Download Apply Properties of Real Numbers

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
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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
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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
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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
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Definition of division
Lesson 1.1 • Algebra 2 Notetaking Guide
3