt is allowed.

It is traditional to use the same verbs as with counting number-phrases and decimal number-phrases, that is: <, >, =, and ^, ^.

a. There are two cases depending on the signs of the two signed number-phrases: • When the signs of the two signed number-phrases are the same

— any two positive number-phrases algebra-compare the same way as their sizes compare Example 22.

CHAPTER 5. SIGNED NUMBER-PHRASES

algebra-more-than

algebra-less-than

is-left-of

is-right-of

size viewpoint

+365.75 Dollars > +219.28 Dollars because 365.75 > 219.28.

— any two negative number-phrases algebra-compare the way opposite to the way their sizes compare

Example 23.

-432.69 Dollars < -184.41 Dollars because 432.69 > 184.41.

• When the sign of the two signed number-phrases are opposite, we can say either that

— any positive number-phrase is algebra-more-than any negative number-phrase

or, dually, that

— any negative number-phrase is algebra-less-than any positive number-phrase

Example 24.

-2386.77 Dollars < +17.871 Dollars because any negative number-phrase is less-than any positive number-phrase. b. In other words, when we picture on a ruler the signed number-phrases involved in an algebraic comparison, an algebraic comparison is about the relative positions of the two signed number-phrases relative to each other:

• is-algebra-less-than is pictured as is-left-of

• is-algebra-more-than is pictured as is-right-of Example 25.

■ The algebra-comparison sentence

-4 Dollars < +2 Dollars

corresponds to the fact that in the graphic

* ' ' ' ' ' •-

-> Dollars

i-

the mark that represents —4 is-left-of the mark that represents +2 The algebra-comparison sentence

-1 Dollars > -4 Dollars

corresponds to the fact that in the graphics

* ' 1 * 1 1 1—

-> Dollars

—oo LI I I 0 + + + + + +oo

the mark that represents —1 is-right-of the mark that represents —4 This illustrates the reason that we can reuse the same verbs with signed number-phrases as we did with counting number-phrases and decimal number-phrases.

2. Prom what we shall call the size viewpoint, the comparison depends

only on the size of the two signed number-phrases and not on the sign. is-larger-in-size-than

a. It is quite usual in the real-world to say that a hundred dollar debt ls "^ ma er ~ m ~

. , ,, nr, i 11 i n , ,ii • iii is-farther-away-from-the-

is larger than a htty dollar debt even though someone owing a hundred ce nter

dollars is-poorer-than a person owing fifty dollars.

So, we will say that:

• A first signed number-phrase is-larger-in-size-than a second signed number-phrase when the size of the first signed number-phrase is larger than the size of the second signed number-phrase.

or, dually, we can say

• A first signed number-phrase is-smaller-in-size-than a second signed number-phrase when the size of the first signed number-phrase is smaller than the size of the second signed number-phrase.

We shall not use symbols and we shall just write the words.

EXAMPLE 26. We have of course that

+365.75 Dollars is-larger-in-size-than + 219.28 Dollars

which corresponds to the fact that 365.75, the size of the first signed number-phrase, is larger than 219.28, the size of the second signed number-phrase. We also have that

-365.75 Dollars is-larger-in-size-than - 219.28 Dollars

which corresponds to the fact that 365.75, the size of the first signed number-phrase, is larger than 219.28, the size of the second signed number-phrase. And we also have that

-365.75 Dollars is-larger-in-size-than + 219.28 Dollars

which corresponds to the fact that 365.75, the size of the first signed number-phrase, is larger than 219.28, the size of the second signed number-phrase.

None of this has anything to do with the fact that, from the algebra viewpoint,

+365.75 Dollars > +219.28 Dollars -365.75 Dollars < -219.28 Dollars -365.75 Dollars < +219.28 Dollars

b. In other words, when we illustrate on a ruler the signed number-phrases involved in a size comparison, the comparison is about which numerator is-farther-away-from-t he-center.

Example 27.

