Download 5-9 Complex Numbers

5-9
Complex Numbers
TEKS FOCUS
VOCABULARY
ĚAbsolute value of a complex number –
TEKS (7)(A) Add, subtract,
and multiply complex
numbers.
The absolute value of a complex number
is its distance from the origin on the
complex number plane.
TEKS (1)(F) Analyze
mathematical relationships
to connect and communicate
mathematical ideas.
ĚComplex conjugates – number pairs of
the form a + bi and a - bi
ĚComplex number – Complex numbers
are the real numbers and imaginary
numbers.
Additional TEKS (1)(D),
(4)(F), (7)(B)
ĚComplex number plane – The complex
number plane is identical to the
coordinate plane, except each ordered
pair (a, b) represents the complex
number a + bi.
ĚImaginary number – any number of
the form a + bi, where a and b are real
numbers and b ≠ 0
ĚImaginary unit – The imaginary unit i
is the complex number whose square
is -1.
ĚPure imaginary number – If a = 0 and
b ≠ 0, the number a + bi is a pure
imaginary number.
ĚAnalyze – closely examine objects, ideas,
or relationships to learn more about
their nature
ESSENTIAL UNDERSTANDING
r The complex numbers are based on a number whose
square is -1.
r Every quadratic equation has complex number solutions
(that sometimes are real numbers).
Key Concept
Example
1-5 = i15
#
#
Lesson 5-9
Complex Numbers
Note
( 1-5)2 = (i15)2 = i 2( 15)2
= -1 5 = -5 (not 5)
#
Complex Numbers
You can write a complex number in
the form a + bi, where a and b are real
numbers.
If b = 0, the number a + bi is a real
number.
If a = 0 and b ≠ 0, the number a + bi is a
pure imaginary number.
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so that when you restrict the operations to the subset of
real numbers, you get the familiar operations on the real
numbers.
Square Root of a Negative Real Number
Algebra
For any positive number a,
1-a = 1-1 a
= 1-1 1a = i1a.
Key Concept
r You can define operations on the set of complex numbers
a
Real
part
bi
Imaginary
part
Complex Numbers (a bi )
Real
Numbers
(a 0i)
Imaginary Numbers
(a bi, b 0)
Pure Imaginary
Numbers
(0 bi, b 0)
Key Concept
Complex Number Plane
In the complex number plane, the point (a, b) represents
the complex number a + bi. To graph a complex number,
locate the real part on the horizontal axis and the imaginary
part on the vertical axis.
imaginary axis
3
real axis
4
2i
2
1i
1i
The absolute value of a complex number is its distance
from the origin in the complex plane.
2i
2i
0 a + bi 0 = 2a2 + b2
3 2i
3 2i 13
Problem 1
P
TEKS Process Standard (1)(E)
Simplifying a Number Using i
Is 1−18 a real
number?
No. There is no real
number that when
multiplied by itself gives
-18. You must use the
imaginary unit i to write
1-18.
How do you write 1−18 by using the imaginary unit i?
H
# 18
= 1-1 # 118
= i # 118
= i # 312
1-18 = 1-1
Multiplication Property of Square Roots
Definition of i
Simplify.
= 3i12
Problem
bl
2
TEKS Process Standard (1)(D)
Graphing in the Complex Number Plane
What are the graph and absolute value of each number?
A −5 + 3i
8i imaginary axis
0 -5 + 3i 0 = 2( -5)2 + 32
6ii
6
= 134
Where is a pure
imaginary number in
the complex plane?
The real part of a pure
imaginary number is 0.
The number must be on
the imaginary axis.
B 6i
0 6i 0 = 0 0 + 6i 0
= 202 + 62
= 136
=6
6 units up
4i
5
5 3i
5 units left,
3 units up
2i
66 4
2
2
real axis
4
2i
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Problem 3
Adding and Subtracting Complex Numbers
How is adding
complex numbers
similar to
adding algebraic
expressions?
