Complex Analysis - Maths KU

1.1 Complex Numbers and Their Properties 5
then its conjugate is ¯z = a − ib.For example, if z =6+3i, then ¯z =6− 3i;
if z = −5 − i, then ¯z = −5 +i.If z is a real number, say, z = 7, then
¯z = 7.From the definitions of addition and subtraction of complex numbers,
it is readily shown that the conjugate of a sum and difference of two complex
numbers is the sum and difference of the conjugates:
z1 + z2 =¯z1 +¯z2, z1 − z2 =¯z1 − ¯z2. (1)
Moreover, we have the following three additional properties:
z1z2 =¯z1¯z2,
� �
z1
=
z2
¯z1
,
¯z2
¯z = z. (2)
Of course, the conjugate of any finite sum (product) of complex numbers is
the sum (product) of the conjugates.
The definitions of addition and multiplication show that the sum and
product of a complex number z with its conjugate ¯z is a real number:
z +¯z =(a + ib)+(a− ib) =2a (3)
z¯z =(a + ib)(a − ib) =a 2 − i 2 b 2 = a 2 + b 2 . (4)
The difference of a complex number z with its conjugate ¯z is a pure imaginary
number:
z − ¯z =(a + ib) − (a − ib) =2ib. (5)
Since a = Re(z) and b = Im(z), (3) and (5) yield two useful formulas:
Re(z) =
z +¯z
2
and Im(z) =
z − ¯z
. (6)
2i
However, (4) is the important relationship in this discussion because it enables
us to approach division in a practical manner.
Division
To divide z 1 by z 2, multiply the numerator and denominator ofz 1/z2 by
the conjugate ofz 2. That is,
z1
z2
= z1
z2
· ¯z2
=
¯z2
z1¯z2
z2¯z2
and then use the fact that z2¯z2 is the sum ofthe squares ofthe real and
imaginary parts ofz 2.
The procedure described in (7) is illustrated in the next example.
EXAMPLE 2 Division
If z1 =2− 3i and z2 =4+6i, find z1/z2.
(7)