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

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

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

Using the Commutative Properties EXAMPLE 1 Using the Commutative Properties Use a commutative property to complete each statement. (a) –3 + 7 = 7 + ( –3 ) (a) –3 + 7 = 7 + ? 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 ).

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

Using the Associative Properties EXAMPLE 2 Using the Associative Properties Use an associative property to complete each statement. (a) 9 + ( –5 + 2 ) = ( 9 + –5 ) + 2 (a) 9 + ( –5 + 2 ) = ( 9 + ? ) + 2 The missing number is –5. (b) [ 4 ( –3 ) ] 8 = 4 ? (b) [ 4 ( –3 ) ] 8 = 4 [ ( –3 ) 8 ] The missing expression is [ ( –3 ) 8 ].

Distinguishing between Associative and Commutative Properties EXAMPLE 3 Distinguishing between Associative and Commutative Properties State the property given in each statement. (a) 3 + ( 4 + 5 ) = ( 3 + 4 ) + 5 (a) 3 + ( 4 + 5 ) = ( 3 + 4 ) + 5 (a) 3 + ( 4 + 5 ) = ( 3 + 4 ) + 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.

Distinguishing between Associative and Commutative Properties EXAMPLE 3 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.

Using Commutative and Associative Properties EXAMPLE 4 Using Commutative and Associative Properties Use the commutative and associative properties to choose pairs of numbers that are easy to add or multiply. (a) 18 + 36 + 17 + 14 + 52 18 + 36 + 17 + 14 + 52 = ( 18 + 52 ) + ( 36 + 14 ) + 17 = 70 + 50 + 17 = 137

Using Commutative and Associative Properties EXAMPLE 4 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 ) = 100 ( 38 ) = 3800

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 + 0 = a and 0 + a = a Addition a • 1 = a and 1 • a = a Multiplication

Using Identity Properties EXAMPLE 5 Using Identity Properties These statements are examples of identity properties. (a) 0 + –8 = –8 Addition (b) –14 • 1 = –14 Multiplication

Using the Identity Element for Multiplication to Simplify Expressions EXAMPLE 6 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

Using the Identity Element for Multiplication to Simplify Expressions EXAMPLE 6 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

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 ) = 0 and ( –a ) + a = 0 Addition 1 a 1 a 1 a a • = 1 and • a = 1 Multiplication ( a ≠ 0 )

Using Inverse Properties EXAMPLE 7 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 ?

Using Inverse Properties EXAMPLE 7 Using Inverse Properties Complete each statement to demonstrate the given inverse property. (c) 9 + = 0 Addition ( –9 ) ? 5 1 – 1 5 – (d) • = 1 5 – = 1 • ? Multiplication Rewrite as a fraction.

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