■ The size-comparison sentence

—4 Dollars is-larger-in-size-than + 1 Dollars

corresponds to the fact that in the graphic

• ' ' ' ' • ' ' ' ' > Dollars

-°ol l L^io + + + + ++oo

the mark that represents —4 Dollars is farther-away-from-the-center-than the mark that represents +1 Dollars.

CHAPTER 5. SIGNED NUMBER-PHRASES

merge

The size-comparison sentence

—4 Dollars is-larger-in-size-than — 3 Dollars corresponds to the fact that in the graphic

£ £ | | | | | | | |

->■ Dollars

—on L ! I I o + + + + + +nn

<*J .k uj N> ^ ° — to w 4* Ui ' ^

the mark that represents —4 farther-away-from-the-center-than the mark that represents — 3

We investigate the second fundamental process involving actions and states.

1. Just as in in the case of collections we could attach a second collection to a first collection, here we can

• follow up a first action with a second action. Example 28.

— a gambler may win forty-five dollars and then follow up with winning sixty-two dollars.

— a gambler may win thirty-one dollars and then follow up with losing forty-four dollars.

— a gambler may lose twenty-one dollars and then follow up with winning fifty-seven dollars.

— a gambler may lose seventy-eight dollars and then follow up with losing thirty-four dollars.

• merge a first state with a second state

Example 29.

— a business that is three thousand dollars in the black may merge with a business that is six hundred dollars in the black.

— a business that is three hundred dollars in the black may merge with a business that is five hundred dollars in the red.

— a business that is two thousand dollars in the red may merge with a business that is seven hundred dollars in the black.

— a business that is seven hundred dollars in the red may merge with a business that is two hundred dollars in the red.

NOTE. English forces us to use a different word order here: while we attached a second collection to a first collection, here we must say that we follow up a first action with a second action. In order to be consistent, and although it is not necessary, we will also say that we merge a first state with a second state.

2. Then, just like adding a counting-number-phrases was the paper procedure to get the result of attaching a collection, adding a signed number-phrase will be the paper procedure to get the result of following up an action

55

and/or merging a state.

In order to distinguish adding signed number-phrases from adding counting number-phrases as we develop the procedure, we shall use for a while the symbol ©. Later, we will just use + and learn to rely on the context.

3. Just like, in Chapter 1, we introduced counting number-phrases with slashes, /, to discuss addition of signed number-phrases, we will use temporarily arrows of two kinds, <— and —>.

EXAMPLE 30. We will use temporarily

—> —> —> —> —> Dollars instead of +5 Dollars and

<— <— <— <— <— Dollars instead of —5 Dollars.

When adding a signed number-phrase, we must distinguish two cases. a. The second signed number-phrase has the same sign as the first signed number-phrase. Then, all the items are of the same kind and so following up is the same as attaching. So, in that case, to get the size of the result, we add the sizes of the two signed number-phrases. Example 31.

In the real-world, when we: deposit five dollars and then

deposit three dollars, altogether this

is the same as when we deposit eight dollars

or

Example 32.

In the real-world, when we withdraw five dollars and then

withdraw three dollars, altogether this

is the same as when we withdraw eight dollars

We write on paper:

-> —> —> —» —» Dollars

Dollars

-*•] Dollars Dollars

+8 Dollars

We write on paper:

— <— <— <— <— Dollars

Dollars

-] Dollars Dollars

-8 Dollars

b. The second signed number-phrase has the opposite sign from the first signed number-phrase. Then, the items are of the same kind and so

following up is the same as attaching. So, in that case, to get the size of the result, we add the sizes of the two signed number-phrases.

Example 33.

• When the two signed number-phrases have the same sign,

— We get the sign of the result by taking the common sign

— We get the size of the result by adding the two sizes.

• When the two signed number-phrase have opposite signs, we must first compare the sizes of the two signed number-phrases and then

— We get the sign of the result by taking the sign of the signed number-phrase whose size is larger,

— We get the size of the result by subtracting the smaller size from the larger size.

