Complex Numbers
Complex Number
A number of the form z = x + iy, where x , y ∈ R, is called a complex
number. Here, the symbol i is used to denote −1 and it is called iota.
The set of complex numbers is denoted by C.
Real and Imaginary Parts of a Complex Number Let z = x + iy
be a complex number, then x is called the real part and y is called the
imaginary part of z and it may be denoted as Re( z ) and Im ( z ),
respectively.
Purely Real and Purely Imaginary Complex Number A complex
number z is a purely real, if its imaginary part is 0.
i.e. Im ( z ) = 0. And purely imaginary, if its real part is 0 i.e. Re ( z ) = 0.
Zero Complex Number A complex number is said to be zero, if its
both real and imaginary parts are zero.
Equality of Complex Numbers
Two complex numbers z1 = a1 + ib1 and z 2 = a2 + ib2 are equal, iff
a1 = a2 and b1 = b2 i.e. Re ( z1 ) = Re ( z 2 ) and Im ( z1 ) = Im ( z 2 ).
Iota
Mathematician Euler, introduced the symbol i (read as iota) for − 1
with property i 2 + 1 = 0. i.e. i 2 = − 1. He also called this symbol as the
imaginary unit. Integral power of iota (i) are given below.
i = −1 , i 2 = − 1, i3 = − i , i 4 = 1
So, i 4n + 1 = i , i 4n + 2 = − 1, i 4n +3 = − i , i 4n + 4 = 1
( −1)n / 2 , if n is an even integer
In other words, i =
n
n −1
( −1) 2 ⋅ i , if n is an odd integer
,Algebra of Complex Numbers
1. Addition of Complex Numbers
Let z1 = x1 + iy1 and z 2 = x2 + iy2 be any two complex numbers, then
their sum will be defined as
z1 + z 2 = ( x1 + iy1 ) + ( x2 + iy2 ) = ( x1 + x2 ) + i( y1 + y2 )
Properties of Addition of Complex Numbers
(i) Closure Property Sum of two complex numbers is also a
complex number.
(ii) Commutative Property z1 + z 2 = z 2 + z1, ∀ z1 , z 2 , z3 ∈ C
(iii) Associative Property ( z1 + z 2 ) + z3 = z1 + ( z 2 + z3 ),
∀ z1 , z 2 , z3 ∈ C
(iv) Existence of Additive Identity z + 0 = z = 0 + z
Here, 0 is additive identity element.
(v) Existence of Additive Inverse z + (− z ) = 0 = (− z + z )
Here, ( −z ) is additive inverse or negative of complex number z.
2. Subtraction of Complex Numbers
Let z1 = x1 + iy1 and z 2 = x2 + iy2 be any two complex numbers, then
the difference z1 − z 2 is defined as
z1 − z 2 = ( x1 + iy1 ) − ( x2 + iy2 )
= ( x1 − x2 ) + i( y1 − y2 )
Note The difference of two complex numbers z1 − z2 , follows the closure
property, but this operation is neither commutative nor associative, like in real
numbers. Also, there does not exist any identity element for this operation and
so inverse element also does not exists.
3. Multiplication of Complex Numbers
Let z1 = x1 + iy1 and z 2 = x2 + iy2 be any two complex numbers, then
their multiplication is defined as
z1z 2 = ( x1 + iy1 ) ( x2 + iy2 ) = ( x1x2 − y1 y2 ) + i( x1 y2 + x2 y1 )
Properties of Multiplication of Complex Numbers
(i) Closure Property Product of two complex numbers is also a
complex number.
(ii) Commutative Property z1z 2 = z 2z1 ∀ z1 , z 2 ∈ C.
(iii) Associative Property ( z1 z 2 ) z3 = z1( z 2 z3 ) ∀ z1 , z 2 , z3 ∈ C.
, (iv) Existence of Multiplicative Identity z ⋅ 1 = z = 1 ⋅ z
Here, 1 is multiplicative identity element of z.
(v) Existence of Multiplicative Inverse For every non-zero
complex number z there exists a complex number z1 such that
z ⋅ z1 = 1 = z1 ⋅ z.
Then, complex number z1 is called multiplicative inverse element
of complex number z.
(vi) Distributive Property For each z1 , z 2 , z3 ∈ C
(a) z1( z 2 + z3 ) = z1z 2 + z1z3 [left distribution]
(b) ( z 2 + z3 )z1 = z 2z1 + z3 z1 [right distribution]
4. Division of Complex Numbers
Let z1 = x1 + iy1 and z 2 = x2 + iy2 be two complex numbers, then their
z
division 1 is defined as
z2
z1 ( x1 + iy1 ) ( x1x2 + y1 y2 ) + i( x2 y1 − x1 y2 )
= =
z 2 ( x2 + iy2 ) x22 + y22
provided, z 2 ≠ 0.
z1
Note The division of two complex numbers , follows the closure property, but
z2
this operation is neither commutative nor associative, like in real numbers. Also,
there does not exist any identity element for this operation and so inverse
element also does not exists.
Identities Related to Complex Numbers
For any complex numbers z1 , z 2, we have
(i) ( z1 + z 2 )2 = z12 + 2 z1z 2 + z 22
(ii) ( z1 − z 2 )2 = z12 − 2z1z 2 + z 22
(iii) ( z1 + z 2 )3 = z13 + 3z12z 2 + 3z1z 22 + z32
(iv) ( z1 − z 2 )3 = z13 − 3z12z 2 + 3z1z 22 − z32
(v) z12 − z 22 = ( z1 + z 2 ) ( z1 − z 2 )
These identities are similar as the algebraic identities in real numbers.
