Properties of the Real Numbers Part I

Symbols in Mathematics As you are aware by now, mathematics uses many symbols Students sometimes feel overwhelmed in math classes beginning with algebra, not only by the symbols, but because they don’t know how to think about them As we proceed through the course, I will emphasize how you should think, in your mind, about the meaning of these symbols You should practice referring to the symbols in the manner that I show you, not only in your own mind, but when talking to someone else about them We will start with how to think about variables

What is a Variable? What do you see below? 2

2 What is a Variable? What do you see below? If you said that this is “two”, well, that’s not quite correct The figure “2” is a symbol that we have chosen to represent, for example, , or , or

What is a Variable? As you are aware by now, we sometimes use, for example 𝑥 where, in the first years of your schooling, you saw number symbols Both “𝑥” and “2” are symbols, but where “2” is a symbol that represents a specific number, we can use “𝑥”, or any other letter (or even any other symbol) to represent an unknown amount or sometimes just any number It is good practice, and one you will see me use often, to state ahead of time what a variable will represent when we use it However, sometimes we will just assume that a letter will be used to represent any number

What is a Variable? As you may have noticed, the back wall of the classroom reminds you that “Math Has to Make Sense” Our first step in making sense of the math you will be learning is to remember this: when we use a variable like 𝑥, you should think or say to yourself that 𝑥 is a number As we review the properties of real numbers, you will see letters being used and you must remind yourself that they are symbols that we use for numbers! *Answer the first question on your notes handout

What Are Real Number Properties? *The Real Number Properties are very basic true statements about numbers under addition and multiplication; fill in the next part of your notes The Properties are “true” because we choose them to be so, but they also correspond to our experiences with number operations Everything you learn in algebra 2 will be true because of these properties; you must learn them and, especially, learn how to use them Over time, we will allow the properties to fade into the background, but we can always return to them if necessary

What Are Real Number Properties? There are 11 properties all together Five tell us things that are true of numbers under addition, five tell us what is true under multiplication, and one property links addition and multiplication We will look at them in pairs because each (except for the one mentioned above at the end) is the same property, in one case for addition and in the other for multiplication The first property is shown on the next slide

The Closure Property *The Closure Property for Addition says: If 𝑎 and 𝑏 represent any real numbers, then 𝑎+𝑏 represents a unique real number *The Closure Property for Multiplication says: If 𝑎 and 𝑏 represent any real numbers, then 𝑎⋅𝑏 represents a unique real number *Before looking at how to interpret these properties, here is a question: what does the word unique mean? Write an answer in your notes

The Closure Property Unique means one of a kind When the property tells us that 𝑎+𝑏 or that 𝑎⋅𝑏 are unique, it means that we can be sure that they represent just one number, not two or more different numbers

The Closure Property How should we interpret the Closure Property? *First, a question: how many numbers are involved in the expression 2+3? *What are the numbers? Write an answer in your notes. The numbers involved are 2, 3, and 5. Why? Because the Closure Property for Addition tells us that 2+3 is, itself, a number

The Closure Property *How many numbers are involved in the expression 2⋅3? *What are the numbers? Write an answer in your notes. The numbers involved are 2, 3, and 6. Why? Because the Closure Property for Multiplication tells us that 2⋅3 is, itself, a number

The Closure Property If 𝑥 and 𝑦 are real numbers, then what numbers can we talk about in the expression 𝑥+𝑦? The numbers are 𝑥, 𝑦, and 𝑥+𝑦 (why?) What numbers can we talk about in the expression 𝑥⋅𝑦? The numbers are 𝑥, 𝑦, and 𝑥⋅𝑦 (why?)

The Closure Property Note that we don’t need to know any actual values here; the Closure Property guarantees that 𝑥+𝑦 and 𝑥⋅𝑦 are themselves numbers *Here is what to remember about the Closure Property: By the Closure Property for Addition, we should think of 𝑥+𝑦 as a number By the Closure Property for Multiplication, we should think of 𝑥⋅𝑦 as a number

The Commutative Property *The Commutative Property for Addition says: If 𝑎 and 𝑏 are real numbers, then 𝑎+𝑏=𝑏+𝑎 *The Commutative Property for Multiplication says: If 𝑎 and 𝑏 are real numbers, then 𝑎⋅𝑏=𝑏⋅𝑎

The Commutative Property One way to think about this property under addition is to imagine having to count two separate sets of objects Suppose that you have two baskets of apples, one of which contains three apples, the other of which contains five apples You could start counting apples in the first basket, “1, 2, 3”, then continue counting the apples in the second basket, “4, 5, 6, 7, 8” OR, you could start counting apples in the second basket, “1, 2, 3, 4, 5”, then continue counting the apples in the second basked, “6, 7, 8” So we have 3+5=5+3