EXAMPLE 35. To identify the specifying-phrase (+3) © (+5) and since (+3) and (+5) have the same sign, we proceed as follows:

■ We get the sign of the result by taking the common sign which gives us +

■ We get the size of the result by adding the sizes 3 and 8 which gives us 8 In symbols,

(+3) ©(+5) = (+[3+ 5]) = (+8)

EXAMPLE 36. To identify the specifying-phrase (+3) © (—5) and since (+3) and (—5) have opposite signs, we must compare the sizes. Since 3 < 5,

■ We get the sign of the result by taking the sign of the number-phrase with the larger size which gives us —

■ We get the size of the result by subtracting the smaller size, 3, from the larger size, 5 which gives us 2

In symbols,

(+3) ®(-5) = (-[5 -3]) = (-2)

5.7 Subtracting a Signed Number-Phrase

We investigate the third fundamental process involving actions and states.

While, in the case of collections, detaching a collection made immediate sense as "un-attaching", in the case of actions "un-following up" and in the case of states "un-merging" do not make immediate sense. So, instead, we shall look at subtraction from the point of view of correction after we have done a long string of signed-additions and realized that there is an incorrect entry, that is a signed number-phrase that we shouldn't have added (for whatever reason), so that the total is incorrect.

1. Up front, things would seem to work out exactly as in the case of un-signed number-phrases.

EXAMPLE 37. Suppose that we work in a bank and that we had added transactions as the day went which gave us the following specifying phrase

day we identified the specifying-phrase which gave us

-132 Dollars

but that we then realized that — 7 Dollars was an outcast (it was not for a transaction add the opposite b u t f or mone y involved in some other matter) with the consequence that —132 Dollars

is incorrect in that it is not the sum total of the transaction for the day.

2. To get the correct total, we have the following two choices for the procedure:

• Procedure A would be to strike out the incorrect signed number-phrase and redo the entire addition:

EXAMPLE 38. In the above example, we would strike out the incorrect entry -7 Dollars

Of course, since Procedure A is going to involve a lot of unnecessary work redoing all that had been done correctly, it is very inefficient.

• Procedure B would be to cancel out the effect of the incorrect entry on the incorrect total by subtracting the incorrect entry from the incorrect total.

EXAMPLE 39. In the above example, we would subtract the incorrect entry — 7 Dollars from the incorrect total —132 Dollars

except that, at this point, we have no procedure for ©! Indeed, at this point, the only procedure we have for subtracting is for subtracting unsigned number-phrases.

On the other hand, the obvious way to cancel out the effect of the incorrect entry on the incorrect total and that it is by adding the opposite of the incorrect entry to the incorrect total. (Accountants call this "entering an adjustment".)

EXAMPLE 40. In the above example, we would add the opposite of the incorrect entry —7 Dollars, that is we would add —7 Dollars to the incorrect total —132 Dollars

3. We now want to see that the two procedures must give us the same result either way For that, we place the specifying-phrases in the two procedures side by side and we see that that the remaining number-phrases are the same either way.

EXAMPLE 41. In the above example, we place the specifying-phrases in the two procedures side by side:

■ The specifying-phrase in Procedure A is:

■ The specifying-phrase in Procedure B is:

loss

5.8. EFFECT OF AN ACTION ON A STATE 59

4. Altogether then: subtract

• Adding the opposite of the incorrect entry (Procedure B): initial state

final state

-132 Dollars © +7 Dollars change

gain

necessarily amounts to exactly the same as

• Striking out the incorrect entry (Procedure A):

-132 Dollars 0 - 7 Dollars

Since Procedure B is much faster than Procedure A, we say that the procedure for subtracting a signed number-phrase will be to add its opposite.

