Complex Numbers 137. z 1 ± Zz = .Z1 ± Zz.
- ZiZz = z1z2.
9. (zi/z2) = zifzz.
10. F(zi, z2, ... , Zn) = F(z1, z2, ... , .Zn), where F(z1, z2, ... , Zn) denotes
a rational algebraic expression in z 1 , z 2 , ••• , Zn·
Proofs The proofs of the properties above follow easily from Definitions 1.2,
except property 10, which is a consequence of properties 7, 8, and 9.
As an illustration we prove property 8. Let z1 = x1 + iy1 and z2 =
Xz + iyz. Sincewe haveBut
z1z2 = (x1 - iy1)(x2 - iy2)
= (x1x2 - Y1Y2) - i(x1y2 + X2Y1)
From (1.3-2) and (1.3-3) it follows that z 1 z 2 = z 1 .Zz.(1.3-2)(1.3-3)Property 2 means that the conjugate of z is z. This is expressed by
saying that complex conjugation is an involutory mapping.
Property 1, together with properties 7 and 8, show that complex con-
jugation is an automorphism of C, i.e., an isomorphism from C onto C.
Clearly, the identical transformation J(z) = z is also an automorphism of
C. It can be shown [25] that they are not the only automorphism of C
and, in fact, that there aresuch automorphisms, where ~o (aleph subzero) denotes the smallest
transfinite number [23].
From properties 4 and 5 it follows that
z+z z-z
x = -2-' y = 2i (1.3-4)
By means of (1.3-4) any equation involving x and y can be expressed in
terms of z and z. For instance, the equation of a straight line in the plane
is given in Cartesian coordinates by
ax+ by+ c = 0
with a^2 + b^2 =f=. 0. By use of (1.3-4), we obtain
(a - bi)z +(a+ bi)z + 2c = 0