Graphics Reference
In-Depth Information
a
+
bi
c
+
di
=
(a
+
bi)(c
−
di)
(c
+
di)(c
−
di)
bdi
2
ac
−
adi
+
bci
−
=
c
2
d
2
+
ac
+
bd
c
2
bc
−
ad
c
2
i.
=
+
+
d
2
+
d
2
Another special case is when
a
=
1 and
b
=
0:
i
1
c
d
di)
−
1
di
=
(c
+
=
−
+
c
2
d
2
c
2
d
2
c
+
+
which is the
inverse
of a complex number.
Let's evaluate the quotient:
4
+
3
i
4
i
.
Multiplying top and bottom by the complex conjugate 3
3
+
−
4
i
we have
4
+
3
i
(
4
+
3
i)(
3
−
4
i)
4
i
=
3
+
(
3
+
4
i)(
3
−
4
i)
12
i
2
12
−
16
i
+
9
i
−
24
25
−
7
25
i.
=
=
25
2.9 The Inverse
Although we have already discovered the inverse of a complex number, let's employ
another strategy by declaring
1
z
z
1
=
where
z
is a complex number.
Next, we divide both sides by the complex conjugate of
z
to create
z
1
z
∗
=
1
zz
∗
.
But we have previously shown that
zz
∗
=|
2
, therefore,
z
|
z
1
z
∗
=
1
|
z
|
2
and rearranging, we have
z
∗
z
1
=
2
.
|
z
|
In general
z
∗
1
z
=
z
−
1
=
2
.
|
z
|