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Complex Numbers
1. C = set of complex numbers
= { x + iy :
x, y   , i   1 }
z = x + iy ; x = Re(z) , real part of z
; y = Im(z) , imaginary part of z
z = iy -- purely imaginary numbers
2. For any n    , 1, i , i 2  1 , i 3  i , i 4  1 , ......
3. Operations on C :
z1  a  bi , z2  c  di
(i)
(ii)
(iii)
(iv)
(v)
Equality
+
–
x
/
4. (C, +) is an commutative (abelian) group.
(C \{0}, * ) is an commutative group.
5. z  a  bi ,
Properties:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
conjugate of z = z = a – bi
6. Agrand diagram: -- Cartesian system -- Geometrical Rep.
-- > z = x +iy ~ P(x, y) where P is the affix of z
7. Polar form:
z = x +iy = r (cos   i sin  )
r = modulus of z
In general,
;
θ = argument (amplitude) of z
arg z + 2k   .
8. Multiplication and Division of z1 , z2
etc .
9. Representation of sum, difference, product and quotient :
P1 ~ z1  r1 (cos1  i sin 1 ) ; P2 ~ z2  r2 (cos 2  i sin  2 )
10. Properties of modulus and argument :
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
11. Exponential form :
Denote ei  cos  i sin  ;
Then
e i  cos  i sin 
Euler’s formulas:
12. De Moivre’s theorem for integral exponents:
( c os  i s i n )n  c o sn  i s i nn
Pf: (Use of MI)
13. De Moivre’s theorem for rational exponents:
p
p
p
( c os  i s i n )  c o s   i s i n 
q
q
q
14. Applications:
c  cos ,
z
s  sin  , z  c  is , z  c  is
1
1
, zn 
z
zn
(i)
(ii) cos nθ =
sin nθ =
tan nθ =
(iii)
eg. Find an expression for tan 7θ in terms of θ .
Hence prove that the roots of x3  21x 2  35 x  7  0 are
tan 2

7
Deduce
, tan 2
sec 4

7
2
3
, t a2 n
.
7
7
 sec 4
2
3
 sec 4
 416 .
7
7
Eg. Express cos7θ as a polynomial in cosθ and deduce
sec 2

14
 sec 2
3
5
 sec 2
8 .
14
14

4
eg. Evaluate
 cos  sin
5
7
 d
0
More properties: For any n  
(iv)
(v)
(vi)
(vii)
15. Roots of Unity:
zn
 1  cos 0  i sin 0
 cos 2k  i sin 2k
,k 
z=
The n distinct roots of the equation are 1, λ ,
where λ =
Observing from the Argand diagrams, some roots of unity are in conjugate pairs.
For odd and even n ,
z n 1 
eg. Cube roots of unity z3 = 1
Geometrical Representation :
 they are points (Pk, k = 0,1, …,n,) on the circumference of a circle of unit
radius’ having center at origin, such that P0 =(1,0) and P0OPk 
Variations :
2k
.
n
zn = -1
eg. Find z if z3 = -1 .
k
 z 1
eg. Show that if n    , 
, k =1,2,..,n-1
  1 , then z =  i cot
n
 z 1
n
eg. Solve ( z  1) n  z n  0 , n    .
Show that Re(z) = -0.5 for any root z of the equation.
16. (A) Roots of Complex Numbers
If z n  r (cos  i sin  ) …….. (*)
r > 0 , n   ,
then z =
(B) Factorization of x 2 n  2a n x n cos n  a n
x 2 n  2a n x n cos n  a n =0
Start at
Therefore xn =
n 1
Eventually, x 2 n  2a n x n cos n  a n =  ( x 2  2ax cos( 
k 0
2k
)  a2 )
n
(C) Special cases:
a) Put a =1 , θ = 0 ,
( x n  1) 2 
b) Put a = 1 ,  

n
,
( x n  1)2 
c) Put a = 1, divided both sides by xn ,
2 (cos n  cos n ) 
eg. With 0   

