the game of mathematics continues… the game of mathematics continues… © 2007 herbert i. gross...

The Game of Mathematics

Continues…

The Game of Mathematics

Continues…

© 2007 Herbert I. Gross

byHerbert I. Gross & Richard A. Medeiros

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Lesson 10

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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Prelude

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.


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The Manual

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. next

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).


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Mathematical Usage

Fortunately, the Game of Mathematics sets out methods for dealing with such

subtleties. next

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.


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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. next

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.


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How We Play the Game

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.next

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?


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That is: do you accept such “facts” as… 5 + 3 = 3 + 5?

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If you do, one of our rules will be…

If you don’t accept this rule, you might have to think about playing a different game.

If a and b denote numbers, a + b = b + a.

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We first talked about axioms in Lesson 9, and discussed those governing the

equality of numbers.


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AXIOMS

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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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.


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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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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.

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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Stated more formally…

© 2007 Herbert I. Gross next

The AXIOMS of CLOSURE

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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.



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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.

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The way we usually state Axioms C1 and C2 in mathematical language is: “our number system is closed with

respect to addition and multiplication”.


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Closure is important because it guarantees us that the sum and product of numbers

are always numbers.

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Notes on Closure


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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.next

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. nextnext© 2007 Herbert I. Gross


The AXIOMS for ADDITION

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a + b = b + a

Axiom A1 tells us that the sum of anytwo 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.

A1: (The Commutative Property of Addition)

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(a + b) + c = a + (b + c)

A more informal version of Axiom A2 is: we don’t need grouping symbols in an

addition problem.

A2: (The Associative Property of Addition)

For example, 9 + 3 + 1 means the same whether we write it as…

(9 + 3) + 1 or as 9 + (3 + 1).

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If an expression involves only addition we do not need to use grouping symbols. next


While Axiom A2 may seem self-evident, it’s important to note that not all operations are

associative.


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For example,

(9 – 3) – 1 ≠ 9 – (3 – 1)

In other words, subtraction does not have the associative property.

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There is a number, denoted by 0, such that for any number a,

a + 0 = a.

0 is called the Additive Identity.

A3: (The Additive Identity Property)

It is called the additive identity because it doesn’t change a number when we add

zero to it.


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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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Important Note



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.

A4: (The Additive Inverse Property)

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.


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By using Axiom A4 in conjunction with our other rules, we can now define

subtraction.

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Definition


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to mean…

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D

a – b

Given any numbers a and b, we define

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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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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means… a – b a + -b

Axiom A4 guarantees that the extended number system is closed with respect to

subtraction.


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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.

However, now that we have written Axiom A4 into our Manual: once the number

3 exists, so also does the number -3.


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Thus, by Axiom C1, the closure property of addition, 2 + -3 must also be a number.

And by our definition of subtraction, 2 – 3 means 2 + -3.

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The AXIOMS for MULTIPLICATION

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a × b = b × a

M1: (The Commutative Property of Multiplication)

In a similar way, there are four corresponding axioms for multiplication.



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(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.

M2: (The Associative Property of Multiplication)



There is a number, denoted by 1, such that for any number a,

a × 1 = a.

1 is called the Multiplicative Identity.

M3: (The Multiplicative Identity Property)

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.


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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

Importance of Closure



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 by1/a or a-1.

M4: (The Multiplicative Inverse Property)

By using multiplication and Axiom M4, we can define division in terms of the “invert and multiply” rule. Namely…


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By the expression

a ÷ b we mean…

a × 1/b (which is often written as ) a/bnext

Definition D

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.© 2007 Herbert I. Gross

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RemarkRemark

Axiom M4 allows us to extend the whole numbers to include the rational numbers (fractions).


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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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So far we have rules/axioms for addition and rules/axioms for multiplication, but we have no rules that combine

these two operations.

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For our present purposes, there is only one such rule/axiom that we need.

Namely…


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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

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.


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For example to see why…

2 × (3 + 4) = (2 × 3) + (2 × 4)

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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.


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| | | | | | |

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| | | | | | |

3 + 4 =

3 + 4 == 2 × (3 + 4)

Demonstration #1

2 × 3 2 × 4 next+( () )

We may use a rectangle to represent 2 × (3 + 4) as an area. Namely…


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Demonstration #2

2 × 3 2 × 4 Area = (2 × 3) + (2 × 4)

3 4

2 Area = 2 × (3 + 4)

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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.

next© 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.


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Demonstration #3

Therefore… 2 × (3 + 4) = (2 × 3) + (2 × 4).

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.


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In concluding this lesson, we present a summary of our axioms. next

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.

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The AXIOMS of EQUALITY

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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).

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© 2007 Herbert I. Grossnext


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.

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A1: a + b = b + a (The commutative property of addition).

A2: ( a + b ) + c = a + (b + c) (The associative property of addition).

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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).

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M1: a × b = b × a (The commutative property of multiplication)

M2: (a × b) × c = a × (b × c) (The associative property of multiplication)

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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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The Distributive Property of Multiplication over Addition

D1: a × (b + c) = (a × b) + (a × c)

the game of mathematics continues… the game of mathematics continues… © 2007 herbert i. gross...

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