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.
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