the game of algebra an introduction the game of algebra an introduction © 2007 herbert i. gross by...

The Game of AlgebraAn

Introduction

The Game of AlgebraAn

Introduction

© 2007 Herbert I. Gross

byHerbert I. Gross & Richard A. Medeiros

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Lesson 9

In Lesson 1, we discussed how to develop a strategy that would allow us to

paraphrase an algebraic equation into the form of a simpler numerical equation. To paraphrase an equation means to change

the wording of the equation, without changing the meaning of the equation.

To make sure of not changing the meaning, we had to conform to various generally

accepted rules of logic (such as “If a = b and b = c then a = c”).


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Whenever rules and strategy are involved in a process, we may view the process as being, in a manner of speaking, a game.

Therefore, in this Lesson we examine algebra in terms of its being a game.


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When you first learn a game, the strategies are relatively simple, but as you become a more advanced player, the strategies you

need in order to win become more complicated. In this sense, this lesson

begins to prepare us to study more complicated algebraic expressions and

equations. So with this in mind, Lesson 9 begins with a discussion of what

constitutes a game.


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For example, chess is a game, baseball is a game, and gin rummy is a game. What is it that these three very different games have in common that allows us to call each of them a game? More generally, what is it

that all games have in common? Our answer, which will be developed in this

Lesson, is…


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♠ Every game has its own “pieces”, and we can't

even begin to play the game unless we know the definitions (vocabulary) that describe these

“pieces”.

© 2007 Herbert I. Gross next

The GameThe Game

♣ Every game has its own rules that tell us how

the various “pieces” that make up the game are related.

♥ And every game has “winning” as its

objective; but where “winning” means that we have to do it in terms of the rules of the game.

♦ The process of applying the definitions and

the rules to arrive at a winning situation is called strategy. nextnextnext


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We define a game to be any system that consists of definitions, rules

and the objective.

Defining a Game

The objective is carried out as an inescapable consequence of the

definitions and rules, by means of strategy.

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In terms of a diagram…


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The Rules of the Game(tells us how the terms are related).

The Definitions or Vocabulary(tells us what the terminology means).

Apply strategy(to the definition and rules).

The Objective(to win).

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The interesting part about this definition is that it allows us to view almost anything as

a game.

© 2007 Herbert I. Grossnext

For example, in any academic subject we have terminology, rules, and an objective.

♥ In creative writing, the objective is to use vocabulary and the rules of grammar to

communicate in exciting ways with our fellow human beings.

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♠ In economics, the objective is to use various accepted principles (which become the accepted rules) to help foster economic

growth and stability.

♦ In psychology, the objective is to explain and/or account for human behavior in terms of certain observations (rules) that people

have developed through the years.

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♣ And in algebra, the objective is to use various “self evident” rules of arithmetic in

order to paraphrase more complicated expressions into simpler ones. next

Even such subjective topics as religion may be viewed as games.


For example, in any religion, the objective is to lead what the religion defines as a “good life” by applying certain rules (often called tenets or dogma) that we

accept in that religion.

In this sense, life itself may be viewed as a game. That is, each of us accepts certain definitions and rules, and our objective is to lead what we define to be a rewarding life. What makes the “game of life” evenmore complicated is that the rules of the game include not only our own but also

those of our family, our church, our society and so on. It often requires

compromise and sacrifice in order to balance all the sets of rules under which

we have to live.© 2007 Herbert I. Gross

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All we ask of our rules is that they be consistent. For example, in baseball

there is a rule that says 3 strikes is an out. This rule was arbitrary in the sense that 4 strikes is an out could have been chosen just as easily. But what can’t be allowed is to have both of these be rules in the same game. Otherwise, the game would be at an impasse the first time a

batter had 3 strikes.

Note


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Since we want the results of our game of mathematics to apply to the “real world”,

we must choose our rules of mathematics to be those which we believe are true

in the “real world”.