EXAMPLE 42. In order to identify the specifying-phrase (+3) G(+5) ,

i. we identify instead the specifying-phrase (+3) ©(— 5) ii. we do the addition which gives us —2

EXAMPLE 43. In order to identify the specifying-phrase (—3) G(—5) ,

i. we identify instead the specifying-phrase (—3) ©(+5) ii. we do the addition which gives us +2

EXAMPLE 44. In order to identify the specifying-phrase (-3) 0(+5) ,

i. we identify instead the specifying-phrase (—3) ©(—5) ii. we do the addition which gives us —8

EXAMPLE 45. In order to identify the specifying-phrase (+3) ©(-5) ,

i. we identify instead the specifying-phrase (+3) ©(+5) ii. we do the addition which gives us +8

5.8 Effect Of An Action On A State

We now look at the connection between states and actions.

1. A state does not exist in isolation but is always one of many. EXAMPLE 46. The state of an account is usually different on different days. Given two states, we shall refer to the first one as the initial state and to the second one as the final state. The change from the initial state to the final state can be up in which case we shall call the change a gain or can be down in which case we shall call the change a loss. On paper, we shall use + for a gain and we shall use — for a loss. Example 47.

■ At the beginning of a month, Jill's account was two dollars in-the-red

■ At the end of the month, Jill's account was three dollars in-the-black

So, during that month Jill's account went up by five dollars and we shall write the gain as +5 Dollars.

Change: +5

States: ' * ' ' ' ' * ' ' ' >- Dollars

-oo l i Jt ! J io + + + i+ +00

Example 48.

■ At the beginning of a month, Jack's account was two dollars in-the-black

■ At the end of the month, Jack's account was five dollars in-the-red

So, during that month Jack's account went down by seven dollars and we shall write the loss as —7 Dollars.

-^^^^^^^H Change: ^^^^| States: ' * ' ' ' ' ' ' * ' > Dollars

-oo i, ilijjioiii +00

THEOREM 2. Regardless of what the sign of the initial state and the sign of the final state are, we have that

change = final state Q initial state

EXAMPLE 49.

■ At the beginning of a month, Jill's account was two dollars in-the-red

■ At the end of the month, Jill's account was three dollars in-the-black

change = +3 Dollars 0 -2 Dollars = +3 Dollars © +2 Dollars = +5 Dollars

EXAMPLE 50.

■ At the beginning of a month, Jack's account was two dollars in-the-black

■ At the end of the month, Jack's account was five dollars in-the-red

change = — 5 Dollars 0 +2 Dollars = -5 Dollars © -2 Dollars = -7 Dollars

2. A change always happens as the result of an action. EXAMPLE 51. On an account,

■ A deposit results in a gain,

■ A withdrawal results in a loss.

In fact, we have exactly

action = change so that, as a consequence of the previous THEOREM, actions and states are related as follows:

THEOREM 3 (Conservation Theorem).

action = final state Q initial state

Example 52.

■ On Monday, Jill's account was five dollars in-the-red,

■ On Tuesday, Jill deposits seven dollars. So, we have:

Action = +7 Dollars

States: ' * ' ' ' ' ' ' * ' > Dollars

-ooi-. i,id J ^io + + + +oo

So, on Wednesday, Jill's account is two dollars in-the-black iii. Then we compute the change:

Change = Final State 0 Initial State = +2 Dollars e -5 Dollars = +2 Dollars © +5 Dollars = +7 Dollars

And we have indeed that

action = final state — initial state What happened is that each state is the result of all prior actions. So, by subtracting the initial state from the final state, we eliminate the effect of all the actions that resulted in the initial state, that is the effect of all the actions except the effect of the last one, namely the seven dollars deposit.

5.9 From Plain To Positive

We now have two kinds of number-phrases: plain number-phrases and signed number-phrases. The two, though, overlap and we want to analyze the connections between the two and what is gained when we go from using plain number-phrases to using signed number-phrases.

CHAPTER 5. SIGNED NUMBER-PHRASES

1. We developed

• plain number-phrases in order to deal with collections of items that are all of one kind,

• signed number-phrases in order to deal with collections of items that are all of one kind or all of another kind—with items of different kinds canceling each other.

