Download Chapter 1 Section 3 - Canton Local Schools

Chapter 1
Equations and
Inequalities
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SECTION 1.3
Complex Numbers
OBJECTIVES
1
2
3
4
Define complex numbers.
Add and subtract complex numbers.
Multiply complex numbers.
Divide complex numbers.
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Definition of i
The square root of −1 is called i.
i  1 so that i  1.
2
The number i is called the imaginary unit.
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Complex Numbers
A complex number is a number of the form
z  a  bi,
where a and b are real numbers and i2 = –1.
The number a is called the real part of z, and
we write Re(z) = a.
The number b is called the imaginary part of z
and we write Im(z) = b.
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Definitions
A complex number z written in the form
a + bi is said to be in standard form.
A complex number with a = 0 and b ≠ 0,
written as bi, is called a pure imaginary
number.
If b = 0, then the complex number a + bi is a
real number. Real numbers form a subset of
complex numbers (with imaginary part 0).
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Square Root of a Negative Number
For any positive number, b
b 
 b i  i
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b.
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EXAMPLE 1
Identifying the Real and the Imaginary
Parts of a Complex Number
Identify the real and the imaginary parts of each
complex number.
1
c. 3i
b. 7  i
a. 2  5i
2
f. 3  25
e. 0
d.  9
Solution
a. 2  5i
real part 2; imaginary part 5
1
b. 7  i
2
c. 3i
1
real part 7; imaginary part 
2
real part 0; imaginary part 3
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EXAMPLE 1
Identifying the Real and the Imaginary
Parts of a Complex Number
Solution continued
d.  9
real part –9; imaginary part 0
e. 0
real part 0; imaginary part 0
f. 3  25
real part 3; imaginary part 5
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Equality of Complex Numbers
Two complex numbers z = a + bi and w = c + di
are equal if and only if
a = c and b = d
That is, z = w if and only if Re(z) = Re(w) and
Im(z) = Im(w).
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EXAMPLE 2
Equality of Complex Numbers
Find x and y assuming that
(3x – 1) + 5i = 8 + (3 – 2y)i.
Solution
Let z = (3x – 1) + 5i and w = 8 + (3 – 2y)i.
Then Re(z) = Re(w) and Im(z) = Im(w).
So,
3x – 1 = 8 and 5 = 3 – 2y.
3x = 8 + 1
x=3
So, x = 3 and y = –1.
5 – 3 = –2y
–1 = y
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ADDITION AND SUBTRACTION OF
COMPLEX NUMBERS
For all real numbers a, b, c, and d, let
z = a + bi and w = c + di.
z  w   a  bi    c  di    a  c    b  d  i
z  w   a  bi    c  di    a  c   b  d  i .
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EXAMPLE 3
Adding and Subtracting Complex Numbers
Write the sum or difference of two complex
numbers in standard form.
a.  3  7i    2  4i 
c.
2 
b.  5  9i    6  8i 
 
9  2  4

Solution
a.  3  7i    2  4i  
  3  2   7   4   i
 5  3i
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EXAMPLE 3
Adding and Subtracting Complex Numbers
Solution continued
b.  5  9i    6  8i  
  5  6   9   8   i
 1  17i
c.
2 
 

9  2  4   2  3i    2  2i 
 2  3i  2  2i
  2  2  3  2 i
 4i
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MULTIPLYING COMPLEX NUMBERS
For all real numbers a, b, c, and d,
 a  bi  c  di    ac  bd    ad  bc  i .
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EXAMPLE 4
Multiplying Complex Numbers
Write the following products in standard form.
a.  3  5i  2  7i 
b.  2i  5  9i 
Solution
F
O
I
L
a.  3  5i  2  7i   6  21i  10i  35i
2
 6  11i  35
 41  11i
b.  2i  5  9i   10i  18i 2
 10i  18  18  10i
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WARNING
Recall that if a and b are positive real numbers,
a b  ab .
However, this property is not true for nonreal
numbers. For example,
9 9   3i  3i   9i  9  1  9,
2
but
 9  9  
Thus
9 9 
81  9.
 9  9 .
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EXAMPLE 5
Multiplication Involving Roots of Negative
Numbers
Perform the indicated operation and write the
result in standard form.
a.
2 8
c.
 2  3 

3 2  3
b.

2


d. 3  2 1  32

Solution
a.
2 8  i 2  i 8  i
i
2
2
2 8
16   1 4   4
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EXAMPLE 5
Multiplication Involving Roots of Negative
Numbers
Solution continued
b.
c.
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EXAMPLE 5
Multiplication Involving Roots of Negative
Numbers
Solution continued
d.
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CONJUGATE OF A
COMPLEX NUMBER
If z = a + bi, then the conjugate (or complex
conjugate) of z is denoted by z and defined by
z  a  bi  a  bi.
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EXAMPLE 6
Multiplying a Complex Number by Its
Conjugate
Find the product zz for each complex number z .
a. z  2  5i
b. z  1  3i
Solution
a. zz   2  5i  2  5i 
 2   5i   4  25i
2
2
2
 4   25   29
b. zz  1  3i 1  3i 
 1   3i   1  9i 2
2
2
 1   9   10
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COMPLEX CONJUGATE
PRODUCT THEOREM
If z = a + bi, then
zz  a  b .
2
2
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DIVIDING COMPLEX NUMBERS
To write the quotient of two complex numbers
w and z (z ≠ 0), and write
w wz

z zz
and then write the right side in standard form.
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EXAMPLE 7
Dividing Complex Numbers
Write the following quotients in standard form.
4  25
b.
2  9
1
a.
2i
Solution
1 2  i 
2i
a.
 2 2
 2  i  2  i  2  1
2i

5
2 1
  i
5 5
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EXAMPLE 7
Dividing Complex Numbers
Solution continued
4  25 4  5i  4  5i  2  3i 
b.


2  3i  2  3i  2  3i 
2  9
8  12i  10i  15i

22  32
8  15  22i 7  22i


49
13
7 22
  i
13 13
2
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