Graphics Reference
In-Depth Information
For example,
12
+
3
i
12
+
3
i
9
+
4
i
108
−
12
27
+
48
96
97
+
75
97
i
4
i
=
4
i
=
16
+
16
i
=
9
−
9
−
4
i
9
+
81
+
81
+
7.3.2 Magnitude of a complex number
If z
=
a
+
b
i
, then its magnitude or absolute value is defined as
√
a
2
z
=
+
b
2
For example,
=
√
144
√
153
12
+
3
i
+
9
=
7.3.3 The complex conjugate
=
+
If z
a
b
i
, then its complex conjugate is defined as
¯
z
=
a
−
b
i
and was used above to simplify the division of two complex numbers.
Just by defining the complex conjugate produces some interesting identities. For example,
a
2
b
2
z
z
¯
=
a
+
b
i
a
−
b
i
=
+
z
+¯
z
=
a
+
b
i
+
a
−
b
i
=
2a
z
1
+
z
2
=
a
1
−
b
1
i
+
a
2
−
b
2
i
=
a
1
+
a
2
−
b
1
+
b
2
i
=
z
1
+
z
2
z
1
z
2
=
a
1
−
b
1
i
a
2
−
b
2
i
=
a
1
a
2
−
b
1
b
2
−
a
1
b
2
+
b
1
a
2
i
=
z
1
z
2
and
2
a
2
b
2
z
¯
z
=
z
=
+
(7.6)
or
√
z
z
=
z
¯
Furthermore, using Eq. (7.6), we find that
2
z
=
z
¯
z
Therefore,
2
=
z
1
z
2
z
1
z
2
z
1
z
2
2
z
1
z
2
=
z
1
z
2
z
1
z
2
2
2
2
=
z
1
z
1
z
2
z
2
=
z
1
z
2
z
1
z
2
or
z
1
z
2
=
z
1
z
2