But then, given collections of items that are all of one kind, it often happens that we can eventually think of another kind of items that cancel the first kind of items.

EXAMPLE 53. We may start counting steps to find out how much we walked. But eventually, we may want to know how far we progressed, being that there are steps backward as well as step forward and, if it doesn't matter what kind of steps they are when it comes to how much we walked, it does matter very much when it comes to how far we progressed and so we need to keep track of the direction of the steps.

2. But then, we can represent the original collection of items in two ways:

• With a plain number-phrase

• With a positive number-phrase

EXAMPLE 54. Given a collection of seven steps (necessarily all in the same direction since all items in a collection have to be the same), we can represent the collection by:

■ the plain number-phrase

7 Steps

■ or we can adopt that direction as standard direction and then represent the collection by the positive number-phrase

+7 Steps

3. We now check that, when we do an addition, we can go either one of two routes:

• We can first replace the two plain number-phrases by positive number-phrases and then oplus the two positive number-phrases,

• We can add the two plain number-phrases and then replace the result of the addition by a positive number-phrase.

Both routes get us to the same result. Example 55.

Replace

7 Steps

Replace

->■ +7 Steps

This works also with subtraction.

5.9. FROM PLAIN TO POSITIVE

63

Example 56.

Replace

5 Steps

Replace

-> +5 Steps

NOTE. The reader should check on her/his own that if, instead of replacing plain number-phrases by positive number-phrases, we were to replace plain number-phrases by negative number-phrases, then things would not always work in the sense that the two routes would not always result with the same number-phrase.

worth unit-worth value unit-value

Chapter 6

Co-Multiplication and Values

We seldom deal with a collection without wanting to know what the (money) worth of the collection is, that is how much money the collection could be exchanged for.

6.1 Co-Multiplication

Since all the items in a collection are the same, to find the worth of that collection, we need only know the unit-worth of the items, that is the amount of money that any one of these items can be exchanged for.

EXAMPLE 1. Given a collection of five apples, and given that the unit-worth of apples is seven cents, the real-world process for finding the worth of the collection is to exchange each apple for seven cents. Altogether, we end up exchanging the whole collection for thirty-five cents which is therefore the worth of the collection.

We now want to develop a paper procedure to get the number-phrase that represents the worth of the given collection, which we will call value, in terms of the number-phrase that represents the unit-worth of the items in the collection, which we will call unit-value.

1. We know how to write the number-phrase that represents the given collection and how to write its value, that is the number-phrase that represents its worth, but what is not obvious is how we should write the unit-value that is the number-phrase that represents the unit-worth.

EXAMPLE 2. In EXAMPLE 1, we represent the collection of five apples by writing the number-phrase 5 Apples and we represent its worth by writing its value, that is the number-phrase 35 Cents.

What is not obvious is how to write the unit-value of the Apples, that is the number-phrase that represents the unit-worth of the apples, that is the fact that "each apple is

co-denominator worth seven cent s".

co-mu tip ication More specifically, we know what the numerator of the unit-value should be

but what we don't know is how to write the denominator of the unit-value

which we will call co-denominator.

Looking at the real-world shows that the procedure for finding the value

must involve multiplication so that the specifying-phrase must look like: Number-phrase for collection x Unit-value = Number-phrase for money

EXAMPLE 3. In EXAMPLE 2, the number-phrase that represents the collection is

5 Apples and the numerator of the unit-phrase that represents the unit-value of the

items is 7 so the specifying-phrase must look like

5 Apples x 7 111 where 111 stands for the co-denominator.

2. The co-denominator should be such that the procedure for going from the specifying phrase to the result should prevent the denominator of the number-phrase for the collection from appearing in the result and, at the same time, be such as to force the denominator of the number-phrase for the value to appear in the result.

