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Page # 31
COMPLEX NUMBER
1. The complex number system
z = a + ib, then a – ib is called congugate of z and is denoted by z .
2. Equality In Complex Number:
 z
1
 = z
2
?????? Re(z
1
) = Re(z
2
)  and  ?
m
 (z
1
) = ?
m
 (z
2
).
3. Properties of arguments
(i) arg(z
1
z
2
) = arg(z
1
) + arg(z
2
) + 2m ? for some integer m.
(ii) arg(z
1
/z
2
) = arg (z
1
) – arg(z
2
) + 2m ? for some integer m.
(iii) arg (z
2
) = 2arg(z) + 2m ? for some integer m.
(iv) arg(z) = 0 ? z is a positive real number
(v) arg(z) = ± ?/2 ? z is purely imaginary and  z ? 0
4. Properties of conjugate
(i) |z| = | z | (ii) z z = |z|
2
(iii)
2 1
z z ? = 
1
z + 
2
z
(iv)
2 1
z z ? = 
1
z – 
2
z (v)
2 1
z z = 
1
z
2
z
(vi)
?
?
?
?
?
?
?
?
2
1
z
z
 = 
2
1
z
z
(z
2
 ? 0)
(vii) |z
1
 + z
2
|
2
 = (z
1
 + z
2
) ) z z (
2 1
?  = |z
1
|
2
 + |z
2
|
2
 + z
1 2
z + 
1
z z
2
(viii) ) z (
1
 = z (ix) If w = f(z), then w = f( z )
(x) arg(z) + arg(
z
)
5. Rotation theorem
If P(z
1
), Q(z
2
) and R(z
3
) are three complex numbers and ?PQR = ?, then
?
?
?
?
?
?
?
?
?
?
2 1
2 3
z z
z z
 = 
2 1
2 3
z z
z z
?
?
e
i ?
6. Demoivre’s Theorem :
If n is any integer then
(i) (cos ? + i sin ? )
n
 = cos n ? + i sin n ?
(ii) (cos ?
1
 + i sin ?
1
) (cos ?
2
 + i sin ?
2
) (cos ?
3
 + i sin ?
2
)
(cos ?
3
 + i sin ?
3
) .....(cos ?
n
 + i sin ?
n
)  = cos ( ?
1
 + ?
2
 + ?
3
 + ......... ?
n
) +
i sin ( ?
1
 + ?
2
 + ?
3
 + ....... + ?
n
)
Page 2


Page # 31
COMPLEX NUMBER
1. The complex number system
z = a + ib, then a – ib is called congugate of z and is denoted by z .
2. Equality In Complex Number:
 z
1
 = z
2
?????? Re(z
1
) = Re(z
2
)  and  ?
m
 (z
1
) = ?
m
 (z
2
).
3. Properties of arguments
(i) arg(z
1
z
2
) = arg(z
1
) + arg(z
2
) + 2m ? for some integer m.
(ii) arg(z
1
/z
2
) = arg (z
1
) – arg(z
2
) + 2m ? for some integer m.
(iii) arg (z
2
) = 2arg(z) + 2m ? for some integer m.
(iv) arg(z) = 0 ? z is a positive real number
(v) arg(z) = ± ?/2 ? z is purely imaginary and  z ? 0
4. Properties of conjugate
(i) |z| = | z | (ii) z z = |z|
2
(iii)
2 1
z z ? = 
1
z + 
2
z
(iv)
2 1
z z ? = 
1
z – 
2
z (v)
2 1
z z = 
1
z
2
z
(vi)
?
?
?
?
?
?
?
?
2
1
z
z
 = 
2
1
z
z
(z
2
 ? 0)
(vii) |z
1
 + z
2
|
2
 = (z
1
 + z
2
) ) z z (
2 1
?  = |z
1
|
2
 + |z
2
|
2
 + z
1 2
z + 
1
z z
2
(viii) ) z (
1
 = z (ix) If w = f(z), then w = f( z )
(x) arg(z) + arg(
z
)
5. Rotation theorem
If P(z
1
), Q(z
2
) and R(z
3
) are three complex numbers and ?PQR = ?, then
?
?
?
?
?
?
?
?
?
?
2 1
2 3
z z
z z
 = 
2 1
2 3
z z
z z
?
?
e
i ?
6. Demoivre’s Theorem :
If n is any integer then
(i) (cos ? + i sin ? )
n
 = cos n ? + i sin n ?
(ii) (cos ?
1
 + i sin ?
1
) (cos ?
2
 + i sin ?
2
) (cos ?
3
 + i sin ?
2
)
(cos ?
3
 + i sin ?
3
) .....(cos ?
n
 + i sin ?
n
)  = cos ( ?
1
 + ?
2
 + ?
3
 + ......... ?
n
) +
i sin ( ?
1
 + ?
2
 + ?
3
 + ....... + ?
n
)
Page # 32
7. Cube Root Of Unity :
(i) The cube roots of unity are 1, 
? ? 1 3
2
i
, 
? ? 1 3
2
i
.
(ii) If ? is one of the imaginary cube roots of unity then 1 + ? + ?² = 0.
In general 1 + ?
r 
+ ?
2r 
= 0; where r ? I but is not the multiple of 3.
8. Geometrical Properties:
Distance formula :  |z
1
 – z
2
|.
Section formula : z = 
n m
nz mz
1 2
?
?
(internal division), z = 
n m
nz mz
1 2
?
?
(external
division)
(1) amp(z) = ? is a ray emanating from the origin inclined at an angle
? to the x ?
 
axis.
(2) ?z ? a ? = ?z ? b ? is the perpendicular bisector of the line joining a
to b.
(3) If 
1
z z
z z
?
?
 = k ? 1, 0, then locus of z is circle.
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