1. Properties of Real Numbers
Commutative
Associative
Distributive
Identity + ×
Inverse + ×

2. Commutative Properties
Changing the order of the numbers in addition or multiplication will not change the result.
Commutative Property of Addition states: = or a + b = b + a.
Commutative Property of Multiplication states: 4 • 5 = 5 • 4 or ab = ba.

3. Associative Properties
Changing the grouping of the numbers in addition or multiplication will not change the result.
Associative Property of Addition states: 3 + (4 + 5)= (3 + 4)+ 5 or a + (b + c)= (a + b)+ c
Associative Property of Multiplication states: (2 • 3) • 4 = 2 • (3 • 4) or (ab)c = a(bc)

4. Distributive Property
Multiplication distributes over addition.

5. Additive Identity Property
There exists a unique number 0 such that zero preserves identities under addition.
a + 0 = a and 0 + a = a
In other words adding zero to a number does not change its value.

6. Multiplicative Identity Property
There exists a unique number 1 such that the number 1 preserves identities under multiplication.
a ∙ 1 = a and 1 ∙ a = a
In other words multiplying a number by 1 does not change the value of the number.

7. Additive Inverse Property
For each real number a there exists a unique real number –a such that their sum is zero.
a + (-a) = 0
In other words opposites add to zero.

8. Multiplicative Inverse Property
For each real number a there exists a unique real number such that their product is 1.

9. Let’s play “Name that property!”

10. State the property or properties that justify the following.
3 + 2 = 2 + 3
Commutative Property

11. State the property or properties that justify the following.
10(1/10) = 1
Multiplicative Inverse Property

12. State the property or properties that justify the following.
3(x – 10) = 3x – 30
Distributive Property

13. State the property or properties that justify the following.
3 + (4 + 5) = (3 + 4) + 5
Associative Property

14. State the property or properties that justify the following.
(5 + 2) + 9 = (2 + 5) + 9
Commutative Property

15. Commutative Property of Addition2.
Which Property?
3 + 7 = 7 + 3
Commutative Property of Addition

16. Identity Property of Addition3.
Which Property?
8 + 0 = 8
Identity Property of Addition

17. Commutative Property of Multiplication5.
Which Property?
6 • 4 = 4 • 6
Commutative Property of Multiplication

18. Inverse Property of Addition6.
Which Property?
17 + (-17) = 0
Inverse Property of Addition

19. Commutative Property of Multiplication7.
Which Property?
2(5) = 5(2)
Commutative Property of Multiplication

20. Associative Property of Addition1.
Which Property?
(2 + 1) + 4 = 2 + (1 + 4)
Associative Property of Addition

21. Distributive Property9.
Which Property?
3(2 + 5) = 3•2 + 3•5
Distributive Property

22. Associative Property of Multiplication
10.
Which Property?
6(7•8) = (6•7)8
Associative Property of Multiplication

23. Identity Property of Multiplication
11.
Which Property?
5 • 1 = 5
Identity Property of Multiplication

24. Distributive Property
13.
Which Property?
(6 – 3)4 = 6•4 – 3•4
Distributive Property

25. Identity Property of Multiplication
14.
Which Property?
1(-9) = -9
Identity Property of Multiplication

26. Inverse Property of Addition
15.
Which Property?
3 + (-3) = 0
Inverse Property of Addition

27. Associative Property of Addition
16.
Which Property?
1 + [-9 + 3]
= [1 + (-9)] + 3
Associative Property of Addition

28. Commutative Property of Multiplication
17.
Which Property?
-3(6) = 6(-3)
Commutative Property of Multiplication

29. Identity Property of Addition
18.
Which Property?
= -8
Identity Property of Addition

30. Distributive Property
19.
Which Property?
3•7 – 3•4 = 3(7 – 4)
Distributive Property

31. Associative Property of Addition
20.
Which Property?
6 + [(3 + (-2)]
= (6 + 3) + (- 2)
Associative Property of Addition

32. Commutative Property of Addition
21.
Which Property?
7 + (-5) =
Commutative Property of Addition

33. Distributive Property
22.
Which Property?
(5 + 4)9 =
Distributive Property

34. Properties Using
Variables

35. Distributive Property
31.
Which Property?
6(3 – 2n) = 18 – 12n
Distributive Property

36. Commutative Property of Addition
32.
Which Property?
2x + 3 = 3 + 2x
Commutative Property of Addition

37. Commutative Property of Multiplication
33.
Which Property?
ab = ba
Commutative Property of Multiplication

38. Identity Property of Addition
34.
Which Property?
a + 0 = a
Identity Property of Addition

39. Associative Property of Multiplication
35.
Which Property?
a(bc) = (ab)c
Associative Property of Multiplication

40. Identity Property of Multiplication
36.
Which Property?
a•1 = a
Identity Property of Multiplication

41. Commutative Property of Addition
37.
Which Property?
a +b = b + a
Commutative Property of Addition

42. Distributive Property
38.
Which Property?
a(b + c) = ab + ac
Distributive Property

43. Associative Property of Addition
39.
Which Property?
a + (b + c) = (a +b) + c
Associative Property of Addition

44. Inverse Property of Addition
40.
Which Property?
a + (-a) = 0
Inverse Property of Addition

45. Properties of Real Numbers
Commutative
Associative
Distributive
Identity + ×
Inverse + ×