EXAMPLE 4. In EXAMPLE 3, since we must have

5 Apples x 7 111 = 35 Cents

the procedure to go from the specifying phrase on the left, that is 5 Apples x 7 111, to the result on the right, that is 35 Cents, must

■ prevent Apples from appearing on the right

■ but force Cents to appear on the right.

3. What we will do is to write the co-denominator just like a fraction with:

• the denominator of the value above the bar

• the denominator of the items below the bar. EXAMPLE 5. In EXAMPLE 4, we write ^nt£ in p | ace 0 f ??? so tnat tne specifying-

phrase becomes

5 Apples x 7 |H

That way, the procedure for identifying such a specifying phrase, called

co-multiplication, is quite simply stated:

i. multiply the numerators

ii. multiply the denominators with cancellation.

EXAMPLE 6. When we carry out the procedure on the specifying phrase in EXAMPLE

5, we get

Cents , , / Cents \

5 APP|6S X ? Apple" = (5 X 7) {^ eSx Apfte) = 35 Cents

extend

6.2. SIGNED-CO-MULTIPLICATION 67

co-number-phrase evaluate which is what we needed to represent the real-world situation in EXAMPLE 1. percentage

4. From now on, in order to remind ourselves that the reason why unit-values are written this way is to make it easy to co-multiply, we shall call them co-number-phrases 1 .

Also, just as we often say "To count a collection" as a short for "To find the numerator of the number-phrase that represents a collection", we shall say "To evaluate a collection" as a short for "To find the numerator of the number-phrase that represents the value of a collection".

NOTE. Co-multiplication is at the heart of a part of mathematics called Dimensional Analysis that is much used in sciences such as Physics, Mechanics, Chemistry and Engineering where people have to "cancel" denominators all the time.

Example 7.

5 Hours x 7^ = (5x7) (itetrrs x gj) = 35 Miles

Example 8.

Pound _ /r ~ T\ /'c«..,K«J m -hsr^ Pound ^

5 Square-Inches x 7 j^b = (5x7) (Sojiare-JnclieT x ^^d. j = 35 Pounds

Co-multiplication is also central to a part of mathematics called Linear Algebra that is in turn of major importance both in many other parts of mathematics and for all sort of applications in sciences such as ECONOMICS.

Example 9.

5 Hours x 7 ^srs = (5 x 7) (uenrs x ^) = 35 Dollars

More modestly, co-multiplication also arises in percentage problems:

Example 10.

5 Dollars x 7 §^| = (5 x 7) (Deters x _§§§.) = 35 Cents

6.2 Effect of Transactions on States: Signed Co-Multiplication

We now want to extend the concept of co-multiplication to signed-numbei-phrases in order to deal with actions and states.

Educologists will of course have recognized number-phrases and co-number-phrases for the vectors and co-vectors that they are—albeit one-dimensional ones.

signed co-number-phrase ]__ \y e begin by looking at the real-world. As before, we want to inves-

tigate the change in a given state, gain or loss, that results from a given transaction, "in" or "out" as before but with two-way collections of "good" items or "bad" items.

EXAMPLE 11. Consider a store where, for whatever reason best left to the reader's imagination, collections of apples can either get in or out of the store. Moreover, the collections are really two-way collections in that the apples can be either good — inasmuch as they will generate a sales profit—or bad —inasmuch as they will have to be disposed of at a cost.

2. We now look at the way we will represent things on paper.

a. To represent collections that can get in or out, we use signed number-phrases and we use a + sign for collections that get in and a — sign for collections that get out.

So, we will represent

• collections getting "in" by positive number-phrases,

• collections getting "out" by negative number-phrases,

EXAMPLE 12. In the above example, we would represent

■ a collection of three apples getting in the store by the number-phrase +3 Apples

■ a collection of three apples getting out of the store by the number-phrase —3 Apples

b. To represent unit-values that can be gains or losses, we use signed co-number-phrase and we use a + sign to represent gains and a — sign to represent losses.