The Commutative Property *The way to think about the Commutative Property for Addition is to say that we can rearrange any addition without changing the meaning of the expression *The way to think about the Commutative Property for Multiplication is to say that we can rearrange any multiplication without changing the meaning of the expression Be sure to fill in the appropriate phrase in your notes

The Associative Property *The Associative Property for Addition says: If 𝑎, 𝑏, and 𝑐 are real numbers, then 𝑎+ 𝑏+𝑐 = 𝑎+𝑏 +𝑐 *The Associative Property for Multiplication says: If 𝑎, 𝑏, and 𝑐 are real numbers, then 𝑎⋅ 𝑏⋅𝑐 = 𝑎⋅𝑏 ⋅𝑐 To this point we haven’t had to talk about Order of Operations, so in a sense we don’t have any But we will add the first Order of Operations: always perform operations within grouping symbols first

The Associative Property To a mathematician, the operations of addition and multiplication are known as binary operators (you don’t need to remember this) You might guess that it has something to do with 2, because the prefix bi- refers to 2; and you’d be correct To say that addition and multiplication are binary operations means that, technically, we can only ever add or multiply two numbers at a time The Associative Property for Addition tells us that, when adding three (or more) numbers, it doesn’t matter where we start

The Associative Property Similarly, the Associative Property for Multiplication tells us that, when multiplying three (or more) numbers, it doesn’t matter where we start But the property also tells us where we can insert parentheses *The way to think about the Associative Property for Addition is that we are allowed to use grouping symbols around any addition of two numbers *The way to think about the Associative Property for Multiplication is that we are allowed to use grouping symbols around any multiplication of two numbers

Mixed Operations Consider the expression 2⋅3+4 Each of the properties you have seen so far deal with either addition alone or with multiplication alone The expression above includes both addition and multiplication Does it matter which operation we perform first?

Mixed Operations *Find the number that results from multiplying first *Next, find the number that results from adding first If we first perform multiplication (indicated by parentheses), we get 2⋅3 +4=6+4=10 If addition is first we get 2⋅ 3+4 =2⋅7=14

Mixed Operations Since we get two different answers, then we clearly cannot ignore this situation We have two options: ALWAYS use parentheses (or any other grouping symbol) around the operation to be performed first Choose one of the two operations to ALWAYS be performed first (although operations in grouping symbols have priority) Because mathematicians prefer to write as little as possible, they have chosen the second option *For reasons we won’t discuss here, they chose the order: 1) Grouping symbols 2) Multiplication 3) Addition

Mixed Operations Therefore, we can write 2⋅3 +4=2⋅3+4, because multiplication is performed first (don’t need parentheses) But 2⋅(3+4) can only be written this one way (well, sort of; you’ll see why soon) If we use variables 𝑥, 𝑦, and 𝑧 to represent any real numbers, then 𝑥⋅𝑦 +𝑧=𝑥⋅𝑦+𝑧=𝑥𝑦+𝑧 𝑥⋅(𝑦+𝑧) cannot, yet, be written without parentheses

Mixed Operations Note that this is not yet the PEMDAS you know so well The reason is that exponents are a special case we will get to a bit later, and division and subtraction don’t follow the properties we have listed so far; but we will add them in due time In the meantime, what I want you to practice is using the properties we have so far, especially in expressions that include both addition and multiplication (recall that 𝑥⋅𝑦=𝑥𝑦; please do not use × for multiplication!)

Guided Practice 1: The Closure Property Tell which numbers can be named in each expression. All variables represent any real number. Remember that multiplication is performed before addition! 4+2⋅3 (HINT: there are 5 numbers) 𝑎+𝑏⋅𝑐 (HINT: there are 5 numbers) 4+2⋅(3+1) (HINT: there are 7 numbers) 𝑚+𝑛⋅(𝑥+𝑦) (HINT: there are 7 numbers) 1+2+3 (HINT: there are 5 numbers, but more than one correct answer is possible) 𝑓+𝑔+ℎ (HINT: there are 5 numbers, but more than one correct answer is possible)

Guided Practice 2: The Commutative Property Use the Commutative Property (CP) as indicated. Use the CP of Addition for 4+2⋅3 Use the CP of Multiplication for 4+2⋅3 Use the CP of Additon for 4+2⋅(3+1) in the first addition Use the CP of Multiplication for 4+2⋅(3+1) Use the CP of addition for 2⋅𝑥+3⋅𝑦 Use the CP of multiplication for 2⋅𝑥+3⋅𝑦 in the second multiplication

Guided Practice 3: The Associative Property Use the Associative Property (AP) as indicated. Use the AP of Addition for 4⋅2+2⋅3 +3⋅4 Use the AP of Multiplication for 4⋅ 2+1 ⋅3 Use the AP of Additon for 𝑎⋅𝑏+(𝑥⋅𝑦+𝑝⋅𝑞) Use the AP of Multiplication for 𝑎⋅ 𝑥+𝑦 ⋅ 𝑛+𝑚

Concentrate!