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Chapter 2: Complex Numbers
2.1 Introduction to complex numbers
A complex number is a number of the form a bi , where a and b are real
numbers and i 1 with property 1
2
i .
Every complex number has two parts: z a bi
Real part , Re( z ) a and imaginary part ,Im( z ) b
Given z 1 4 6 i , then Re( z 1 ) 4 andIm( z 1 ) 6
z 2 2 i , then Re( z 2 ) 0 and Im( z 2 ) 2
Power of IOTA i
i i 1 i i i 1 i i
5 4
2 i 1 1 ^1
6 4 2 i i i
i i i 1 i i 3 2 i i i 1 i i 7 4 3
1 1 1
4 2 2
i i i 1 2 1
8 42 i i
2.2 Operations on complex numbers
Addition: a bi u vi a u b v i
Subtraction: a bi u vi a u b v i
Multiplication: 2 a bi u vi au avi ubi bvi
au avi ubi bv 1
au avi ubi bv
au bv av ub i
Conjugate of z : if z 1 a bi then z 1 a bi a bi
if z 2 a bi then z 2 a bi a bi
Therefore, 2 2 a bi a bi a b
Division:
u vi
u vi
u vi
a bi
u vi
a bi
2 2 u v
a bi u vi
[Tutorial 5]
2.3 The Argand diagram
Complex numbers are represented as points on a plane called the Argand
diagram.
The Argand diagram consists of two perpendicular axes.
a) The horizontal axis is the real axis.
b) The vertical axis is the imaginary axis.
A complex number z a bi can be represented as an ordered pair a , b ,
called Cartesian coordinates on the Argand diagram.
The Argand diagram below shows four complex numbers:
z (^) 1 4 3 i z (^) 2 2 i z (^) 3 5 4 i z (^) 4 1 i
2.5 Multiple Complex Numbers in Polar form
Let z 1 r 1 cos 1 i sin 1 and z 2 r 2 cos 2 i sin 2
1. Multiplication
z 1 z 2 r 1 r 2 cos^ ^1 2 ^ i sin^1 2
Modulus of z r 1^ r 2 z 1 z 2
Argument of z Arg z 1 z 2 1 2
2. Division
2
1
2
1 cos i sin
r
r
z
z
Modulus of z
2
1
2
1 z
z
r
r
Argument of z 1 2
2
1
z
z
Arg
3. Power of n
n m z (^) 1 andz 2
Multiplication:
n m z r 1 r 2
Arg z n 1 m 2
Division: m
n
r
r z 2
1
Arg z n 1 m 2
[Tutorial 7]
2.6 De Moivre’s Theorem
For all real values of n ,
i n i n
n cos sin cos sin
Let z cos i sin
then, cos sin
z i z
therefore, 2 cos
z
z and 2 sin
i z
z
Using De Moivre’s Theorem:
n
z
z n
n 2 cos
and i n
z
z n
n 2 sin
Binomial Theorem:
n (^) n n n n ab n
n a b
n a b
n ab
n a b 0 11 2 2 0 ... 0 1 2
where n is positive
Using Pascal’s Triangle in Binomial expansion:
[Tutorial 8]
2.7 Roots of complex numbers
A complex number w is a n-th root of the complex number z if w z
n
or
w zn
1
.Hence
