1. The Game of Mathematics Herbert I. Gross & Richard A. Medeiros
Continues…
Lesson 10 by
Herbert I. Gross & Richard A. Medeiros
© Herbert I. Gross
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2. 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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3. 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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4. 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.
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5. 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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6. “self-evident” so that they make sense to you.
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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7. 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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8. If a and b denote numbers, a + b = b + a.
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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9. 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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10. 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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11. 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, = 8). In fact, the sum of two odd numbers is always an even number.
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12. C1: If a and b are any numbers, then
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, and are also numbers. We’ll discuss this in more detail shortly.
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13. C2: If a and b are any numbers, then
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.
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14. 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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15. For example, 2 and 3 are whole numbers, but
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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16. 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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17. a + b = b + a 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.
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18. (a + b) + c = a + (b + c) 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, 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.
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19. In other words, subtraction does not have the associative property.
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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20. There is a number, denoted by 0, such that for any number a,
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.
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21. 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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22. 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.
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23. 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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24. a – b a + -b D Definition Given any numbers a and b, we define
to mean…
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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25. a – b
a + -b
means…
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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26. That is: prior to Axiom A4, the number
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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27. 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
Thus, by Axiom C1, the closure property of addition, must also be a number.
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28. a × b = b × a The AXIOMS for MULTIPLICATION
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
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29. (a × b) × c = a × (b × c) 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.
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30. There is a number, denoted by 1, such that for any number a,
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.
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31. 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 is also a number. We name it 2. Then is also a number. We name it 3, etc.
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32. 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.
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33. a × 1/b (which is often written as ) a/b
By using multiplication and Axiom M4, we can define division in terms of the “invert and multiply” rule. Namely…
Definition D
By the expression
a ÷ b
we mean…
a × 1/b (which is often written as ) a/b
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34. 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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35. 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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36. but we have no rules that combine these two operations.
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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37. D1: a × (b + c) = (a × b) + (a × c) or using our earlier agreements…
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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38. 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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39. We may use tally marks to represent
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 =
| | | | | | |
= 2 × (3 + 4)
3 + 4 =
| | | | | | | (
2 × 3 ) + (
2 × 4 )
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40. We may use a rectangle to represent
Demonstration #2
We may use a rectangle to represent
2 × (3 + 4) as an area. Namely… 3 4
2 × 3
2 × 4
Area = 2 × (3 + 4)
Area = (2 × 3) + (2 × 4) 2
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41. a, b, and c are whole numbers.
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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42. Demonstration #3 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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43. how well we learn to apply strategy to arrive at a winning situation.
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 summary of our axioms.
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44. E3: If a = b and if b = c, then a = c (the transitive property)
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).
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45. 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.
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46. 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).
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47. 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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48. The Distributive Property of Multiplication over Addition
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The Distributive Property of Multiplication over Addition
D1: a × (b + c) = (a × b) + (a × c)
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