Graphics Reference
In-Depth Information
z
1
=
+
3
4
i
z
2
=
5
−
2
i
z
1
z
2
=
(
3
+
4
i)(
5
−
2
i)
8
i
2
=
15
−
6
i
+
20
i
−
=
15
+
14
i
+
8
=
23
+
14
i.
Remember that the addition, subtraction and multiplication of complex numbers
obey the normal axioms of algebra. Also, the multiplication of two complex num-
bers, and their addition always results in a complex number, that is, the two opera-
tions are closed.
2.7 The Complex Conjugate
A special case exists when we multiply two complex numbers together where the
only difference between them is the sign of the imaginary part:
a
2
b
2
i
2
+
−
=
−
+
−
(a
bi)(a
bi)
abi
abi
=
a
2
+
b
2
.
As this real value is such an interesting result,
a
−
bi
is called the
complex conjugate
of
a
+
bi
. In general, the complex conjugate of
z
=
a
+
bi
z
symbol or an asterisk
z
∗
¯
is written either with a bar
as
z
∗
=
a
−
bi
and implies that
zz
∗
=
a
2
b
2
2
.
+
=|
z
|
2.8 Division of Two Complex Numbers
The complex conjugate provides us with a mechanism to divide one complex num-
ber by another. For instance, consider the quotient
a
+
bi
di
.
This can be resolved by multiplying the numerator and denominator by the complex
conjugate
c
c
+
−
di
to create a real denominator:
