Instructor: Ms. Hoa Nguyen
([email protected])
1 Reminder of the Complex Numbers
•
Conjugate: If
z
=
a
+
bi
, then the conjugate of
z
is ¯
z
=
a

bi
.
•
i
2
=

1 because
√

1 =
i
.
•
z
¯
z
=
a
2
+
b
2
=

z

2
. Why? Because
z
¯
z
= (
a
+
bi
)(
a

bi
) =
a
2

abi
+
abi

b
2
i
2
=
a
2
+
b
2
.
So the multiplication of a complex number
z
with its conjugate ¯
z
is equal to the sum
of the real part
a
squared and the imaginary part
b
squared.
•
Division: If
z
=
a
+
bi
and
w
=
c
+
di
, then
z
w
=
z
·
¯
w
w
¯
w
=
z
·
¯
w
c
2
+
d
2
because
w
¯
w
=
c
2
+
d
2
.
Example 1
This is the division of complex numbers
z
w
where
z
= 2 +
i
and
w
= 1

3
i
. As suggested in
the reminder, multiply the conjugate ¯
w
= 1 + 3
i
to the numerator and denominator of the
fraction, i.e.,
z
w
=
z
·
¯
w
w
¯
w
=
(2+
i
)(1+3
i
)
1
2
+(

3)
2
=

1+7
i
10
because
z
·
¯
w
= (2+
i
)(1+3
i
) = 2+6
i
+
i
+3
i
2
=

1 + 7
i
and
w
¯
w
= (1

3
i
)(1 + 3
i
) = 1 + 3
i

3
i
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 Spring '11
 Nuegyen
 Complex Numbers, Quadratic equation, abi − b2

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