1. Properties of Real Numbers
SOL 7.16

2. Vocabulary
Addend: a number that is added to another
Sum: The answer to an addition problem
Factor: a number that is being multiplied
Product: The answer to a multiplication problem
Difference: The answer to a subtraction problem
Reciprocal: the multiplicative inverse of a number
Identity Element: numbers that combine with other numbers without changing the other numbers
Inverse: numbers that combine with other numbers and result in identity elements
Make sure to learn these important terms…we will use them from now on! 

3. Commutative Property of…
Addition
Changing the order of the addends does not change the sum.
5+2=2+5
For Example:
m+𝑛=𝑛+𝑚
3+(2∙3)= 2∙3 + 3
Multiplication
Changing the order of the factors does not change the product.
3∙6=6∙3
For Example:
a𝑏=𝑏𝑎

4. Let’s think about the Commutative Property!
What Changed from one side of the equal sign to the other? What should we look for to help us identify this property?
The order of the addends or factors changed.
I think of commuters when I think of the commutative property. You start one way and end, but to start again you must leave where you are and go back to where you started.
Write down what you think of!
Complete the examples using the commutative property.
Can you make your own example?
a) =
b) 8 (3 - 2) =
c) 6- ( 5+ 9) =
d) fg=
5 + 4
(3 - 2)8
6-(9 + 5) gf

5. Associative Property of…
Addition
Regrouping the addends does not change the sum.
5+3 +2=5+(3+2)
For Example:
𝑥+(m+𝑛)=(𝑥+𝑚)+𝑛
Multiplication
Regrouping the factors does not change the product.
For Example:
2 3∙6 = 2∙3 6
a(𝑏∙𝑐)= 𝑎∙𝑏 𝑐

6. Let’s think about the Associative Property!
What Changed from one side of the equal sign to the other? How is the associative property different from the commutative?
The number inside of the grouping symbols or parentheses changed.
In the commutative property the order of the numbers changed, but the numbers that were in the parentheses changed with the associative.
How will you remember this property?
Complete the examples using the associative property.
Make your own example!
a) 2+(4 + 5) =
b) 8(3 • 2) =
c) 6+( 5+ 9) =
d) a(b•c) =
(2 + 4) + 5
(8 •3) •2
(6+5) + 9
(a•b)c

7. What about… Subtraction? Division? 5−2__ 2−5 2− 3−4 __ 2−3 −4
Is subtraction commutative? Associative?
For Example:
5−2__ 2−5 ≠
2− 3−4 __ 2−3 −4 ≠
Division?
Is division commutative? Associative?
3÷6 ___6÷3
For Example: ≠
10 20÷4 ___ 20÷4 10 ≠

8. The Distributive Property
The product of a number and the sum (or difference) of two other numbers equals the sum (or difference) of the
products of the number and each other number.
For Example:
3 5+2 = 3∙5 +(3∙2)
3 5−2 = 3∙5 −(3∙2)
3(6∙4)=72

9. Let’s think about the Distributive Property!
What Changed from one side of the equal sign to the other? What should we look for to help us identify this property?
The number on the outside of the parentheses was passed out to the numbers inside the parentheses.
I think of distributing flyers to the class. You start with one stack of flyers, then once you distribute the stack to the class each person has that same flyer.
Write down what you think of!
Complete the examples using the distributive property.
Can you make your own example?
a) 2(4 + 5) =
b) 8 (3 - 2) =
c) 6( 5+ 9) =
d) e(f-g)=
(2•4) + (2•5)
(8 •3)- (8•2)
(6•5) + (6•9)
(e•f) – (e•g)

10. Additive … Identity Inverse 5+0=5 0+𝑛=𝑛 3+(−3)=0 −a+𝑎=0
The sum of any real number and zero is equal to the given real number.
5+0=5
For Example:
0+𝑛=𝑛
Inverse
The sum of a number and its additive inverse (opposite) always equals zero.
3+(−3)=0
For Example:
−a+𝑎=0

11. Multiplicative… Identity Inverse 5×1=5 1∙𝑛=𝑛 3∙ 1 3 =1 𝑎∙ 1 𝑎 =1
The product of any real number and one is equal to that given number.
5×1=5
For Example:
1∙𝑛=𝑛
Inverse
The product of a number and its multiplicative inverse (reciprocal) always equals one.
3∙ 1 3 =1
For Example:
𝑎∙ 1 𝑎 =1

12. Let’s think about the Identity Properties!
What are the identity elements? How can we tell the identity properties from the inverse properties?
The additive identity element is 0 and the multiplicative identity element is one.
The identity properties result in the same real number that was in the problem. Also, in the identity properties 0 and 1 are being added or multiplied. They are the answers in the inverse properties!
Complete the examples using the identity properties.
Make your own example of each.
a) =
b) 8 (1) =
c) 0+( 5+ 9) = 4 8
(5+9)

13. Let’s think about the Inverse Properties!
Is there a number with no additive inverse? What is the difference between the additive inverse and the multiplicative inverse?
Zero does not have an opposite. It is neither positive or negative to have an additive inverse.
With the additive inverse you are adding the opposite, but with multiplicative inverse you are multiplying the reciprocal, not the opposite.
Complete the examples using the inverse properties.
a) 4 +__ = 0
b) 8 (__) = 1
c) 1 2 ∙___ =1 -4
Make your own example of each.
1 8 2

14. The Multiplicative Property of Zero!
The product of any real number and zero is zero!
5+3 ∙0=0
For Example:
a∙0=0
0=2∙0
0=0∙197

15. Let’s think about the Multiplicative Property of Zero!
Both the Multiplicative Property of Zero and the Additive Identity both have an answer of zero. How will you remember which property is which?
The multiplicative property of zero has a product of zero and there is a zero both on the right and the left of the equal sign.
The additive identity has a sum of zero and there is only a zero on one side of the equal sign.
Complete the examples using the multiplicative property of zero.
Can you make your own example?
a) 2(0) =
b) 0 = 0 •
c)( 5- 9)• ___ = 0
any real number

16. Let’s see if you can identify these properties!
Properties Examples
Let’s see if you can identify these properties!

17. Commutative Property of Multiplication
3∙2=2∙3
Commutative Property of Multiplication

18. Multiplicative Inverse
8∙ 1 8 =1
Multiplicative Inverse

19. Commutative Property of Addition
3 2+3 =3 3+2
Commutative Property of Addition

20. Associative Property of Multiplication
4(3∙2)= 4∙3 2
Associative Property of Multiplication

21. 3∙0=0
Property of Zero

22. Multiplicative Identity
3=1∙3
Multiplicative Identity

23. Associative Property of Addition
4+2 +1=4+(2+1)
Associative Property of Addition

24. Distributive Property
3(2+1)=9
Distributive Property

25. 8+(−8)=0
Additive Inverse

26. Distributive Property
2 1+3 =2∙1+2∙3
Distributive Property

27. Commutative Property of Multiplication
5 2+3 =(2+3)∙5
Commutative Property of Multiplication

28. Multiplicative Inverse
1= 1 2 ∙2
Multiplicative Inverse

29. (5𝑥+1)+0=(5𝑥+1)
Additive Identity

30. Associative Property of Addition
9+1 +8=9+(1+8)
Associative Property of Addition

31. Distributive Property
3 𝑚+𝑛 =3𝑚+3𝑛
Distributive Property