Section 3.5 Complex Numbers
The complex number
i
is the solution of the equation
x
2
=

1, i.e.
i
2
=

1. We write
i
=
√

1.
The complex number system consists of all numbers of the form
a
+
bi
where
a
,
b
are real numbers.
The usual realnumber algebraic operations (addition, subtraction, multiplication, division) also hold
for complex numbers. For example,
(3 + 2
i
)+(2

i
)=5+
i
and
(3 + 2
i
)(2

i
)=6

3
i
+4
i

2
i
2
=8+
i
If
z
=
a
+
bi
is a complex number, then
a
is called the
real part
of
z
and
b
is called the
imaginary
part
of
z
.
The number ¯
z
=
a

bi
is called the
complex conjugate
of
z
=
a
+
bi
.
Example.
3+2
i
=3

2
i
Example.
2

3
i
=2+3
i
If
z
=
a
+
bi
,then
•
z
+¯
z
=2
a
•
z

¯
z
bi
•
z
¯
z
=
a
2
+
b
2
=

z

2
Example.
Simplify
2

i
3+
i
.
Solution.
2

i
3+
i
=
2

i
3+
i
3

i
3

i
=
6

2
i

3
i
+
i
2
9+1
=
5

5
i
10
=
1

i
2
=
1
2

1
2
i
Some basics of complex conjugates:
•
¯
¯
z
=
z
•
z
+
w
=¯
z
w
•
z
·
w
z
·
¯
w
Powers of
i
•
i
2
=

1
•
i
3
=
i
2
i
=

i
•
i
4
=(
i
2
)
2
=1
•
i
5
=
i
4
i
=
i
Solving Quadratic Equations Involving Complex Numbers:
Example.
Solve for
x
:
x
2
+2
x
+4=0.
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 Fall '11
 Kutter
 Algebra, Real Numbers, Complex Numbers, Quadratic equation, Complex number

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