So, we will represent

• the unit-value of "good" items by positive co-number-phrases,

• the unit-value of "bad" items by negative co-number-phrases,

EXAMPLE 13. In the above example, we would represent

■ the unit-value of apples that will generate a sales profit of seven cents per apples by the co-number-phrase +7 §^

■ the unit-value of apples that will generate a disposal cost of seven cents per apple by the co-number-phrase — 7 £^

3. Looking at the effect that transactions (of two-way collections) can have on (money) states, that is at the fact that:

• A two-way collection of "good" items getting "in" makes for a "good" change.

• A two-way collection of "good" items getting "out" makes for a "bad" change.

• A two-way collection of "bad" items getting "in" makes for a "bad" change.

• A two-way collection of "bad" items getting "out" makes for a "good" change.

6.2. SIGNED-CO-MULTIPLICATION

69

we can now write the procedure for signed co-multiplication for which signed co-multiplication

we will use the symbol ®:

i. multiply the denominators (with cancellation).

ii. multiply the numerators according to the way gains and losses occur:

• (+) <S> (+) gives (+)

Example 14.

Three apples get in the store.

The apples have a unit-value of seven cents-per-apple gain.

+3 Apples

+7

Cents Apple

The specifying phrase is

We co-multiply We get a twenty-one cent gain.

(+) <8> (-) gives (+)

Example 15.

Three apples get in the store.

The apples have a unit-value of seven cents-per-apple loss.

The specifying phrase is

We co-multiply We get a twenty-one cent loss.

(-) <8> (+) gives (+)

Example 16.

Three apples get out of the store.

The apples have a unit-value of seven cents-per-apple gain.

The specifying phrase is

We co-multiply We get a twenty-one cent loss.

(-) <8> (-) gives (+)

Example 17.

Three apples get out of the store.

The apples have a unit-value of seven cents-per-apple loss.

The specifying phrase is

We co-multiply We get a twenty-one cent gain.

[+3 Apples] ® [+7 gg]

[(+3) ® (+7)] [Awtes x jgg = +21 Cents

+3 Apples

[+3 Apples]

n Cents Apple

7 Cents |

Apple J

[(+3) 0 (-7)] [Apples x H|] = -21 Cents

-3 Apples

+7

-3 Apples

]® [+7

Cents Apple

Cents Apple

[(-3) ® (+7)] [Apples x JH = -21 Cents

-3 Apples

-3 Apples]

-7

Cents Apple

y Cents |

Apple J

[(-3) ® (-7)] [Apples x H|] = +21 Cents

NOTE. The choice of symbols, + to represent good and — to represent bad, was not an arbitrary choice because of the way they interact with the symbols for in and out. We leave it as an exercise for the reader to investigate

CHAPTER 6. CO-MULTIPLICATION AND VALUES

what happens when other choices are made.

4. Just as with addition and subtraction, in the case of co-multiplication too, we can replace plain number-phrases by positive number-phrases .

Example 18.

Replace

8 Steps

Seconds Step

Replace

+8 Steps

<8>

-> +3

Seconds Step

24 Seconds

Replace

-> +24 Seconds

Part II

Inequations & Equations Problems

(Looking For That Collection)

Chapter 7

Basic Problems 1:

(Counting Numerators)

select

requirement

meet

enter

noun

blank

form

instruction

nonsense

In the real world, we often select collections on the basis of requirements that these collections must meet. After introducing some more mathematical language and discussing real-word situations, we will develop a paper world approach and introduce what will be our general procedure when dealing with such problems.

7.1 Forms, Data Sets And Solution Subsets

We begin by looking at the way we deal in ENGLISH with the selection of collections in the real world.

1. Essentially, what we use are "incomplete sentences" like those we encounter on certain exams or when we have to enter a noun in the blanks of a form.

Example 1. The following

is a past President of the United States.

is a form in which the box is the blank in which we are supposed to enter a noun.