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In other words, the rules must be consistent in every game. In addition,

when we come to a game that is based on helping to explain the “real world”, our rules must also be chosen so that they

conform to what we believe to be reality.next

The problem here is that what one person calls “reality” may not be what another

person calls “reality”.


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So whenever possible, what we do is to choose as our rules only those things that

everyone is willing to accept.

In certain situations, this is not always possible. (Perhaps this is why people often say that it’s a bad idea to discuss

religion or politics.)

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© 2007 Herbert I. Gross nextnext

The “pieces” are numbers, and the rules tell us how we may manipulate these numbers. In mathematics, our rules are usually called

axioms.

A major objective of algebra is to solve equations.

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In terms of our general definition of a game; algebra is a game in which…

The strategy is usually to paraphrase, if possible, complicated mathematical

relationships into simpler, but equivalent ones. next

There are times when two relationships may look alike and still be different while at other times, two relationships may look different and yet be equivalent. For example, look at

the following three “recipes”…

© 2007 Herbert I. Gross next

1. Start with (x)

2. Add 3

3. Multiply by 2

4. The answer is (y)

Program 1

1. Start with (x)

2. Multiply by 2

3. Add 3


Program 2

1. Start with (x)

2 Multiply by 2

3. Add 6


Program 3

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A Major Difficulty

At first glance, Programs 1 and 2 may seem to look alike. In fact, the only

difference between them is that we have interchanged the order of steps (2) and (3).


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1. Start with (x)

2. Add 3

3. Multiply by 2


Program 1

1. Start with (x)

2. Multiply by 2

3. Add 3


Program 2

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On the other hand, Program 3 looks neither like Program 1 nor Program 2. For

example, Program 3 contains the command Add 6; a command that is not part of either Program 1 or Program 2.


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1. Start with (x)

2. Add 3

3. Multiply by 2


Program 1

1. Start with (x)

2. Multiply by 2

3. Add 3


Program 2

1. Start with (x)

2 Multiply by 2

3. Add 6


Program 3

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Add 6


1. Start with (x)

2. Add 3

3. Multiply by 2


Program 1

1. Start with (x)

2. Multiply by 2

3. Add 3


Program 2

1. Start with (x)

2 Multiply by 2

3. Add 6


Program 3

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To see why they are not, let’s see what happens when we replace x by 7 in both

programs.next

7

10

20

20

7

14

17

17

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Although Programs 1 and 2 may seem to be equivalent, in reality they are not.


1. Start with (x) 7

2. Add 3 10

3. Multiply by 2 20

4. The answer is (y) 20

Program 1

1. Start with (x) 7

2. Multiply by 2 14

3. Add 3 17


Program 2

1. Start with (x)

2 Multiply by 2

3. Add 6


Program 3

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Equivalent Equivalent

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If Programs 1 and 2 were equivalent, we would not have been able to get different outputs (20 and 17) for the same input (7).

However, when x is replaced by 7 in Program 3, we obtain the same result as

we did in Program 1.

20 17

7

14

20

2020

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However, what we shall eventually show is that by applying the traditional rules of

arithmetic, we can prove that Program 1 and Program 3 are equivalent, and therefore

it was not a coincidence that we obtained the same output in both these programs

when we started with 7 as the input. next

The fact that we got the same output in Programs 1 and 3 when we started with 7

could have been a coincidence.

CautionCaution

In order for two programs to be equivalent, they must be two different ways of saying the same thing. Therefore, if even one input gives us a different output in the two programs, the

two programs are not equivalent.

Summary


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In other words, in order for two programs to be equivalent, the output we get in one

program for each input must always be the same as the output we get in the other

program, for the same input.next


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On the other hand, the fact that we got the same output in Programs 1 and 3 when the

input was 7 is not sufficient evidence to prove that Programs 1 and 3 are

equivalent.