Adding the real parts
and imaginary parts
separately is like adding
like terms.
What is each sum or difference?
W
A (4 − 3i) + (−4 + 3i)
4 + ( -4) + ( -3i) + 3i
Use the commutative and associative properties.
0+0=0
4 - 3i and - 4 + 3i are additive inverses.
B (5 − 3i) − (−2 + 4i)
5 - 3i + 2 - 4i
To subtract, add the opposite.
5 + 2 - 3i - 4i
Use the commutative and associative properties.
7 - 7i
Simplify.
Problem
bl
4
Multiplying Complex Numbers
What is each product?
A (3i)(−5 + 2i)
How do you multiply
two binomials?
Multiply each term
of one binomial by
each term of the other
binomial.
-15i + 6i 2
Distributive Property
-15i + 6( -1)
Substitute - 1 for i 2.
-6 - 15i
Simplify.
B (4 + 3i)(−1 − 2i)
-4 - 8i - 3i -
C (−6 + i)(−6 − i)
36 + 6i - 6i - i 2
6i 2
-4 - 8i - 3i - 6( -1)
2 - 11i
Substitute
1 for i 2.
36 + 6i - 6i - ( -1)
37
Problem
bl
5
Dividing Complex Numbers
What is each quotient?
W
What is the goal?
Write the quotient in the
form a + bi.
A
9 + 12i
3i
9 + 12i
3i
# -- 3i3i
- 27i - 36i 2
- 9i 2
- 27i - 36( - 1)
- 9( - 1)
36 - 27i
9
4 - 3i
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Lesson 5-9
Complex Numbers
Multiply numerator
and denominator
by the complex
conjugate of the
denominator.
Substitute
1 for i 2.
2 + 3i
B 1 − 4i
2 + 3i
1 - 4i
# 11 ++ 4i4i
2 + 8i + 3i + 12i 2
1 + 4i - 4i - 16i 2
2 + 8i + 3i + 12( - 1)
1 + 4i - 4i - 16( - 1)
- 10 + 11i
17
10
11
- 17 + 17i
Problem 6
TEKS Process Standard (1)(F)
Factoring Using Complex Conjugates
Is the expression
factorable using real
numbers?
No. Look for factors using
complex numbers.
What is the factored form of 2x2 + 32?
W
2x2 + 32
2(x2 + 16)
Factor out the GCF.
2(x + 4i )(x - 4i )
Use a2 + b2 = (a + bi )(a - bi ) to factor (x 2 + 16).
Check
2(x2 + 4xi - 4xi - 16i 2)
2(x2
- 16( -1))
Multiply the binomials.
i2 = -1
2(x2 + 16)
Simplify within the binomial.
2x2 + 32
Multiply.
Problem
bl
7
Finding Imaginary Solutions
What are the solutions of 2x2 − 3x + 5 = 0?
Use the Quadratic
Formula with
a = 2, b = -3, and
c = 5.
Simplify.
x=
− b t 2b2 − 4ac
2a
=
− ( − 3) t 2( − 3)2 − 4(2)(5)
2(2)
=
3 t 29 − 40
4
=
3 t 2 − 31
4
=
231
3
t
i
4
4
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HO
ME
RK
O
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PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Simplify each number by using the imaginary number i.
1. 1-4
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go to PearsonTEXAS.com.
2. 1-7
3. 1-15
4. 1-50
Plot each complex number and find its absolute value.
5. 2i
6. 5 + 12i
7. 2 - 2i
8. 3 - 6i
Simplify each expression.
9. (2 + 4i) + (4 - i)
10. ( -3 - 5i) + (4 - 2i)
11. (7 + 9i) + ( -5i)
12. (12 + 5i) - (2 - i)
13. ( -6 - 7i) - (1 + 3i)
14. (8 + i)(2 + 7i)
15. ( -6 - 5i)(1 + 3i)
16.
( -6i)2
17. (9 + 4i)2
Write each quotient as a complex number.
