Download Introduction to the Game of Algebra

The Game of Algebra
An
Introduction
Lesson 9
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross
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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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© 2007 Herbert I. Gross
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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© 2007 Herbert I. Gross
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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© 2007 Herbert I. Gross
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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© 2007 Herbert I. Gross
The Game
♠ 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”.
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
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strategy.
© 2007 Herbert I. Gross
Defining a Game
We define a game to be any system that
consists of definitions, rules
and the objective.
The objective is carried out as an
inescapable consequence of the
definitions and rules, by means of strategy.
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© 2007 Herbert I. Gross
In terms of a diagram…
The Rules of the Game
(tells us how the terms are related).
Apply strategy
The Objective
(to the definition and rules).
(to win).
The Definitions or Vocabulary
(tells us what the terminology means).
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© 2007 Herbert I. Gross
The interesting part about this definition is
that it allows us to view almost anything as
a game.
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.
© 2007 Herbert I. Gross
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♦
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.
In economics, the objective is to use
various accepted principles (which become
the accepted rules) to help foster economic
growth and stability.
And in algebra, the objective is to use
various “self evident” rules of arithmetic in
order to paraphrase more complicated
expressions into simpler ones.
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♠
♣
© 2007 Herbert I. Gross
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.
© 2007 Herbert I. Gross
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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” even
more 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.
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© 2007 Herbert I. Gross
Note
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.
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© 2007 Herbert I. Gross
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”.
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.
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© 2007 Herbert I. Gross
The problem here is that what one person
calls “reality” may not be what another
person calls “reality”.
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
In terms of our general definition of a
game; algebra is a game in which…
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.
The strategy is usually to paraphrase,
if possible, complicated mathematical
relationships into simpler, but equivalent
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ones.
© 2007 Herbert I. Gross
A Major Difficulty
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”…
Program 1
Program 2
Program 3
1. Start with (x)
1. Start with (x)
1. Start with (x)
2. Add 3
2. Multiply by 2
2 Multiply by 2
3. Multiply by 2
3. Add 3
3. Add 6
4. The answer is (y)
4. The answer is (y)
4. The answer is (y)
© 2007 Herbert I. Gross
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Program 1
Program 2
1. Start with (x)
1. Start with (x)
2. Add 3
2. Multiply by 2
3. Multiply by 2
3. Add 3
4. The answer is (y)
4. The answer is (y)
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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© 2007 Herbert I. Gross
Program 1
Program 2
Program 3
1. Start with (x)
1. Start with (x)
1. Start with (x)
2. Add 3
2. Multiply by 2
2 Multiply by 2
3. Multiply by 2
3. Add 3
3. Add 6
4. The answer is (y)
4. The answer is (y)
4. The answer is (y)
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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© 2007 Herbert I. Gross
Program 2
Program 1
Program 3
7
1. Start with (x)
7
1. Start with (x)
2. Add 3
10
2. Multiply by 2
14
2 Multiply by 2
3. Multiply by 2
20
3. Add 3
17
3. Add 6
1. Start with (x)
4. The answer is (y) 20
4. The answer is (y) 17
4. The answer is (y)
Although Programs 1 and 2 may seem to
be equivalent, in reality they are not.
To see why they are not, let’s see what
happens when we replace x by 7 in both
programs.
© 2007 Herbert I. Gross
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Program 2
Program 1
Program 3
7
1. Start with (x)
7
1. Start with (x)
7
2. Add 3
10
2. Multiply by 2
14
2 Multiply by 2
14
3. Multiply by 2
20
3. Add 3
17
3. Add 6
20
1. Start with (x)
4. The answer is (y) 20
Equivalent
4. The answer is (y) 17
4. The answer is (y) 20
20
Equivalent
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
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we did in Program 1.
© 2007 Herbert I. Gross
Caution
The fact that we got the same output in
Programs 1 and 3 when we started with 7
could have been a coincidence.
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.
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© 2007 Herbert I. Gross
Summary
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.
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.
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© 2007 Herbert I. Gross
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.
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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© 2007 Herbert I. Gross
In the meantime, there are some fairly
straightforward demonstrations that most of
us would accept for showing that
Programs 1 and 3 are equivalent.
Program 1
1. Start with (x)
Program 3
7
1. Start with (x)
7
2. Add 3
10
2. Multiply by 2
14
3. Multiply by 2
20
3. Add 6
20
4. The answer is (y) 20
4. The answer is (y) 20
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© 2007 Herbert I. Gross
One such way is quite visual. Namely, we
can use
to stand for whatever number
to stand for 1.
we want x to be, and
In this way…
♠
x + 3=
2(x + 3) =
♠♠♠
♠♠♠ ♠♠♠
Program 1
2x + 6 =
Program 3
© 2007 Herbert I. Gross
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Comparing Programs 1 and 3
♠♠♠♠♠♠)
(
with Program 2
(
♠♠♠)
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.
© 2007 Herbert I. Gross
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Important Note
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.
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© 2007 Herbert I. Gross
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
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
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.
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© 2007 Herbert I. Gross
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
AXIOMS OF EQUALITY
(In these axioms, a, b, and c stand for numbers)
E1:
a = a (the reflexive property).
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
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age as him/herself.
© 2007 Herbert I. Gross
AXIOMS OF EQUALITY
E2:
If a = b then b = a
(the symmetric property).
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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© 2007 Herbert I. Gross
AXIOMS OF EQUALITY
E2:
If a = b then b = a
(the symmetric property).
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
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John.
© 2007 Herbert I. Gross
AXIOMS OF EQUALITY
E3: If a = b and if b = c, then a = c
(the transitive property).
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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© 2007 Herbert I. Gross
AXIOMS OF EQUALITY
E3: If a = b and if b = c, then a = c
(the transitive property).
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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© 2007 Herbert I. Gross
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.
Thus, for example, equality (more
specifically “is equal to”) is an example of
an equivalence relation.
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© 2007 Herbert I. Gross
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.
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© 2007 Herbert I. Gross
AXIOMS OF EQUALITY
More specifically, we use the following
property as a shortcut summary
of E1, E2, and E3.
E4:
(the equivalence property)
If a = b, then a and b can be used
interchangeably in any mathematical
relationship that involves equality.
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© 2007 Herbert I. Gross
An Important Warning
When we talk about two things being
equivalent, we always mean with
respect to a given relationship.
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”.
© 2007 Herbert I. Gross
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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!
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. Gross
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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!
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.
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© 2007 Herbert I. Gross
Notes
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…
x=y
…and we decide to add 3 to x. The left hand
side of the equation becomes…
© 2007 Herbert I. Gross
x+3
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Notes
When x = y, by E4 we may replace x by y
in any mathematical relationship.
Replacing x by y in x + 3 gives us the
equivalent expression…
x+3
y
The mathematical way of saying that x + 3
and y + 3 are equivalent is to write…
x+3=y+3
© 2007 Herbert I. Gross
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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
© 2007 Herbert I. Gross
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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.
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. Gross
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Notes
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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In other words…
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)…
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.
© 2007 Herbert I. Gross
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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.
© 2007 Herbert I. Gross
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