1. Properties of the Real Numbers Part I

2. 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

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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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?)

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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?

22. 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

23. 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

24. 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

25. 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!)

26. 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)

27. 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

28. 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

29. Concentrate!