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```Division :
P.M./Complex/p.3
z1
x + iy 1 x2 - iy 2
= 1
ï
z2
x2 + iy 2 x2 - iy 2
ï
=
(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
iï± ï¶
ï¦
ï§r e 1ï·
ï¦z ï¶
ï·
arg ï§ 1 ï· = arg ï§ 1
ï§z ï·
iï± ï·
ï§
ï¨ 2ï¸
ï§r e 2ï·
ï¨ 2
ï¸
ï¦ r i (ï± - ï± ) ï¶
= arg ï§ 1 e 1 2 ï· = arg z1 - arg z 2
ï§r
ï·
ï¨ 2
ï¸
(c) Conjugation :
z = x + iy
ï
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
ï
-y
= - arg z
x
(-y) = 0 ï x = 0
ï
x = y = 0
â´
(v)
(vi)
(vii)
z = 0 iff z = 0
ï y = -y
z = z ï x + iy = x - iy
â´ z is self-conjugate , i.e. z = z iff z is real .
z ï z = (x + iy )(x - iy ) = x2 + y2 = z 2
z ï± z = z ï± z
1
2
1
2
(viii) z ï z = z ï z
1 2
1 2
(ix)
(x)
ï¦z ï¶
z
ï§ 1ï· = 1
ï§z ï·
z
ï¨ 2ï¸
2
ï¬ z + z = 2 Re z
ï­
ï® z - z = 2i Im z
ï