Complex Numbers 21
can be written as lz 212 (zi/z 2 ) 2:: 0, or simply as zif z 2 2:: 0. Hence, under
the assumption z 2 -:/= 0, the equality in (1.6-5) holds iff the ratio zif z 2 is
real and nonnegative.
We observe, in passing, that property 4 is just a special case of
property 7. Replacing z 2 by -z 2 in (1.6-5), we also obtain
To prove property 8, consider the identitiesz1 = (z1 - z2) + z2,
and apply property 7 to obtainlz1I = l(z1 - z2) + z2I:::; lz1 -z2I + lz2I
andHence
and
lz2l - lz1I:::; lz2 -z1I = lz1 - z2IThe last two inequalities show that
Replacing z2 by -z2, we also have
(1.6-6)(1.6-7)(1.6-8)Property 9 is a consequence of property 6, and 10 follows from (1.6-4)
for n > 0. The case n = 0 is trivial, and if n = -m (m > 0), we have
An alternate proof of (1.6-5) can be obtained by using property 9
and (1.6-3) as follows:lz1I + lz2I
lz1 + z2I
Re ( z1 ) + Re (-z2 ) = Re ( z1 + z2 ) = 1
- Z1 + Z2 Z1 + Z2 Z1 + Z2
=1
assuming z1 + z2 -:/= 0. The case z1 + z2 = 0 is trivial.