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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”).
© 2007 Herbert I. Gross
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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.
© 2007 Herbert I. Gross
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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.
© 2007 Herbert I. Gross
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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…
© 2007 Herbert I. Gross
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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
© 2007 Herbert I. Gross
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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…
© 2007 Herbert I. Gross
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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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© 2007 Herbert I. Grossnext
♠ 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.
© 2007 Herbert I. Grossnext
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
© 2007 Herbert I. Gross
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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”.
© 2007 Herbert I. Gross
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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”.
© 2007 Herbert I. Gross
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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
4. The answer is (y)
Program 2
1. Start with (x)
2 Multiply by 2
3. Add 6
4. The answer is (y)
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).
© 2007 Herbert I. Gross
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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
4. The answer is (y)
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.
© 2007 Herbert I. Gross
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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
4. The answer is (y)
Program 2
1. Start with (x)
2 Multiply by 2
3. Add 6
4. The answer is (y)
Program 3
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Add 6
© 2007 Herbert I. Grossnext
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
4. The answer is (y)
Program 2
1. Start with (x)
2 Multiply by 2
3. Add 6
4. The answer is (y)
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.
© 2007 Herbert I. Grossnext
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
4. The answer is (y) 17
Program 2
1. Start with (x)
2 Multiply by 2
3. Add 6
4. The answer is (y)
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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© 2007 Herbert I. Gross
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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
© 2007 Herbert I. Gross
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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
© 2007 Herbert I. Gross
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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.
© 2007 Herbert I. Gross
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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
4. The answer is (y) 20
Program 1
1. Start with (x) 7
2. Multiply by 2 14
3. Add 6 20
4. The answer is (y) 20
Program 3
© 2007 Herbert I. Grossnext
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…
♠
next
© 2007 Herbert I. Grossnext
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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© 2007 Herbert I. Gross
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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
© 2007 Herbert I. Gross
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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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© 2007 Herbert I. Gross
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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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© 2007 Herbert I. Gross
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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
© 2007 Herbert I. Gross
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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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© 2007 Herbert I. Gross
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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
© 2007 Herbert I. Gross
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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.
© 2007 Herbert I. Gross
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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.
© 2007 Herbert I. Gross
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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.
© 2007 Herbert I. Gross
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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.
© 2007 Herbert I. Gross
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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.
© 2007 Herbert I. Gross
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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
© 2007 Herbert I. Gross
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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.
© 2007 Herbert I. Grossnext
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.
© 2007 Herbert I. Grossnext
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
© 2007 Herbert I. Grossnext
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
© 2007 Herbert I. Grossnext
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
© 2007 Herbert I. Grossnext
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.
© 2007 Herbert I. Grossnext
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.
© 2007 Herbert I. Grossnext
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…
© 2007 Herbert I. Grossnext
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.
© 2007 Herbert I. Grossnext
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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