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The Game of
Mathematics
Continues…
Lesson 10
by
Herbert I. Gross & Richard A. Medeiros
© 2007 Herbert I. Gross
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Prelude
To play the Game of Mathematics you will
have to get used to “math-think” and
“math-speak”. It’s a bit like an “Operating
Manual” that you create as you go along.
In this context our “manual” starts from
scratch. In a sense, what we’ve taken for
granted previously in our study of
mathematics, from kindergarten on, no
longer counts.
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© 2007 Herbert I. Gross
The Manual
The first time we encounter a mathematical
concept (either a new one or one we’ve
seen before), we redefine it in an
unambiguous way that shows evidence
that what we are writing agrees with
what we believe to be true.
Be patient and attentive because as you get
deeper into algebra (and higher math), the
Manual will help to clarify many
not-so-obvious results, and thus protect
you from making errors.
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© 2007 Herbert I. Gross
Mathematical Usage
Talking about math concepts can be tricky.
For example, try to define distance
without using the concept of distance
in the definition, or try defining time
without using the concept of time
in the definition. It can’t be done (at least
on an elementary level).
Fortunately, the Game of Mathematics
sets out methods for dealing with such
subtleties.
© 2007 Herbert I. Gross
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For example, as we mentioned in Lesson 9,
some people view whole numbers as
lengths and some people view them as tally
marks. Other people may view them in still
different ways.
In order not to rely on one specific
viewpoint, we will not try to define numbers
or the various operations we perform on
numbers. Instead, we will list the rules that
we believe these concepts obey, and we will
leave it to you to decide if these rules agree
with your own perceptions.
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© 2007 Herbert I. Gross
How We Play the Game
In any game, players have to agree to abide by
the rules. If they don’t, they can’t play the
game. Hence, in the Game of Algebra you will
have to accept (agree to) the rules that we
shall set forth. These rules have to be
“self-evident” so that they make sense to you.
In return for your acceptance of the rules, you
are promised that any new claims we make
about numbers follow inescapably from the
definitions and rules that you agree to accept.
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© 2007 Herbert I. Gross
For example, think about what
words such as “number” and “addition”
mean to you. Are you willing to accept as
a rule that when you add two numbers,
the sum does not depend on the order
in which you add the two numbers?
That is: do you accept such “facts” as…
5 + 3 = 3 + 5?
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© 2007 Herbert I. Gross
If you do, one of our rules will be…
If a and b denote numbers,
a + b = b + a.
If you don’t accept this rule, you might have
to think about playing a different game.
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© 2007 Herbert I. Gross
AXIOMS
We first talked about axioms in Lesson 9,
and discussed those governing the
equality of numbers.
In this lesson, we will develop the axioms
that govern addition and multiplication,
and show how they can be used to
paraphrase numerical and algebraic
expressions.
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© 2007 Herbert I. Gross
After defining addition, we will then go on
to define subtraction in terms of addition.
That is: subtraction is performed by using
the “add the opposite” rule.
And after defining multiplication, we will
then go on to define division in terms of
multiplication.
That is: division is performed by using the
“invert and multiply” rule .
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© 2007 Herbert I. Gross
Self-Evident?
Perhaps the simplest observation
we are willing to accept is that when we
add or multiply two numbers, the answer
is always a number.
Facts such as this are not self-evident.
For example, the sum of two odd numbers
(such as 3 and 5) is not an odd number (for
example, 5 + 3 = 8). In fact, the sum of two
odd numbers is always an even number.
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© 2007 Herbert I. Gross
Stated more formally…
The AXIOMS of CLOSURE
C1: If a and b are any numbers, then
a + b is also a number.
To a mathematician: although a and b are
different letters, they may represent the
same as well as different numbers.
For example, Axiom C1 tells us that …
if a = 2 and b = 3, then the sums, 2 + 2 and
2 + 3 are also numbers. We’ll discuss this
in more detail shortly.
