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The Game of Mathematics Continues… The Game of Mathematics Continues… © 2007 Herbert I. Gross by Herbert I. Gross & Richard A. Medeiros next Lesson 10

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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. © 2007 Herbert I. Gross next Prelude

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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. © 2007 Herbert I. Gross next 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

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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). © 2007 Herbert I. Gross next Mathematical Usage Fortunately, the Game of Mathematics sets out methods for dealing with such subtleties. next

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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. © 2007 Herbert I. Gross next 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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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. © 2007 Herbert I. Gross next 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

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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? © 2007 Herbert I. Gross next That is: do you accept such “facts” as… = 3 + 5?

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© 2007 Herbert I. Gross next 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. next

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We first talked about axioms in Lesson 9, and discussed those governing the equality of numbers. © 2007 Herbert I. Gross next 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. next

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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. © 2007 Herbert I. Gross next 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 next 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, = 8). In fact, the sum of two odd numbers is always an even number. next

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Stated more formally… © 2007 Herbert I. Gross next 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, and are also numbers. We’ll discuss this in more detail shortly.

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© 2007 Herbert I. Gross next 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. next

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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”. © 2007 Herbert I. Gross next Closure is important because it guarantees us that the sum and product of numbers are always numbers. Notes on Closure

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© 2007 Herbert I. Gross next 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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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. next © 2007 Herbert I. Gross

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next The AXIOMS for 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. A1: (The Commutative Property of Addition)

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next The AXIOMS for 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. A2: (The Associative Property of Addition) For example, means the same whether we write it as… (9 + 3) + 1 or as 9 + (3 + 1). next If an expression involves only addition we do not need to use grouping symbols. next © 2007 Herbert I. Gross

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While Axiom A2 may seem self-evident, it’s important to note that not all operations are associative. next 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 next The AXIOMS for ADDITION 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)

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It is called the additive identity because it doesn’t change a number when we add zero to it. © 2007 Herbert I. Gross next 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. Important Note

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© 2007 Herbert I. Gross next The AXIOMS for ADDITION 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)

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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. © 2007 Herbert I. Gross next By using Axiom A4 in conjunction with our other rules, we can now define subtraction.

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Definition © 2007 Herbert I. Gross next to mean… 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. © 2007 Herbert I. Gross next means… a – b a + - b

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Axiom A4 guarantees that the extended number system is closed with respect to subtraction. © 2007 Herbert I. Gross next 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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However, now that we have written Axiom A4 into our Manual: once the number 3 exists, so also does the number - 3. © 2007 Herbert I. Gross next Thus, by Axiom C1, the closure property of addition, must also be a number. And by our definition of subtraction, 2 – 3 means

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© 2007 Herbert I. Gross next The AXIOMS for MULTIPLICATION 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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© 2007 Herbert I. Gross next The AXIOMS for 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. M2: (The Associative Property of Multiplication)

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© 2007 Herbert I. Gross next The AXIOMS for 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)

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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. © 2007 Herbert I. Gross next 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 is also a number. We name it 2. Then is also a number. We name it 3, etc. Importance of Closure

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© 2007 Herbert I. Gross next The AXIOMS for MULTIPLICATION 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. M4: (The Multiplicative Inverse Property)

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By using multiplication and Axiom M4, we can define division in terms of the “invert and multiply” rule. Namely… © 2007 Herbert I. Gross next By the expression a ÷ b we mean… a × 1 / b (which is often written as ) a / b Definition D

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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 next Remark

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Axiom M4 allows us to extend the whole numbers to include the rational numbers (fractions). © 2007 Herbert I. Gross next 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 next 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 next 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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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. © 2007 Herbert I. Gross next For example to see why… 2 × (3 + 4) = (2 × 3) + (2 × 4)

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We may use tally marks to represent by | | | | | | | and to represent 2 × (3 + 4), we write | | | | | | | twice. This is illustrated in the rectangular array below. © 2007 Herbert I. Gross next | | | | | | | next | | | | | | | = = 2 × (3 + 4) Demonstration #1 2 × 32 × 4 next + ( ( ) )

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We may use a rectangle to represent 2 × (3 + 4) as an area. Namely… © 2007 Herbert I. Gross next Demonstration #2 2 × 32 × 4 Area = (2 × 3) + (2 × 4) Area = 2 × (3 + 4) next

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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

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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. next Demonstration #3 Therefore… 2 × (3 + 4) = (2 × 3) + (2 × 4).

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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. © 2007 Herbert I. Gross next In concluding this lesson, we present a summary of our axioms. 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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© 2007 Herbert I. Gross next 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). next

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© 2007 Herbert I. Gross next 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. next

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© 2007 Herbert I. Gross next 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). next 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). next

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© 2007 Herbert I. Gross next 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) next 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) next

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© 2007 Herbert I. Gross next The Distributive Property of Multiplication over Addition D1: a × (b + c) = (a × b) + (a × c)

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