section 7chapter 8. 1 copyright © 2012, 2008, 2004 pearson education, inc. objectives 2 6 5 3 4...
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Section 7Chapter 8
1
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objectives
2
6
5
3
4
Complex Numbers
Simplify numbers of the form where b > 0.
Recognize subsets of the complex numbers.
Add and subtract complex numbers.
Multiply complex numbers.
Divide complex numbers.
Find powers of i.
,b
8.7
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Simplify numbers of the form where b > 0.
,b
Objective 1
Slide 8.7- 3
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Imaginary Unit i
The imaginary unit i is defined as
That is, i is the principal square root of –1.
21, where 1.i i
Slide 8.7- 4
Simplify numbers of the form where b > 0.
,b
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
For any positive real number b, .b i b
b
Slide 8.7- 5
Simplify numbers of the form where b > 0.
,b
It is easy to mistake for with the i under the radical. For this reason,
we usually write as as in the definition of
2i 2i
2i 2,i .b
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Write each number as a product of a real number and i.
25 25i 5i
81 81i 9i
7
44 44i 4 11i 2 11i
7i
Slide 8.7- 6
CLASSROOM EXAMPLE 1
Simplifying Square Roots of Negative Numbers
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Multiply.
6 5 6 5i i
2 6 5i
( 1) 30
8 6
30
8 6i i 2 8 6i 2 48i2 16 3i
4 3
5 7 5 7i
35i
Slide 8.7- 7
CLASSROOM EXAMPLE 2
Multiplying Square Roots of Negative Numbers
Solution:
16 25 16 25i i 4 5i i
220i
20 1
20
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Divide.
80
5
80
5
i
i
80
5
16
4
40
10
40
10
i
40
10i
4i
2i
Slide 8.7- 8
CLASSROOM EXAMPLE 3
Dividing Square Roots of Negative Numbers
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Recognize subsets of the complex numbers.
Objective 2
Slide 8.7- 9
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Complex Number
If a and b are real numbers, then any number of the form a + bi is called a complex number. In the complex number a + bi, the number a is called the real part and b is called the imaginary part.
Slide 8.7- 10
Recognize subsets of the complex numbers.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
For a complex number a + bi, if b = 0, then a + bi = a, which is a real number.
Thus, the set of real numbers is a subset of the set of complex numbers.
If a = 0 and b ≠ 0, the complex number is said to be a pure imaginary number.
For example, 3i is a pure imaginary number. A number such as 7 + 2i is a nonreal complex number.
A complex number written in the form a + bi is in standard form.
Slide 8.7- 11
Recognize subsets of the complex numbers.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
The relationships among the various sets of numbers.
Slide 8.7- 12
Recognize subsets of the complex numbers.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Add and subtract complex numbers.
Objective 3
Slide 8.7- 13
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Add.
( 1 8 ) (9 3 )i i ( 1 9) ( 8 3)i
8 11i
( 3 2 ) (1 3 ) ( 7 5 )i i i
[ 3 1 ( 7)] [2 ( 3) ( 5)]i
9 6i
Slide 8.7- 14
CLASSROOM EXAMPLE 4
Adding Complex Numbers
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Subtract.
( 1 2 ) (4 )i i ( 1 4) (2 1)i 5 i
(8 5 ) (12 3 )i i (8 12) [ 5 ( 3)]i
4 2i (8 12) ( 5 3)i
Slide 8.7- 15
CLASSROOM EXAMPLE 5
Subtracting Complex Numbers
Solution:
( 10 6 ) ( 10 10 )i i [ 10 ( 10)] (6 10)i
0 4i 4i
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Multiply complex numbers.
Objective 4
Slide 8.7- 16
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Multiply.
6 (4 3 )i i 6 (4) 6 (3 )i i i 224 18i i
24 18( 1)i
18 24i
Slide 8.7- 17
CLASSROOM EXAMPLE 6
Multiplying Complex Numbers
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
(6 4 )(2 4 )i i 6(2) 6(4 ) ( 4 )(2) ( 4 )(4 )First Outer Inner Last
i i i i
212 24 8 16i i i
12 16 6 )11 (i
12 16 16i
28 16i
Slide 8.7- 18
CLASSROOM EXAMPLE 6
Multiplying Complex Numbers (cont’d)
Multiply.
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
(3 2 )(3 4 )i i 3(3) 3(4 ) (2 )(3) (2 )(4 )First Outer Inner Last
i i i i
29 12 6 8i i i
9 18 8 )1(i
9 18 8i
1 18i
Slide 8.7- 19
CLASSROOM EXAMPLE 6
Multiplying Complex Numbers (cont’d)
Multiply.
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
The product of a complex number and its conjugate is always a real number.
(a + bi)(a – bi) = a2 – b2( –1)
= a2 + b2
Slide 8.7- 20
Multiply complex numbers.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Divide complex numbers.
Objective 5
Slide 8.7- 21
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Find the quotient.
23
3
i
i
(23 )(3 )
(3 )(3 )
i i
i i
2
69 23 3 1
3 1
i i
70 20
10
i
10(7 2 )
10
i 7 2i
Slide 8.7- 22
CLASSROOM EXAMPLE 7
Dividing Complex Numbers
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
5 i
i
(5 )( )
( )
i i
ii
2
2
5i i
i
5 ( 1)
( 1)
i
5 1
1
i 1 5i
Slide 8.7- 23
CLASSROOM EXAMPLE 7
Dividing Complex Numbers (cont’d)
Find the quotient.
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Find powers of i.
Objective 6
Slide 8.7- 24
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Because i2 = –1, we can find greater powers of i, as shown below.
i3 = i · i2 = i · ( –1) = –i
i4 = i2 · i2 = ( –1) · ( –1) = 1
i5 = i · i4 = i · 1 = i
i6 = i2 · i4 = ( –1) · (1) = –1
i7 = i3 · i4 = (i) · (1) = –I
i8 = i4 · i4 = 1 · 1 = 1
Slide 8.7- 25
Find powers of i.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Find each power of i.
28i 74i 7 11
19i 16 3i i 44 3i i 41 ( ) ii
9i 9
1
i
8
1
i i
24
1
i i
2
1
1 i
1
i
1( )
( )
i
i i
2
i
i
( 1)
i
1i
i
Slide 8.7- 26
CLASSROOM EXAMPLE 8
Simplifying Powers of i
Solution:
22i 22
1
i
20 2
1
i i
54
1
( 1)i
5
1
1 ( 1)
1
11