© 2007 Herbert I. Gross
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The AXIOMS of CLOSURE
C2: If a and b are any numbers, then
a × b (which we shall usually write as ab)
is also a number.
Since equality is a relationship between
numbers, we have to accept that a + b
and a × b are numbers. Otherwise,
statements such as a + b = b + a and
a × b = b × a would have no meaning.
That is: we accept Axiom C2 in order to be
able to play the game of mathematics.
© 2007 Herbert I. Gross
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Notes on Closure
The way we usually state Axioms C1
and C2 in mathematical language is:
“our number system is closed with
respect to addition and multiplication”.
Closure is important because it guarantees
us that the sum and product of numbers are
always numbers.
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© 2007 Herbert I. Gross
Notice that we talk about closure with respect
to a particular operation with numbers.
Thus, while the whole numbers are closed
with respect to addition, they are not closed
with respect to subtraction.
For example, 2 and 3 are whole numbers, but
2 – 3 is not a whole number. That is, the whole
numbers are defined as 0, 1, 2, … Since this
definition does not include negative numbers,
it means that 0 is the least whole number.
So, there is no whole number that we can
add to 3 and obtain 2 as the sum.
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© 2007 Herbert I. Gross
Once we accept that our number system is
closed with respect to addition; it then
makes sense to talk about the axioms
(rules) for addition. Again, even though we
shall state the axioms more formally and
give them more technical names, keep in
mind that most likely you already knew
these rules.
In fact, it is important to remember that no
matter how anyone visualizes a number,
the rules have to be so obvious that every
“player” in the game of mathematics will be
willing to accept them.
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© 2007 Herbert I. Gross
The AXIOMS for ADDITION
A1: (The Commutative Property of Addition)
a +b=b+a
Axiom A1 tells us that the sum of any
two numbers a and b does not depend on
the order in which we add them.
Don’t confuse Axiom A1 with the
Symmetry Property, which tells us that
if a + b = b + a, then b + a = a + b.
© 2007 Herbert I. Gross
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The AXIOMS for ADDITION
A2: (The Associative Property of Addition)
(a + b) + c = a + (b + c)
A more informal version of Axiom A2 is:
we don’t need grouping symbols in an
addition problem.
For example, 9 + 3 + 1 means the same
whether we write it as…
(9 + 3) + 1 or as 9 + (3 + 1).
If an expression involves only addition we
do not need to use grouping symbols. next
© 2007 Herbert I. Gross
While Axiom A2 may seem self-evident, it’s
important to note that not all operations are
associative.
For example,
(9 – 3) – 1 ≠ 9 – (3 – 1)
In other words, subtraction does not
have the associative property.
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© 2007 Herbert I. Gross
The AXIOMS for ADDITION
A3: (The Additive Identity Property)
There is a number, denoted by 0,
such that for any number a,
a + 0 = a.
0 is called the Additive Identity.
© 2007 Herbert I. Gross
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It is called the additive identity because it
doesn’t change a number when we add
zero to it.
Important Note
This is the first time the axioms themselves
mention the existence of a specific number
(here, zero) by name.
That is, we have talked about the properties
of numbers, but up to now there had been
no mention of a specific number in our
game.
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© 2007 Herbert I. Gross
The AXIOMS for ADDITION
A4: (The Additive Inverse Property)
Given any number a there exists a
number b for which a + b = 0.
b is called the additive inverse of a
and is usually denoted by -a.
© 2007 Herbert I. Gross
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Axiom A4 is in actuality a restatement of
something mentioned in our discussion
of signed numbers. At that time, we
referred to -a as the opposite of a.
By using Axiom A4 in conjunction with
our other rules, we can now define
subtraction.
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© 2007 Herbert I. Gross
Definition
Given any numbers a and b, we define
a–b
to mean…
a+
-b
In the definition, a stands for the first number
and b stands for the second number.
So, for example, b – a would mean b + -a.
