10.7 Complex Numbers
Objective 1
Simplify numbers of the form where b > 0.
,b
Slide 10.7- 2
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 10.7- 3
Simplify numbers of the form where b > 0.
,b
For any positive real number b, .b i b
b
Slide 10.7- 4
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
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 10.7- 5
CLASSROOM EXAMPLE 1 Simplifying Square Roots of Negative Numbers
Solution:
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 10.7- 6
CLASSROOM EXAMPLE 2 Multiplying Square Roots of Negative Numbers
Solution:
16 25 16 25i i 4 5i i
220i
20 1
20
Divide.
805
805
ii
805
16
4
4010 40
10i
4010
i
4i
2i
Slide 10.7- 7
CLASSROOM EXAMPLE 3 Dividing Square Roots of Negative Numbers
Solution:
Objective 2
Recognize subsets of the complex numbers.
Slide 10.7- 8
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 10.7- 9
Recognize subsets of the complex numbers.
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 10.7- 10
Recognize subsets of the complex numbers.
The relationships among the various sets of numbers.
Slide 10.7- 11
Recognize subsets of the complex numbers.
Objective 3
Add and subtract complex numbers.
Slide 10.7- 12
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 10.7- 13
CLASSROOM EXAMPLE 4 Adding Complex Numbers
Solution:
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 10.7- 14
CLASSROOM EXAMPLE 5 Subtracting Complex Numbers
Solution:
( 10 6 ) ( 10 10 )i i [ 10 ( 10)] (6 10)i
0 4i 4i
Objective 4
Multiply complex numbers.
Slide 10.7- 15
Multiply.
6 (4 3 )i i 6 (4) 6 (3 )i i i 224 18i i
24 18( 1)i
18 24i
Slide 10.7- 16
CLASSROOM EXAMPLE 6 Multiplying Complex Numbers
Solution:
(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 10.7- 17
CLASSROOM EXAMPLE 6 Multiplying Complex Numbers (cont’d)
Multiply.
Solution:
(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 10.7- 18
CLASSROOM EXAMPLE 6 Multiplying Complex Numbers (cont’d)
Multiply.
Solution:
The product of a complex number and its conjugate is always a real number.
(a + bi)(a – bi) = a2 – b2( –1) = a2 + b2
Slide 10.7- 19
Multiply complex numbers.
Objective 5
Divide complex numbers.
Slide 10.7- 20
Find the quotient.
233
ii
(23 )(3 )(3 )(3 )
i ii i
2
69 23 3 13 1i i
70 2010
i
10(7 2 )10
i 7 2i
Slide 10.7- 21
CLASSROOM EXAMPLE 7 Dividing Complex Numbers
Solution:
5 ii (5 )( )
( )i iii
2
2
5i ii
5 ( 1)( 1)i
5 11i
1 5i Slide 10.7- 22
CLASSROOM EXAMPLE 7 Dividing Complex Numbers (cont’d)
Find the quotient.
Solution:
Objective 6
Find powers of i.
Slide 10.7- 23
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 10.7- 24
Find powers of i.
Find each power of i.
28i 74i 7 11
19i 16 3i i 44 3i i 41 ( ) ii
9i 9
1i
8
1i i
24
1
i i
2
11 i
1i
1( )( )i
i i
2
ii
( 1)
i
1ii
Slide 10.7- 25
CLASSROOM EXAMPLE 8 Simplifying Powers of i
Solution:
22i 22
1i
20 2
1i i
54
1
( 1)i
5
11 ( 1)
11
1
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