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.