Download Complex Numbers Simplify numbers of the form where b > 0

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Complex Numbers
Simplify numbers of the form b , where
b > 0.
Recognize complex numbers.
Add and subtract complex numbers.
Multiply complex numbers.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective
1
Simplify numbers of the form b , where
b > 0.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8.7- 2
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.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8.7- 3
b
For any positive real number b, b  i b .
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8.7- 4
EXAMPLE 1
Write each number as a product of a real number and i.
a.
25  i 25  5i
b.  81  i 81  9i
c.
7  i 7
d.
44  i 44  i 4 11  2i 11
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8.7- 5
CAUTION 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 .
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8.7- 6
EXAMPLE 2
Multiply.
a.
b.
6  5  i 6  i 5
c.
5  7
 i2 6  5
i 5 7
 (1) 30   30
 i 35
8  6  i 8  i 6
 i2 8  6
 i 2 48
 i 2 16  3  4 3
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Slide 8.7- 7
Objective
2
Recognize complex numbers.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8.7- 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.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8.7- 9
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.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8.7- 10
Objective
3
Add and subtract complex numbers.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8.7- 11
EXAMPLE 4
Add.
a. (1  8i)  (9  3i)  (1  9)  (8  3)i
 8  11i
b. (3  2i)  (1  3i)  (7  5i)
 [3  1  (7)]  [2  (3)  (5)]i
 9  6i
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Slide 8.7- 12
EXAMPLE 5
Subtract.
a. (1  2i)  (4  i)
 (1  4)  (2  1)i
 5  i
b. (8  5i)  (12  3i)  (8  12)  [5  (3)]i
 (8  12)  (5  3)i
 4  2i
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Slide 8.7- 13
continued
c. (10  6i)  (10  10i)
 [10  (10)]  (6  10)i
 0  4i
 4i
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Slide 8.7- 14
Objective
4
Multiply complex numbers.
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Slide 8.7- 15
EXAMPLE 6
Multiply.
a. 6i(4  3i)  6i(4)  6i(3i)
 24i  18i 2
 24i  18(1)
 18  24i
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Slide 8.7- 16
continued
b. (6  4i)(2  4i)  6(2)  6(4i)  (4i)(2)  (4i)(4i)
First
Outer
Inner
Last
 12  24i  8i  16i 2
 12  16i  16(1)
 12  16i  16
 28  16i
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Slide 8.7- 17
continued
c. (3  2i)(3  4i)  3(3)  3(4i)  (2i)(3)  (2i)(4i)
First
Outer
Inner
 9  12i  6i  8i
Last
2
 9  18i  8(1)
 9  18i  8
 1  18i
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8.7- 18
The product of a complex number and its conjugate is
always a real number.
(a + bi)(a – bi) = a2 – b2( –1)
= a 2 + b2
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8.7- 19
Example
d.
x 40
2
x  4
2
x  4
2
x  2i
e. x  25  0
2
x  25
2
x   25
2
x  5i
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Slide 8.7- 20
Homework
Page 280 20-40 even – skip 34
Simplify.
20.  100
22. (3i )(7i )(2i )
24. i11
26. (10  7i )  (6  9i )
28. (12  5i )  (9  2i )
30. (1  2i )(1  2i )
32. (4  i )(6  6i )
Solve each equation.
2
2
36. 4 x  4  0
38. 2 x  50  0
40. 6 x 2  108  0
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Slide 8.7- 21