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1.1 Apply Properties of Real Numbers Goal Your Notes p Study properties of real numbers. VOCABULARY Opposite Reciprocal SUBSETS OF REAL NUMBERS numbers The real numbers consist of the and the numbers. Two subsets of the rational numbers are the (0, 1, 2, 3...) and the (... , 23, 22, 21, 0, 1, 2, 3, ...). Rational Numbers Irrational Numbers • Can be written as quotients of integers • Cannot be written as quotients of integers • Can be written as decimals that terminate or repeat • Cannot be written as decimals that terminate or repeat Example 1 Graph real numbers on a number line } 13 Graph the real numbers 2} and Ï6 on a number line. 5 Solution 13 5 . Use a calculator to approximate Note that 2} 5 } } 13 Ï6 to the nearest tenth: Ï6 ø . So, graph 2} 5 24 Copyright © Holt McDougal. All rights reserved. 23 } and graph Ï 6 between and between and . 22 21 0 1 2 3 4 Lesson 1.1 • Algebra 2 Notetaking Guide 1 1.1 Apply Properties of Real Numbers Goal Your Notes p Study properties of real numbers. VOCABULARY Opposite The opposite, or additive inverse, of any number b is 2b. Reciprocal The reciprocal, or multiplicative inverse, 1 . of any nonzero number b is } b SUBSETS OF REAL NUMBERS The real numbers consist of the rational numbers and the irrational numbers. Two subsets of the rational numbers are the whole numbers (0, 1, 2, 3...) and the integers (... , 23, 22, 21, 0, 1, 2, 3, ...). Rational Numbers Irrational Numbers • Can be written as quotients of integers • Cannot be written as quotients of integers • Can be written as decimals that terminate or repeat • Cannot be written as decimals that terminate or repeat Graph real numbers on a number line Example 1 } 13 Graph the real numbers 2} and Ï6 on a number line. 5 Solution 13 5 22.6 . Use a calculator to approximate Note that 2} 5 } } 13 Ï6 to the nearest tenth: Ï6 ø 2.4 . So, graph 2} 5 } between 23 and 22 and graph Ï 6 between 2 and 3 . 2 24 Copyright © Holt McDougal. All rights reserved. 23 13 5 6 22 21 0 1 2 3 4 Lesson 1.1 • Algebra 2 Notetaking Guide 1 Your Notes PROPERTIES OF ADDITION AND MULTIPLICATION Let a, b, and c be real numbers. Property Addition Multiplication a 1 b is a real number. ab is a real number. Commutative a1b5 ab 5 ba Associative (a 1 b) 1 c 5 a 1 (b 1 c) (ab)c 5 Identity a 1 0 5 a, 5a a p 1 5 a, 5a Inverse a 1 (2a) 5 Distributive a(b 1 c) 5 1 ap} a 5 1, a Þ 0 The following property involves both addition and multiplication. 1 Identify properties of real numbers Example 2 Identify the property that the statement illustrates. a. (6 p 3) p 2 5 6 p (3 p 2) property of b. 21 1 (221) 5 0 property of Checkpoint Complete the following exercises. 3 } 1 2 1 1. Graph the numbers 23.2, } , Ï 5 , 22, 2} . 2 4 24 23 22 21 0 3 4 2. Identify the property that 10(6 1 8) 5 10(6) 1 10(8) illustrates. 2 Lesson 1.1 • Algebra 2 Notetaking Guide Copyright © Holt McDougal. All rights reserved. Your Notes PROPERTIES OF ADDITION AND MULTIPLICATION Let a, b, and c be real numbers. Property Addition Multiplication a 1 b is a real number. ab is a real number. Commutative a1b5 b1a ab 5 ba Associative (a 1 b) 1 c 5 a 1 (b 1 c) (ab)c 5 a(bc) Identity a 1 0 5 a, 01a 5a a p 1 5 a, 1pa 5a Closure 1 Inverse a 1 (2a) 5 0 Distributive a(b 1 c) 5 ab 1 ac ap} a 5 1, a Þ 0 The following property involves both addition and multiplication. Identify properties of real numbers Example 2 Identify the property that the statement illustrates. a. (6 p 3) p 2 5 6 p (3 p 2) Associative property of multiplication b. 21 1 (221) 5 0 Inverse property of addition Checkpoint Complete the following exercises. 3 } 1 1. Graph the numbers 23.2, } , Ï 5 , 22, 2} . 2 4 24 23.2 22 23 22 1 3 4 22 21 0 5 1 2 3 4 2. Identify the property that 10(6 1 8) 5 10(6) 1 10(8) illustrates. Distributive property 2 Lesson 1.1 • Algebra 2 Notetaking Guide Copyright © Holt McDougal. All rights reserved. Your Notes DEFINING SUBTRACTION AND DIVISION . The Subtraction is defined as opposite, or , of any number b is 2b. If b is positive, then 2b is negative. If b is negative, then 2b is positive. a 2 b 5 a 1 (2b) Definition of subtraction . Division is defined as The reciprocal, or , of any 1 nonzero number b is } . b 1 a4b5ap} ,bÞ0 b Example 3 Definition of division Use properties and definitions of operations Show that 9 1 (b 2 9) 5 b. 9 1 (b 2 9) 5 9 1 [b 1 ( 5 9 1 [( )] ) 1 b] 5 [9 1 (29)] 1 b 5 1b 5 Definition of subtraction Commutative property of addition property of addition Inverse property of addition Identity property of addition Checkpoint Use properties and definitions of operations to show that the statement is true. 3. 5(a 4 5) 5 a Homework Copyright © Holt McDougal. All rights reserved. Lesson 1.1 • Algebra 2 Notetaking Guide 3 Your Notes DEFINING SUBTRACTION AND DIVISION Subtraction is defined as adding the opposite . The opposite, or additive inverse , of any number b is 2b. If b is positive, then 2b is negative. If b is negative, then 2b is positive. a 2 b 5 a 1 (2b) Definition of subtraction Division is defined as multiplying by the reciprocal . The reciprocal, or multiplicative inverse , of any 1 nonzero number b is } . b 1 a4b5ap} ,bÞ0 b Definition of division Use properties and definitions of operations Example 3 Show that 9 1 (b 2 9) 5 b. 9 1 (b 2 9) 5 9 1 [b 1 ( 29 )] Definition of subtraction 5 9 1 [( 29 ) 1 b] Commutative property of addition 5 [9 1 (29)] 1 b Associative property of addition 5 0 1b Inverse property of addition 5 b Identity property of addition Checkpoint Use properties and definitions of operations to show that the statement is true. 3. 5(a 4 5) 5 a 1 5(a 4 5) 5 51 a p } 2 5 Homework 1 5 51 } p a2 Commutative prop. of 3 1 5 15 p } 2a Associative prop. of 3 51pa Inverse prop. of 3 5a Identity prop. of 3 5 5 Copyright © Holt McDougal. All rights reserved. Definition of division Lesson 1.1 • Algebra 2 Notetaking Guide 3