1. Columbus State Community College
Chapter 2 Section 2A
Properties of Real Numbers

2. Properties of Real Numbers
Use the commutative properties.
Use the associative properties.
Use the identity properties.
Use the inverse properties.

3. The Commutative Properties
The commutative properties say that if two numbers are added or multiplied in any order, they give the same results.
a + b = b + a
Addition
a b = b a
Multiplication

4. Using the Commutative Properties
EXAMPLE Using the Commutative Properties
Use a commutative property to complete each statement.
(a) – = ( –3 )
(a) – = ?
By the commutative property for addition, the missing number is –3 because –3 + 7 = 7 + ( –3 ).
(b) –2 ( 8 ) = ? ( –2 )
(b) –2 ( 8 ) = 8 ( –2 )
By the commutative property for multiplication, the missing number is 8 because –2 ( 8 ) = 8 ( –2 ).

5. The Associative Properties
The associative properties say that when we add or multiply three numbers, we can group them in any manner and get the same answer.
( a + b ) + c = a + ( b + c )
Addition
( a b ) c = a ( b c )
Multiplication

6. Using the Associative Properties
EXAMPLE Using the Associative Properties
Use an associative property to complete each statement.
(a) ( – ) = ( –5 ) + 2
(a) ( – ) = ( ? ) + 2
The missing number is –5.
(b) [ 4 ( –3 ) ] 8 = ?
(b) [ 4 ( –3 ) ] 8 = 4 [ ( –3 ) 8 ]
The missing expression is [ ( –3 ) 8 ].

7. Distinguishing between Associative and Commutative Properties
EXAMPLE Distinguishing between Associative and Commutative Properties
State the property given in each statement.
(a) ( ) = ( ) + 5
(a) ( ) = ( ) + 5
(a) ( ) = ( ) + 5
The order of the three numbers is the same on both sides.
The only change is in the grouping, or association, of the numbers. Therefore, this is an example of the associative property.

8. Distinguishing between Associative and Commutative Properties
EXAMPLE Distinguishing between Associative and Commutative Properties
State the property given in each statement.
(b) 5 ( 2 • 6 ) = 5 ( 6 • 2 )
(b) 5 ( 2 • 6 ) = 5 ( 6 • 2 )
(b) 5 ( 2 • 6 ) = 5 ( 6 • 2 )
(b) 5 ( 2 • 6 ) = 5 ( 6 • 2 )
On the left, however, 2 appears first in ( 2 • 6 ). On the right, 6 appears first.
The same numbers, 2 and 6, are grouped on each side.
Since the only change involves the order of the numbers, this statement is an example of the commutative property.

9. Using Commutative and Associative Properties
EXAMPLE Using Commutative and Associative Properties
Use the commutative and associative properties to choose pairs of numbers that are easy to add or multiply.
(a)
= ( ) + ( ) =
= 137

10. Using Commutative and Associative Properties
EXAMPLE Using Commutative and Associative Properties
Use the commutative and associative properties to choose pairs of numbers that are easy to add or multiply.
(b) 5 ( 38 ) ( 20 )
5 ( 38 ) ( 20 )
= 5 ( 20 ) ( 38 )
= ( 38 )
= 3800

11. The Identity Properties
The identity properties say that the sum of 0 and any number equals that number, and the product of 1 and any number equals that number.
a = a and a = a
Addition
a • 1 = a and • a = a
Multiplication

12. Using Identity Properties
EXAMPLE Using Identity Properties
These statements are examples of identity properties.
(a) –8
= –8 Addition
(b) –14 • 1
= – Multiplication

13. Using the Identity Element for Multiplication to Simplify Expressions
EXAMPLE Using the Identity Element for Multiplication to Simplify Expressions
Simplify each expression. 40 48
Factor.
5 • 8
6 • 8 =
(a)
Write as a product. 5 6 = 8 • 5 6 = • 1
Property of 1 5 6 =
Identity property

14. Using the Identity Element for Multiplication to Simplify Expressions
EXAMPLE Using the Identity Element for Multiplication to Simplify Expressions
Simplify each expression.
(b) 1 6 + 11 48
Identity property = 1 6 + 11 48 •
Use 1 = 8 • = 1 6 + 11 48
Multiply. = 8 48 + 11
Add. = 19 48

15. The Inverse Properties
The inverse properties of addition and multiplication lead to the additive and multiplicative identities, respectively.
1) The opposite of a, –a, is the additive inverse of a.
2) The reciprocal of a, , is the multiplicative inverse of the nonzero number a.
a + ( –a ) = and ( –a ) + a = 0
Addition 1 a 1 a 1 a
a • = and • a = 1
Multiplication
( a ≠ 0 )

16. Using Inverse Properties
EXAMPLE Using Inverse Properties
Complete each statement to demonstrate the given inverse property.
(a)
Multiplication 3 4 • = 1 4 3 ?
(b)
Addition 3 4 + = – 3 4 ?

17. Using Inverse Properties
EXAMPLE Using Inverse Properties
Complete each statement to demonstrate the given inverse property.
(c) = 0
Addition
( –9 ) ? 5 1 – 1 5 –
(d) • = 1 5 – = 1 • ?
Multiplication
Rewrite as a fraction.

18. Properties of Real Numbers
Chapter 2 Section 2A – Completed
Written by John T. Wallace