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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
|
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