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© 2007 Herbert I. Gross
a–b
means…
a+
-b
While this definition might not seem too
familiar at first glance, notice that it is
simply a restatement of the
“add the opposite” rule that we presented
in Lessons 3 and 4, when we discussed
how we add and subtract signed numbers.
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© 2007 Herbert I. Gross
Axiom A4 guarantees that the extended
number system is closed with respect to
subtraction.
That is: prior to Axiom A4, the number
2 – 3 did not exist in our Manual
because there is no whole number that
can be added to 3 to yield 2 as the sum.
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© 2007 Herbert I. Gross
However, now that we have written
Axiom A4 into our Manual: once the number
3 exists, so also does the number -3.
And by our definition of subtraction,
2 – 3 means 2 + -3.
Thus, by Axiom C1, the closure property of
addition, 2 + -3 must also be a number.
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© 2007 Herbert I. Gross
In a similar way, there are four
corresponding axioms for multiplication.
The AXIOMS for MULTIPLICATION
M1: (The Commutative Property of
Multiplication)
a×b=b×a
© 2007 Herbert I. Gross
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The AXIOMS for MULTIPLICATION
M2: (The Associative Property of
Multiplication)
(a × b) × c = a × (b × c)
Notice that these two rules are similar to the
corresponding two rules for addition
(Axioms A1 and A2). In essence, all that is
different is that the multiplication sign has
replaced the addition sign.
© 2007 Herbert I. Gross
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The AXIOMS for MULTIPLICATION
M3: (The Multiplicative Identity Property)
There is a number, denoted by 1,
such that for any number a,
a × 1 = a.
1 is called the
Multiplicative Identity.
© 2007 Herbert I. Gross
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Notice that this is only the second number
we’ve specifically defined in our game. It is
called the “multiplicative identity” because
it doesn't change a number when it is
multiplied by 1.
Importance of Closure
The closure properties allow us to “reinvent”
the number system in terms of our axioms.
Namely, by the Closure Property for Addition
the fact that 1 is a number means that 1 + 1 is
also a number. We name it 2. Then 2 + 1 is
also a number. We name it 3, etc. next
© 2007 Herbert I. Gross
The AXIOMS for MULTIPLICATION
M4: (The Multiplicative Inverse Property)
Given any non-zero number a,
there exists a number b for which a
× b = 1. b is called the
multiplicative inverse of a and is
usually denoted by
1/a or a-1.
© 2007 Herbert I. Gross
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By using multiplication and Axiom M4, we
can define division in terms of the “invert
and multiply” rule. Namely…
Definition
By the expression
a÷b
we mean…
a × 1/b (which is often written as ) a/b
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© 2007 Herbert I. Gross
Remark
You may wonder why we made the
restriction that a could not be 0. The
reason is that what we now are calling the
“multiplicative inverse" is a more formal
way of describing what we called the
“reciprocal” in our study of fractions.
Since we already know that the only
number that doesn’t have a reciprocal is 0
(that is, we are not allowed to divide by 0),
we exclude 0 in Axiom M4.
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© 2007 Herbert I. Gross
Axiom M4 allows us to extend the
whole numbers to include the
rational numbers (fractions).
For example, 2 and 3 are numbers.
Therefore, Axiom M4 tells us that 1/3
(that is, 3-1 ) is also a number.
Thus, 2 × 1/3 (which means the same as 2 ÷
3)
is also a number, which we denote by 2/3.
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© 2007 Herbert I. Gross
So far we have rules/axioms for addition
and rules/axioms for multiplication,
but we have no rules that combine
these two operations.
For our present purposes, there is only
one such rule/axiom that we need.
Namely…
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© 2007 Herbert I. Gross
THE DISTRIBUTIVE PROPERTY OF
MULTIPLICATION OVER ADDITION
D1:
a × (b + c) = (a × b) + (a × c)
or using our earlier agreements…
D1:
a(b + c ) = ab + ac
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© 2007 Herbert I. Gross
Although Axiom D1 is probably the
least self-evident of our rules, we can
demonstrate its plausibility by using
tally marks and/or areas of rectangles.