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In this sense, this lesson and the next will describe how in certain cases (such as

with Programs 1 and 3) we can tell for sure whether two programs are equivalent, without our having to resort to either

hunches or trial and error.


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In the meantime, there are some fairly straightforward demonstrations that most of

us would accept for showing that Programs 1 and 3 are equivalent.

1. Start with (x) 7

2. Add 3 10

3. Multiply by 2 20


Program 1

1. Start with (x) 7

2. Multiply by 2 14

3. Add 6 20


Program 3


One such way is quite visual. Namely, we can use

x + 3 =

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2(x + 3) = ♠ ♠ ♠ ♠ ♠ ♠♠ ♠ ♠ ♠ ♠ ♠

2x + 6 =Program 1

Program 3next

♠ ♠ ♠

to stand for whatever number we want x to be, and to stand for 1. In this way…

♠

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we see that for any given input, the output in Programs 1 and 3 is always 3 more than the output in Program 2. This agrees with

our earlier result when we got 20 as the output in Programs 1 and 3; but 17 as the output in Program 2 when the input was 7.

Comparing Programs 1 and 3

( ♠ ♠ ♠ ♠ ♠ ♠ )with Program 2

( ♠ ♠ ♠ )

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However, not all programs are this simple; and as the programs

become more complicated, so also would their visual

representations. This is one reason why our game of algebra is

so important.

Important Note


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By using the rules of the game of algebra, we have a relatively simple, logical way to

decide when two relationships are equivalent and when they aren't.

Moreover, these same rules often help us reduce a complicated relationship to a simpler relationship that is easier for

us to analyze.

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We shall show what this means in more detail throughout the rest of this course.

More specifically, we will begin to define the rules for the game of algebra in this lesson and conclude this discussion in Lesson 10. In Lesson 11, we shall apply the results obtained in Lessons 9 and 10

toward solving more complicated algebraic equations.

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The first thing we have to notice is that people view numbers in different ways. One person may view a whole number

as being a certain number of tally marks (for example, |||).

Another person may view it as being a length (for example, ————) .

What we must do is to be sure that any rules that we accept are independent of

how we view a number.next


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For instance, when we say that two numbers are equal, we must make sure that

this means the same thing to people whether they use tally marks, lengths or

anything else to visualize numbers.

So, we define equality by what we believe are its properties. As you look at the rules, ask yourself whether your own

definition of equality meets the conditions stated in our rules (axioms).

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(In these axioms, a, b, and c stand for numbers)

E1: a = a (the reflexive property).

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AXIOMS OF EQUALITY

In “plain English”, every number is equal to itself. However, not every relationship is reflexive. For example, the relationship “is older than”, is not reflexive because it

is false that a person is older than him/ herself. However, “is the same age as” is reflexive, because a person is the same

age as him/herself. next


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E2: If a = b then b = a (the symmetric property).

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AXIOMS OF EQUALITY

In “plain English”, if the first number equals the second number, it’s also true that the second number is equal to the

first number. For example, the fact that

3 + 2 = 5 means that 5 = 3 + 2.


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E2: If a = b then b = a (the symmetric property).

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AXIOMS OF EQUALITY

Not every relationship is symmetric. For example, “is the father of” is not

symmetric because if John is the father of Bill, Bill is not the father of John.

But the relationship “is the same height as” is symmetric, because if John is the same

height as Bill, then Bill is the same height as John.


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E3: If a = b and if b = c, then a = c (the transitive property).

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AXIOMS OF EQUALITY

In “plain English”, if the first number is equal to the second number and the second number is equal to the third

number, then the first number is also equal to the third number.


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E3: If a = b and if b = c, then a = c (the transitive property).

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AXIOMS OF EQUALITY

The relationship “is taller than" is transitive. For example, if John is taller than Bill, and Bill is taller than Mary, then John

is also taller than Mary. On the other hand, for example,

“is the father of” is not transitive. That is, if John is the father of Bill, and Bill is the father of Mary, then John is the grandfather (not the

father) of Mary.


