NEELKANTH VIDYAPEETH
BCA (IV-Semester) Mathematics-III
UNIT-I COMPLEX NUMBERS
Complex Numbers: - “A Number of he form x+iy where i  (1) and x,y are both
real number, is called complex number. A complex number is also defined as an ordered pair
(x,y) of real numbers and represented by z  x  iy where x= real part and y= imaginary part .
 z1  z2
Complex Number z  a  iy
R(z) = real part of z = a
I(z) = imaginary part of z = b
Note:
1. A complex number is said to be purely real if its imaginary part is zero.
2. A complex number is said to be imaginary if its real part is zero.
3. A complex number 2+0i may be written as 2.
4. Set of complex number is denoted by C.
Equality of complex number:Two complex numbers z1  x1  iy1 or z2  x2  iy2 are said to be equal if x1 =
x2 and y1 = y2. Hence two complex number are equal if and only if the real part of one is equal
to the real part of other and the imaginary part of one is equal to the imaginary part of the other.
Addition of complex number:-
If z1  x1  iy1 and z2  x2  iy2 are the two complex
number, then the sum of z1 and z2 is written as z1 + z2 and defined as
z1  z2  ( x1  iy1 )  ( x2  iy2 ) = ( x1  x2 )  i( y1  y2 )
Or
( z1  z2 )  ( x1 , y1 )  ( x2 , y2 )  ( x1  x2 ,y1  y2 ).
Properties of the Addition of complex numbers: Theorem 1-(Commutative of addition)
To show that
Z1 + Z2 = Z2 + Z1
Where Z1 and Z2 are any complex numbers.
Proof:Let z1 = (x1,y1) and z2 = (x2,y2)
Where x1,x2,y1,y2 are real numbers
Then
z1  z2  ( x1 , y1 )  ( x2 , y2 )
 ( x1  x2 , y1  y2 )
 ( x2  x1 , y2  y1 )
 ( x2 , y2 )  ( x1 , y1 )
 z1  z2 .
Theorem 2- (Associatively of addition)
To show that ( z1  z2 )  z3  z1  ( z2  z3 ) for all complex numbers Z1, z2 and z3.
Proof:
Let z1 = (x1,y1) , z2 = (x2,y2) and z3 = (x3,y3)
Where x1,y1, x2,y2 x3,y3 are real numbers.
Then
(z1 + z2) + z3= [(x1, y1) + (x2,y2) ]+ (x3,y3)
= [x1 + x2, y1 + y2)] + (x3,y3)
= [(x1 + x2) + x3, (y1 + y2) + y3]
= [x1 + ( x2+ x3 ), y1 + (y2 + y3)]
= (x1,y1) + (x2 + x3, y2 + y3)
= (x1,y1) + {(x2,y2) +(x3,y3)}
= z1+ (z2 + z3)
Multiplication of complex numbers: If z1= (x1 + iy1) or (x1,y1) and z2= (x2 + iy2) or (x2,y2) are any two
complex numbers then product is
z1 z2= (x1 + iy1) (x2 + iy2) = (x1 x2 - y1 y2) + i(x1 y2 + x2 y1)
Properties of the multiplication of complex numbers: Theorem 3 - (Commutative of multiplication)
To show that
Z1 Z2 = Z2 Z1
Where Z1 and Z2 are any complex numbers
Proof:
Let z1 = (x1,y1) , z2 = (x2,y2)
Where x1,y1, x2,y2 x3,y3 are real numbers
Z1 Z2 = (x1,y1) (x2,y2)
= (x1 x2 - y1 y2, x1 y2 + x2 y1)
= (x2 x1 – y2 y1) , y2 x1 + y1 x2
= (x2,y2) (x1,y1)
= Z2 Z1
Hence Z1 Z2 = Z2 Z1 for all complex numbers Z1 and Z2 .
Division in c :A complex number (a,b) is said to be divisible by a complex number (c,d) if
there exists a complex number (x,y)
Such that
(x,y) (c,d) = (a,b)


( xc  yd , xd  yc)  (a, b)
xc  yd  a and xd  yc  b
Solve these equations we get
x
ac  bd
bc  ad
,y 2
2
2
c d
c  d2
If Z1 Z2 be two complex numbers then
z1
 z1 ( z2 ) 1 .
z2
Conjugate of a complex number:
The conjugate of a complex number z = x + iy is obtain by
Putting i = -i and – i = i it is denoted by
_
z  x  iy
Thus if
_
Z = -3 +4i  z   3  4i
The following results are for conjugate
1. Two complex numbers are equal if aonly if their conjugates are


Equal z1  z2  z1  z2
z  z
2. z1  z2  z1  z2 , z1  z2  z1  z2 , z1 z2  z1 z2 and  1   1 provided z2  0
 z2  z2
3.
zz
_
4. Let z = x + iy, z  x  iy
Thus, sum of two conjugate complex number, equal to twice the real part of each.
Separation of real and imaginary Part of Complex number:Let the complex number =
=
a  ib
c  id
(a  ib)(c  id )
(c  id )(c  id )
[Multiplying Nr and Dr by conjugate of Dr]
=
(ac  bd )  i(bc  ad )
(c 2  d 2 )
=
(ac  bd ) (bc  ad )
i 2
(c 2  d 2 )
(c  d 2 )
(ca  bd )
(bc  ad )
,B  2
2
2
(c  d )
(c  d 2 )
a  ib
A is a real part B is the imaginary part of
.
c  id
= A + iB Where A 
Modulus of a complex number:If z = (x,y) or z = (x+iy) be any complex number, then the non-negative real
number
x  y 2 is called the modulus or absolute value of complex number z and it is denoted
2
by z or mod( z )
Therefore
2  3i  (2)2  (3)2 =
4  9  13
2  3i  (2)2  (3)2
=
4  9  13
= ei  cos   i sin   cos 2   sin 2 
1 1
=
Hence, the modulus of a complex number is equal to the positive square root of the sum of the
squares of the real and imaginary parts of that complex number.
Theorem-1 :The modulus of the product of two complex numbers is the product of their
moduli.
Proof: - We have
z1 z2  ( z1 z2 )( z1 z2 )
2
 ( z1 z2 )( z1 z2 )
 ( z1 z1 ) ( z2 z2 )
 z1 z2
2
So that
z1 z2  z1 z2
Or
z1 z2  z1 z2
In general
2
2
2
2
z1 z2 ..........zm  z1 z2 ................ zn
Theorem-2 :The modulus of the sum of two complex numbers is always less than or equal to
the sum of their moduli.
z1  z2  z1 z2
Or
Proof: - We have to show that
z1  z2  z1 z2
Now
z1  z2  ( z1  z2 ) ( z1  z2 )
2
 ( z1  z2 ) ( z1  z2 )
 ( z1 z1  z2 z2 )( z2 z1  z2 z2 )
 z1  z2  2 R ( z1 z 2 )
2
2
 z1  z2  2 z1 z 2
2
2
[ z1 z 2  z1 ]
 z1 z 2  z1 z2  2R( z1 z1 )  2 z1 z 2
Also

z1  z2  z1  z2  2 z1 z 2
2
2
2
 z1  z2  2 z1 z2
2
2
z1  z2  ( z1  z2 ) 2
2
z1  z2  z1  z2
[ z2  z2 ]
2