1.3 the complex number system
DESCRIPTION
MATH 17 - COLLEGE ALGEBRA AND TRIGONOMETRYTRANSCRIPT
CHAPTER 1.3
THE COMPLEX
NUMBER SYSTEM
1
Solvable Equations in R
0 is always solvable in .
ax b R
bx
a
2
Solvable Equations in R
2
2
9 is solvable in .
The solutions are 3 and 3
3 is solvable in .
The solutions are 3 and 3.
x R
x R
3
Solvable Equations in R
2
If 0, the solutions to the
equation are
and .
x p
p
p
p
4
Solvable Equations in R
2Is 1 solvable in ?x R
5
i
2
Assume that
1i
6
Complex Numbers
2
A complex number is of the form
, and are real numbers
and .
: is the real part.
is the imag
1
inary part.
x yi
i
x yi x
y
x y
7
Example 1.3.1
Determine the real part and imaginary
parts of the following numbers.
1. 2 3
real part: 2
imaginary part: 3
2. 2 0 2
real part: 0
imaginary part: 2
i
i i
8
Example 1.3.1
3. 1 1 0
real part: 1
imaginary part: 0
4. 0 0 0
real part: 0
imaginary part: 0
i
i
9
Example 1.3.1
5. 1
real part:
imaginary part:
i i
i
10
Complex Numbers
Real Numbers: 0
Pure Imaginary: 0 , 0
x yi
x i
yi y
11
Conjugate of a Number
The conjugate of a complex number
is the complex number .x yi x yi
12
Example 1.3.2
Give the conjugate of each of
the following complex numbers.
1. 2 3
Conjugate: 2 3
2. 2 0 2
Conjugate: 0 2 2
i
i
i i
i i
13
Example 1.3.2
3. 1 1 0
Conjugate: 1 0 1
4. 0
Conjugate: 0
5. 1
Conjugate:
i
i
i i
i
14
Complex Numbers
2, and 1C x yi x y R i
C
R
15
Complex Numbers
if and only if
and
a bi c di
a c b d
16
Solvable Equations in C
2
2
2
2 2
Is 1 solvable in ? YES
The solutions for 1 are
: 1 and
: 1
x C
x
i i
i i i i i
17
Solvable Equations in C
2
2
2
Is 4 solvable in ? NO
Is it solvable in ? YES
The solutions are 2 and 2 .
2 4 1 4
2 4 1 4
x R
C
i i
i
i
18
Example 1.3.3
2
2
2
2 2
2 2
Give the complex solution of the
following equations.
1. 9 3 , 3
2. 5 0
5 5 , 5
3. 0
,
x i i
x
x i i
x
x i i
19
Example 1.3.3
2 3
33 6 3
33 6 3
4. 4
4 4 4 2
4 4 4 2
x
i i i i
i i i i
20
Addition and Multiplication
a bi c di a c b d i
a bi c di ac bd ad bc i
21
Example 1.3.4
Perform the indicated operations.
1. 2 3 4 5 6 2
2. 2 3 4 5 8 15 10 12
23 2
i i i
i i i
i
22
Example 1.3.4
3 2
4 2 2
3. 1
4. 1 1 1
i i i i i
i i i
23
Properties of [C,+,∙]
1. is closed under + and .
2. and are associative in .
3. and are commutative in .
4. is distrubutive over + in .
5. 0 is the additive identity in .
6. 1 is the multiplicative identity in .
C
C
C
C
C
C
24
Additive Inverses
What is the additive inverse of ?
0 0
0 0
0 0
Therefore, is the
additive inverse of .
a bi
a bi x yi i
a x b y i i
a x b y
x a y b
a b i
a bi
25
Additive Inverses
Every complex number has an
additive inverse.
26
Example 1.3.5
Find the additive inverse of the
following.
1. 2 3 2 3
2. 5 2 5 2
3. 1 1
4.
i i
i i
i i
27
Multiplicative Inverses
2 2 2 2
1 a bi
a bi a b a b
28
Multiplicative Inverses
Every nonzero complex number
has a multiplicative inverse.
29
Example 1.3.6
2 2
2 2
Find the multiplicative inverse of the
following.
1. 2 3
2 3 13
2 3
13 13
2. 5 2
5 2 29
5 2
29 29
i
a b a b
i
i
a b a b
i
30
Example 1.3.6
2 2
2 2 2
2
3. 1 1 0
1 0 1
1
4.
0
10
i
a b a b
i
a b a b
i i
31
Subtraction and Division
1
, 0
a bi c di a bi c di
a bia bi c di
c di c di
32
Example 1.3.7
Perform the indicated operation.
1. 2 3 5 2 3 5
2 3 12. 2 3
5 2 5 2
5 22 3
29 29
i i i
ii
i i
i i
33
Example 1.3.7
2 3 5 2
2. 2 35 2 29 29
10 6 4 15
29 29 29 29
4 19
29 29
ii i
i
i
i
34