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 =
(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
Versions of these inequlities hold in general Euclidian space
The proofs we give here are specializations to C of the standard proofs of these
inequalities in
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 +
= (zt + w)(zt + w) =
+ 2 Re(zw)t +
for all values of t. This implies that the quadratic polynomial in t given by
+ 2 Re(zw)t +
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,

Part (g) follows immediately from this.
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