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A relationship that has all of the above three properties (that is, it is reflexive,

symmetric, and transitive) is a very special relationship, and it is given a special name. Namely, it is called an equivalence relation,

and the set of members that obey this relationship is called an equivalence class

with respect to this relationship.

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Thus, for example, equality (more specifically “is equal to”) is an example of

an equivalence relation.


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In informal terms, if you’ve seen one member of an equivalence class you’ve seen them all. That is, with respect to

this course, “If a = b, then a and b can be used interchangeably in any

mathematical expression that involves equality”.

In effect, this property encompasses the reflexive, symmetric, and transitive properties. So we often use it as a

replacement for the other three properties.next


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E4: (the equivalence property) If a = b, then a and b can be used interchangeably in any mathematical

relationship that involves equality.

AXIOMS OF EQUALITY

More specifically, we use the following property as a shortcut summary

of E1, E2, and E3.

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When we talk about two things being equivalent, we always mean with respect to a given relationship.


An Important WarningAn Important Warning

Example: when the Declaration of Independence talks about “all men are

created equal”, it does not mean that all men look alike or that all men have the

same height or the same amount of money. Rather, its meaning is something like

“equal in the eyes of God”.next

Rather, they are equivalent with respect to naming the number that

we must multiply by 2 in order to obtain 6 as the product.


In a similar way, in arithmetic when we say such things as…

6 ÷ 2 = 12 ÷ 4, we do not mean that these

expressions look alike. They don’t!

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As a non-mathematical example, suppose we write…

Mark Twain = Samuel Clemens


This does not mean that the two names look alike, but rather that they name the same

person; Mark Twain is the pen name of Samuel Clemens!

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In other words, any statement that is true about the man, Mark Twain, is also true about

the man, Samuel Clemens. Thus, the statements “Mark Twain wrote Huckleberry

Finn” and “Samuel Clemens wrote Huckleberry Finn” are equivalent statements. next


E4 is the logical reason behind the earlier rules we accepted, such as “equals added to

equals are equal". For example, suppose that we know that…

Notes

x = y…and we decide to add 3 to x. The left hand

side of the equation becomes…

x + 3 next


When x = y, by E4 we may replace x by y in any mathematical relationship.

Notes

x + 3The mathematical way of saying that x + 3 and y + 3 are equivalent is to write…

y

Replacing x by y in x + 3 gives us the equivalent expression…

x + 3 = y + 3nextnext

If we now compare… x = y and x + 3 = y + 3,

we see that in effect we simply added“equals to equals” to obtain equal results.


x = y+ 3 + 3=

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Of course, the rule that “equals added to equals are equal” may have seemed obvious to us without having to talk

about E4. What E4 does for us, however, is to demonstrate that this rule is

consistent with the rules in what we are calling the game of algebra.


Now that we’ve proved by E4 that “equals added to equals are equal”, we may use this rule in our new game, just as

we did in Lesson 1.

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As the algebraic expressions and equations in our course become

increasingly more complicated, some of our strategies will become less obvious.

If at such a point a strategy is not obvious to all of the “game players”, it is our

obligation to show them that the strategy is indeed a logical (that is, inescapable)

consequence of the rules we’ve accepted. © 2007 Herbert I. Gross

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Notes

In other words…


And it is the obligation of the instructor to ensure that each strategy employed by

either the instructor or the students follows inescapably as a consequence of these

definitions and the rules.

It is the obligation of each student to accept the definitions and rules of

arithmetic (and that’s why we try to make them as self-evident as possible)…

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If we are to play the game of algebra correctly, the next challenge that confronts

us is to define the properties (rules) that govern addition, subtraction,

multiplication, and division of numbers.


More crucially, we have yet to answer the

question… “What is a number?”

All of this will be the topic of our next lesson.

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