n
n 1
, show sin n  2n 1  sin(  
k 0
k
).
n
Eg. Let Pk , k =0, 1,…,n-1 be the vertices of a regular n-gon inscribed in a circle,
with radius r and center at O. If P is a point such that POP0   , show
n 1
that
 PP
k
 r 2 n  2r n (OP ) n cos n  OP 2 n .
k 0
17. Geometry of Complex Numbers:
Pk  affix of zk
(i)
(ii)
z1  z2  P1P2 = distance between P1 & P2 .
arg (z1 – z2 ) = angle of turn from OX towards P2 P1
(a) anti-clockwise direction
(b) clockwise direction
(iii)
  arg(
towards
(iv)
arg( z – z ) > 0
arg ( z – z ) < 0
z1  z2
) represents the angle of turn from P2 P3
z3  z 2
P2 P1
3 points P1, P2, P3 are collinear
eqv arg(
eqv
(v)
(
(
z1  z2
)  0 or 
z3  z 2
z1  z2
)   ,  
z3  z 2
z1  z2
)  i , for    \ {0} , then P2 P1  P2 P3 .
z3  z 2
(vi) P1, P2, P3 ,P4 are 4 distinct points representing z1 , z2, z3, z4 .
( z  z )( z  z )
(a) If P1, P2, P3 ,P4 are concyclic, then 1 3 2 4  k , k  0 .
( z2  z3 )( z1  z4 )
(b) If
( z1  z3 )( z2  z4 )
  , then P1, P2, P3 ,P4 are either concyclic or
( z2  z3 )( z1  z4 )
collinear.
Worked examples:
1. (a) Prove ΔP1P2P3 is equilateral iff z1  z2  z3  z1z2  z2 z3  z3 z1
2
2
2
(b) Show that the roots of the equation z 3  6 z 2  12 z  5  0 are the vertices of an
equilateral triangle in the Argand diagram.
2.
Let    , show that if t is any root of the equation
z n cos  z n 1 cos(n  1)  ......  z c o s  1 , then
1
.
2
| t |
3. (a) Prove that if the z’s are any complex numbers and λ is positive, then
| z1  z2 |2  (1   ) | z1 |2 (1 
1
) | z2 |2 .

Under what condition does the sign of the equality hold ?
(b) Prove also that, if the a’s are positive numbers such that
Then
4. Let
z
1
1
 ..... 
1 ,
a1
an
| z1  z2  ....  zn |2  a1 | z1 |2 ...  an | zn |2 .
1  cos  i sin 
,
1  cos  i sin 
(a) Find the modulus and argument of z when 0     .
(b) What are the modulus and argument when     2 ?
5. In the Argand diagram, PQR is an equilateral triangle of which the circumcentre is at
the origin. If P represents the complex number z1  2  i ,
Find the complex number represented by Q and R .
6. The vertices P, Q, R of an isosceles triangle whose equal sides are PQ and PR
represent the complex numbers z1 , z2 , z3 respectively and the angle QPR is α .
(a) Show that ( z1  z2 ) 2  ( z2  z3 ) 2  2( z1  z2 )( z1  z3 ) cos .
(b) Deduce that
2
2
2
z1  z2  z3  z1z2  z2 z3  z3 z1 iff ΔABC is equilateral.
7. Let P, Q, R, S represent the complex numbers
(a) If arg( z1  z2 )=arg( z3  z4 ) , show that
z1, z2 , z3 , z4 respectively.
z1  z2
is real .
z3  z 4
z1  ikz2
, where k is any non-zero real number.
1  ik
Show that, ΔPQR is a right-angled triangle.
(b) If z3 
8. Let
p and q
be non-zero, distinct complex numbers such that | p – q | =| p + q |
(a) Show that p q  pq  0 .
(b) Let O, P, Q be 3 points representing 0, p and q respectively. By considering the
p
argument of
, or otherwise show that OP ⊥ OQ .
q