2. The instruction to enter some given noun in the blank of a form may result in: • nonsense, that is words that say nothing about the real world.

EXAMPLE 2. The instruction to enter the data,

in the blank of the form

Mathematics

CHAPTER 7. BASIC PROBLEMS 1 COUNTING NUMERATORS

sentence TRUE

FALSE

data set curly brackets

{ } problem

results in

Mathematics

is a past President of the United States, is a past President of the United States.

which is nonsense.

a sentence, that is words that say something about the real world but that, like something we may write on a exam, can be true or false. EXAMPLE 3. Given the form

is a past President of the United States.

The instruction to enter the noun,

Jennifer Lopez in the blank of the form results in

Jennifer Lopez is a past President of the United States.

which is a sentence that (unfortunately) happens to be false. The instruction to enter the noun

Bill Clinton

in the blank of the form results in

is a past President of the United States.

Bill Clinton

which is a sentence that happens to be true. 3. In order to avoid having to deal with nonsense, that is in order to make sure that when we enter a noun we always get a sentence, regardless of whether that sentence turns out to be true or false, we will always have a data set from which to take the nouns.

We shall write the data set by writing the data within a pair of curly brackets { } EXAMPLE 4. Given the form

is a past President of the United States.

the following could be a data set

{Bill Clinton, Ronald Reagan, Jennifer Lopez, John Kennedy, Henry Ford} but the following could not be a data set

{Bill Clinton, Ronald Reagan, Jennifer Lopez, Mathematics , Henry Ford}

4. A problem will consist of a form together with a data set. EXAMPLE 5. The form

is a past President of the United States.

and the data set

{Bill Clinton, Ronald Reagan, Jennifer Lopez, John Kennedy, Henry Ford} solution

make up a problem. non-solution

solution subset a. Given a problem, that is given a data set and a form, select

• a solution (of the given problem) is a noun such that, when we enter se ^ 0 f se l ec table collections this noun into the blank of the form, the result in a sentence that is require

TRUE gauge collection

• a non-solution (of the given problem) is a noun such that, when we select subset enter this noun into the blank of the form, the result in a sentence that

is FALSE EXAMPLE 6. Given the problem consisting of the form

is a past President of the United States.

and the data set

{Bill Clinton, Ronald Reagan, Jennifer Lopez, John Kennedy, Henry Ford}

■ The solutions of the problem are

Bill Clinton, Ronald Reagan, John Kennedy

■ The non-solutions of the problem are

Jennifer Lopez, Henry Ford b. Given a problem, that is given a data set and a form, the solution subset for the problem consists of all the solutions.

We write a solution subset the same way as we write s data set, that is we write the solutions between brackets { }. EXAMPLE 7. Given the problem consisting of the form

is a past President of the United States.

and the data set

{Bill Clinton, Ronald Reagan, Jennifer Lopez, John Kennedy, Henry Ford} the solution subset of that problem is

{Bill Clinton, Ronald Reagan, John Kennedy}

7.2 Collections Meeting A Requirement

The simplest way to select collections from a given set of selectable collections is to require them to compare in a given way to a given gauge collection which we do by matching the collections one-to-one with the gauge collection. (See Chapter 2.) The result is what we will call the select subset.

EXAMPLE 8. Jack has the following collection of one-dollar bills

CHAPTER 7. BASIC PROBLEMS 1 COUNTING NUMERATORS

So the bids that he can at all make in an auction (set of selectable collections) are:

■o.y c

f ®. i i

If the starting bid (gauge collection) for a particular object is three dollars (a selectable collection), the bids that Jack could make (select subset) would then be:

E&E3

1. The gauge collection may or may not be a selectable collection. EXAMPLE 9. Jack has the following collection of one-dollar bills

Wm

So the bids that he can at all make in an auction (set of selectable collections) are:

Eia Esa iBl WMWm

■ If the starting bid for a particular object is three dollars (a selectable collection), then the bids that he could make (select subset) would be:

o 4s j a