6 1. The Complex Numbers

.

.

.

z = x+iy

−z = −x−iy

.

z = x−iy

−z = −x+iy

Figure 1.1.2. Plot of the Complex Numbers z, z, −z, and −z.

(b) zz =

|z|2;

(c) z + w = z + w;

(d) zw = z w;

(e) |zw| = |z||w| and |z| = |z|;

(f) |z| is a non-negative real number and is 0 if and only if z = 0;

(g) | Re(zw)| ≤ |z||w|;

(h) |z + w| ≤ |z| + |w|.

Proof. We will prove (g) and (h). The other parts are elementary computations

or observations and will be left as exercises.

Parts (g) and (h) are the Cauchy-Schwarz inequality and the triangle inequality

for the vector space

R2.

Versions of these inequlities hold in general Euclidian space

Rn.

The proofs we give here are specializations to C of the standard proofs of these

inequalities in

Rn.

To prove (g), we begin with the observation that (a) and (d) imply that

zw + zw = 2 Re(zw).

We then let t be an arbitrary real number and note that, by Parts (c), (d), and (f),

0 ≤ |zt +

w|2

= (zt + w)(zt + w) =

|z|2t2

+ 2 Re(zw)t +

|w|2

for all values of t. This implies that the quadratic polynomial in t given by

|z|2t2

+ 2 Re(zw)t +

|w|2

is never negative and, therefore, has at most one real root. This is only possible if

the expression under the radical in the quadratic formula is negative or zero. Thus,

4(Re(zw))2

−

4|z|2|w|2

≤ 0.

Part (g) follows immediately from this.