For example to see why…
2 × (3 + 4) = (2 × 3) + (2 × 4)
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© 2007 Herbert I. Gross
Demonstration #1
We may use tally marks to represent
3 + 4 by | | | | | | | and to represent
2 × (3 + 4), we write | | | | | | | twice. This
is illustrated in the rectangular array below.
3+4= |||
||||
3+4= |||
||||
( 2 × 3)+ (2 × 4)
© 2007 Herbert I. Gross
= 2 × (3 + 4)
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Demonstration #2
We may use a rectangle to represent
2 × (3 + 4) as an area. Namely…
3
2 2×3
4
2×4
Area = 2
(2××(3
3)++4)
(2 × 4)
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© 2007 Herbert I. Gross
Important Note
The area model can be used for
all numbers a, b, and c, whereas
the tally mark model is
restricted to the case in which
a, b, and c are whole numbers.
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© 2007 Herbert I. Gross
Suppose you are
selling candy bars
for $2 each. On Monday you sell 3 bars, and
on Tuesday you sell 4 bars. All in all, you
sold (3 + 4) candy bars; for which you
received a total of 2 × (3 + 4) dollars. And by
looking at how much money you received by
focusing on the daily income: for Monday,
you received (2 × 3) dollars; and for Tuesday,
you received (2 × 4) dollars. So, your total
income is (2 × 3) + (2 × 4) dollars.
Demonstration #3
Therefore… 2 × (3 + 4) = (2 × 3) + (2 × 4).
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© 2007 Herbert I. Gross
While rules and definitions are important in
any game, the purpose of the game lies in
how well we learn to apply strategy to arrive
at a winning situation.
In terms of the game of mathematics, our
goal is to use the rules and definitions to
develop other “facts” about our game.
In Lesson 11 we will apply this idea to
paraphrasing more complicated expressions
and to solving algebraic equations.
In concluding this lesson, we present a
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summary of our axioms.
© 2007 Herbert I. Gross
The AXIOMS of EQUALITY
E1:
a = a (the reflective property)
E2:
If a = b then b = a (the symmetric property)
E3: If a = b and if b = c, then a = c (the transitive property)
E4: If a = b then a and b can be used interchangeably in
any mathematical relationship. That is, if a = b we can
interchange a and b whenever we wish in any
mathematical relationship to give us a relationship
different but equivalent relationship (the equivalence
property).
© 2007 Herbert I. Gross
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The AXIOMS of CLOSURE
C1: If a and b are any two numbers, then
a + b is also a number.
C2: If a and b are any two numbers, then
ab (or, a × b) is also a number.
© 2007 Herbert I. Gross
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The AXIOMS for ADDITION
A1: a + b = b + a
(The commutative property of addition).
A2: ( a + b ) + c = a + (b + c)
(The associative property of addition).
A3: There exists a number, denoted by 0, such that for
any number a, a + 0 = a. 0 is called the additive identity.
(The additive identity property).
A4: Given any number a there exists a number b for which
a + b = 0. b is called the additive inverse of a and is
usually denoted by -a.
(The additive inverse property).
© 2007 Herbert I. Gross
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The AXIOMS for MULTIPLICATION
M1:
a×b= b×a
(The commutative property of multiplication)
M2:
(a × b) × c = a × (b × c)
(The associative property of multiplication)
M3: There is a number, denoted by 1, such that for any
number a, a × 1 = a. 1 is called the multiplicative identity.
(The multiplicative identity property)
M4: Given any non zero number a, there exists a number
b for which a × b 1. b is called the multiplicative inverse
of a and is usually denoted by 1/a or a-1.
(The multiplicative inverse property)
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© 2007 Herbert I. Gross
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The Distributive Property of
Multiplication over Addition
D1:
© 2007 Herbert I. Gross
a × (b + c) = (a × b) + (a × c)