Download 10.7 Complex Numbers

10.7 Complex
Numbers
Objective 1
Simplify numbers of the form
where b > 0.
b ,
Slide 10.7- 2
Simplify numbers of the form b , where b > 0.
Imaginary Unit i
The imaginary unit i is defined as
i  1,
where
i 2  1.
That is, i is the principal square root of –1.
Slide 10.7- 3
Simplify numbers of the form b , where b > 0.
b
For any positive real number b,
b  i b.
It is easy to mistake 2i for 2i with the i under the radical. For this reason,
we usually write 2i as i 2, as in the definition of
b .
Slide 10.7- 4
CLASSROOM
EXAMPLE 1
Simplifying Square Roots of Negative Numbers
Write each number as a product of a real number and i.
Solution:
25
 i 25
 5i
 81
 i 81
 9i
7
i 7
44
 i 44
 i 4 11
 2i 11
Slide 10.7- 5
CLASSROOM
EXAMPLE 2
Multiply.
Multiplying Square Roots of Negative Numbers
Solution:
16  25  i 16  i 25
 i  4i 5
8  6
 i2 8  6
 20i 2
 i 2 48
 20  1
 i 2 16  3
 20
6  5
 i 6 i 5
 i2 6  5
 i 8 i 6
 4 3
5  7
i 5 7
 i 35
 (1) 30
  30
Slide 10.7- 6
CLASSROOM
EXAMPLE 3
Dividing Square Roots of Negative Numbers
Divide.
Solution:
80
5
i 80

i 5
40
10
i 40

10
80

5
40
i
10
 16
i 4
4
 2i
Slide 10.7- 7
Objective 2
Recognize subsets of the complex
numbers.
Slide 10.7- 8
Recognize subsets of the complex numbers.
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
Objective 3
Add and subtract complex numbers.
Slide 10.7- 12
CLASSROOM
EXAMPLE 4
Add
.
Adding Complex Numbers
Solution:
(1  8i)  (9  3i)  (1  9)  (8  3)i
 8 11i
(3  2i)  (1  3i)  (7  5i)
 [3  1  (7)]  [2  (3)  (5)]i
 9  6i
Slide 10.7- 13
CLASSROOM
EXAMPLE 5
Subtracting Complex Numbers
Subtract.
Solution:
(1  2i )  (4  i)
 (1  4)  (2  1)i
 5  i
(8  5i)  (12  3i)  (8  12)  [5  (3)]i
 (8  12)  (5  3)i
 4  2i
(10  6i)  (10  10i)  [10  (10)]  (6  10)i
 0  4i
 4i
Slide 10.7- 14
Objective 4
Multiply complex numbers.
Slide 10.7- 15
CLASSROOM
EXAMPLE 6
Multiplying Complex Numbers
Multiply.
Solution:
6i (4  3i )
 6i(4)  6i(3i)
 24i  18i 2
 24i  18(1)
 18  24i
Slide 10.7- 16
CLASSROOM
EXAMPLE 6
Multiplying Complex Numbers (cont’d)
Multiply.
Solution:
(6  4i )(2  4i)  6(2)  6(4i)  (4i)(2)  (4i)(4i)
First
Outer
Inner
 12  24i  8i  16i
Last
2
 12  16i  16(1)
 12 16i 16
 28  16i
Slide 10.7- 17
CLASSROOM
EXAMPLE 6
Multiplying Complex Numbers (cont’d)
Multiply.
Solution:
(3  2i)(3  4i)
 3(3)  3(4i)  (2i)(3)  (2i)(4i)
First
Outer
Inner
Last
 9  12i  6i  8i 2
 9  18i  8(1)
 9  18i  8
 1  18i
Slide 10.7- 18
Multiply complex numbers.
The product of a complex number and its conjugate is always a real number.
(a + bi)(a – bi) = a2 – b2( –1)
= a 2 + b2
Slide 10.7- 19
Objective 5
Divide complex numbers.
Slide 10.7- 20
CLASSROOM
EXAMPLE 7
Dividing Complex Numbers
Find the quotient.
Solution:
23  i
3i
(23  i)(3  i)

(3  i)(3  i)
69  23i  3i  1

2
3 1
70  20i

10
10(7  2i )

 7  2i
10
Slide 10.7- 21
CLASSROOM
EXAMPLE 7
Dividing Complex Numbers (cont’d)
Find the quotient.
Solution:
5i
i
(5  i)(i)

i(i)
5i  i 2

2
i
5i  (1)

(1)
5i  1

1
 1  5i
Slide 10.7- 22
Objective 6
Find powers of i.
Slide 10.7- 23
Find powers of i.
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
CLASSROOM
EXAMPLE 8
Simplifying Powers of i
Find each power of i.
Solution:
i
28
 i
i
19
 i i
i
i
9
22

4 7
16
3
1 1
7
 i

4 4
1
1
 9
 8
i
i i
i
1(i )
 2

i  (i ) i
1
 22
i
i
3
 14  (i)  i

1
i 
4 2
i
i

(1)
1
 2
1 i
1

i
i
  i
1
1
1
1
1
 20 2 
 5



1
5
4
i i
i
   (1) 1  (1) 1
Slide 10.7- 25