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Division :
P.M./Complex/p.3
z1
x + iy 1 x2 - iy 2
= 1

z2
x2 + iy 2 x2 - iy 2

=
(x1x2 + y1y2 ) + i(x2 y1 - y2 x1 )
x22 + y22
=
(x1x2 + y1y2 )
(x y - y2 x1 )
+ i 2 1
x22 + y22
x22 + y22
z1
1
=
(x1x2 + y1y2 )2 + (x2 y1 - y2 x1 )2
2
2
z2
x2 + y2
=
=
1
1
2
z
2
=
(x1x2 )2 + (y1y2 )2 + (x2 y1)2 + (y2 x1)2
x2 2 + y2 2
(x 2 + y 2)(x 2 + y 2 )
2
2
1
1
z1
z2
i 

r e 1
z 

arg  1  = arg  1
z 
i 

 2
r e 2
 2

 r i ( -  ) 
= arg  1 e 1 2  = arg z1 - arg z 2
r

 2

(c) Conjugation :
z = x + iy

conjugate of z , z = x - iy
Properties :
x2 + (-y)2 = z
(i)
z =
(ii)
arg z = tan -1
(iii)
z = (x - iy) = x + iy = z
(iv)
z = 0

-y
= - arg z
x
(-y) = 0  x = 0

x = y = 0
∴
(v)
(vi)
(vii)
z = 0 iff z = 0
 y = -y
z = z  x + iy = x - iy
∴ z is self-conjugate , i.e. z = z iff z is real .
z  z = (x + iy )(x - iy ) = x2 + y2 = z 2
z  z = z  z
1
2
1
2
(viii) z  z = z  z
1 2
1 2
(ix)
(x)
z 
z
 1 = 1
z 
z
 2
2
 z + z = 2 Re z

 z - z = 2i Im z


n
 n
 z1  = z1



y = 0
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