3 - 2i
5i
i+2
21. i - 2
18.
- 2i
4 - 3i
19. 1 + i
20. - 1 - 4i
22. 2 -4 3i
23.
3 + 2i
(1 + i)2
Find the factored forms of each expression. Check your answer.
24. x2 + 25
25. x2 + 1
26. 3s2 + 75
1
27. x2 + 4
28. 4b2 + 1
29. -9x2 - 100
Find all solutions to each quadratic equation.
30. x2 + 2x + 3 = 0
31. -3x2 + x - 3 = 0
32. 2x2 - 4x + 7 = 0
33. x2 - 2x + 2 = 0
34. x 2 + 5 = 4x
35. 2x(x - 3) = -5
36. a. Name the complex number represented by each point
on the graph at the right.
imaginary axis
F
4i
b. Find the additive inverse of each number.
2i
c. Find the complex conjugate of each number.
d. Find the absolute value of each number.
37. Connect Mathematical Ideas (1)(F) In the complex
number plane, what geometric figure describes the
complex numbers with absolute value 10?
A
2i
E
38. Solve (x + 3i)(x - 3i) = 34.
Simplify each expression.
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Lesson 5-9
39. (8i)(4i)( -9i)
40. (2 + 1-1) + ( -3 + 1-16)
41. (8 - 1-1) - ( -3 + 1-16)
42. 2i(5 - 3i)
43. -5(1 + 2i) + 3i(3 - 4i)
44. (3 + 1-4)(4 + 1-1)
Complex Numbers
3
4i
B
D
C
3
real
axis
45. Analyze Mathematical Relationships (1)(F) In the equation x2 - 6x + c = 0, find
values of c that will give:
a. two real solutions
b. two imaginary solutions
c. one real solution
46. A student wrote the numbers 1, 5, 1 + 3i, and 4 + 3i to represent the vertices
of a quadrilateral in the complex number plane. What type of quadrilateral has
these vertices?
The multiplicative inverse of a complex number z is 1z where z ≠ 0. Find the
multiplicative inverse, or reciprocal, of each complex number. Then use complex
conjugates to simplify. Check each answer by multiplying it by the original number.
47. 2 + 5i
48. 8 - 12i
49. a + bi
Find the sum and product of the solutions of each equation.
50. x2 - 2x + 3 = 0
51. 5x2 + 2x + 1 = 0
52. -2x2 + 3x - 3 = 0
b
For ax2 + bx + c = 0, the sum of the solutions is − a and the product of the
c
solutions is a. Find a quadratic equation for each pair of solutions. Assume a = 1.
53. -6i and 6i
54. 2 + 5i and 2 - 5i
55. 4 - 3i and 4 + 3i
Two complex numbers a + bi and c + di are equal when a = c and b = d.
Solve each equation for x and y.
56. 2x + 3yi = -14 + 9i
57. 3x + 19i = 16 - 8yi
58. -14 - 3i = 2x + yi
59. Show that the product of any complex number a + bi and its complex conjugate
is a real number.
60. For what real values of x and y is (x + yi)2 an imaginary number?
61. Explain Mathematical Ideas (1)(G) True or false: The conjugate of the additive
inverse of a complex number is equal to the additive inverse of the conjugate of
that complex number. Explain your answer.
TEXAS Test Practice
T
62. How can you rewrite the expression (8 - 5i)2 in the form a + bi?
A. 39 + 80i
B. 39 - 80i
C. 69 + 80i
D. 69 - 80i
63. How many solutions does the quadratic equation 4x2 - 12x + 9 = 0 have?
F. two real solutions
H. two imaginary solutions
G. one real solution
J. one imaginary solution
64. What are the solutions of 3x2 - 2x - 4 = 0?
A.
1 { 113
3
B.
1 { i 111
3
C.
- 1 { 113
3
D.
- 1 { i 111
3
65. Using factoring, what are all four solutions to x4 - 16 = 0? Show your work.
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