1.1. Definition and Simple Properties 7
Part (h) follows directly from Part (g) and the other parts of the theorem. In
fact,
|z +
w|2
= (z + w)(z + w)
=
|z|2
+ 2 Re(zw) +
|w|2

|z|2
+ 2|z||w| +
|w|2
= (|z| +
|w|)2.
On taking square roots, we conclude |z + w| |z| + |w|, which is Part (h).
The inequalities in the following theorem are used extensively particularly in
the next section. The proofs are very simple and are left as an exercise (Exercise
1.1.12).
Theorem 1.1.8. If z = x + iy, then max{|x|, |y|} |z| |x| + |y|.
Inversion and Division. Recall that the inverse of a non-zero complex number
z = x + iy is
z−1
=
x iy
x2 + y2
=
z
|z|2
=
z
zz
.
Stating this in the last form makes the identity zz−1 = 1 obvious.
This also suggests the right way to do complex division problems in general: to
express w/z as a complex number in standard form (as a real number plus i times
a real number), simply multiply both numerator and denominator by z. That is,
w
z
=
wz
zz
=
wz
|z|2
.
The number wz is then easily put in standard form and the problem is finished by
dividing by the real number
|z|2.
Example 1.1.9. Express
1
2 + 3i
in the standard form x + yi.
Solution:
1
2 + 3i
=
2 3i
(2 + 3i)(2 3i)
=
2 3i
|2 + 3i|2
=
2
13

3
13
i.
Example 1.1.10. Express
3 + 4i
3 4i
and
3 4i
3 + 4i
in standard form.
Solution:
3 + 4i
3 4i
=
(3 + 4i)2
|3 + 4i|2
=
7
25
+
24
25
i,
3 4i
3 + 4i
=
(3 4i)2
|3 4i|2
=
7
25

24
25
i.
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