Conjugate of a Complex Number
If z = x + iy is a complex number, then conjugate of z is denoted by z,
i.e. z = x − iy
Complex Number
A number of the form z = x + iy, where x , y ∈ R, is called a complex
number. Here, the symbol i is used to denote −1 and it is called iota.
The set of complex numbers is denoted by C.
Real and Imaginary Parts of a Complex Number Let z = x + iy
be a complex number, then x is called the real part and y is called the
imaginary part of z and it may be denoted as Re( z ) and Im ( z ),
respectively.
Purely Real and Purely Imaginary Complex Number A complex
number z is a purely real, if its imaginary part is 0.
i.e. Im ( z ) = 0. And purely imaginary, if its real part is 0 i.e. Re ( z ) = 0.
Zero Complex Number A complex number is said to be zero, if its
both real and imaginary parts are zero.
Equality of Complex Numbers
Two complex numbers z1 = a1 + ib1 and z 2 = a2 + ib2 are equal, iff
a1 = a2 and b1 = b2 i.e. Re ( z1 ) = Re ( z 2 ) and Im ( z1 ) = Im ( z 2 ).
Iota
Mathematician Euler, introduced the symbol i (read as iota) for − 1
with property i 2 + 1 = 0. i.e. i 2 = − 1. He also called this symbol as the
imaginary unit. Integral power of iota (i) are given below.
i = −1 , i 2 = − 1, i3 = − i , i 4 = 1
So, i 4n + 1 = i , i 4n + 2 = − 1, i 4n +3 = − i , i 4n + 4 = 1
( −1)n / 2 , if n is an even integer
In other words, i =
n
n −1
( −1) 2 ⋅ i , if n is an odd integer
,Algebra of Complex Numbers
1. Addition of Complex Numbers
Let z1 = x1 + iy1 and z 2 = x2 + iy2 be any two complex numbers, then
their sum will be defined as
z1 + z 2 = ( x1 + iy1 ) + ( x2 + iy2 ) = ( x1 + x2 ) + i( y1 + y2 )
Properties of Addition of Complex Numbers
(i) Closure Property Sum of two complex numbers is also a
complex number.
(ii) Commutative Property z1 + z 2 = z 2 + z1, ∀ z1 , z 2 , z3 ∈ C
(iii) Associative Property ( z1 + z 2 ) + z3 = z1 + ( z 2 + z3 ),
∀ z1 , z 2 , z3 ∈ C
(iv) Existence of Additive Identity z + 0 = z = 0 + z
Here, 0 is additive identity element.
(v) Existence of Additive Inverse z + (− z ) = 0 = (− z + z )
Here, ( −z ) is additive inverse or negative of complex number z.
2. Subtraction of Complex Numbers
Let z1 = x1 + iy1 and z 2 = x2 + iy2 be any two complex numbers, then
the difference z1 − z 2 is defined as
z1 − z 2 = ( x1 + iy1 ) − ( x2 + iy2 )
= ( x1 − x2 ) + i( y1 − y2 )
Note The difference of two complex numbers z1 − z2 , follows the closure
property, but this operation is neither commutative nor associative, like in real
numbers. Also, there does not exist any identity element for this operation and
so inverse element also does not exists.
3. Multiplication of Complex Numbers
Let z1 = x1 + iy1 and z 2 = x2 + iy2 be any two complex numbers, then
their multiplication is defined as
z1z 2 = ( x1 + iy1 ) ( x2 + iy2 ) = ( x1x2 − y1 y2 ) + i( x1 y2 + x2 y1 )
Properties of Multiplication of Complex Numbers
(i) Closure Property Product of two complex numbers is also a
complex number.
(ii) Commutative Property z1z 2 = z 2z1 ∀ z1 , z 2 ∈ C.
(iii) Associative Property ( z1 z 2 ) z3 = z1( z 2 z3 ) ∀ z1 , z 2 , z3 ∈ C.
, (iv) Existence of Multiplicative Identity z ⋅ 1 = z = 1 ⋅ z
Here, 1 is multiplicative identity element of z.
(v) Existence of Multiplicative Inverse For every non-zero
complex number z there exists a complex number z1 such that
z ⋅ z1 = 1 = z1 ⋅ z.
Then, complex number z1 is called multiplicative inverse element
of complex number z.
(vi) Distributive Property For each z1 , z 2 , z3 ∈ C
(a) z1( z 2 + z3 ) = z1z 2 + z1z3 [left distribution]
(b) ( z 2 + z3 )z1 = z 2z1 + z3 z1 [right distribution]
4. Division of Complex Numbers
Let z1 = x1 + iy1 and z 2 = x2 + iy2 be two complex numbers, then their
z
division 1 is defined as
z2
z1 ( x1 + iy1 ) ( x1x2 + y1 y2 ) + i( x2 y1 − x1 y2 )
= =
z 2 ( x2 + iy2 ) x22 + y22
provided, z 2 ≠ 0.
z1
Note The division of two complex numbers , follows the closure property, but
z2
this operation is neither commutative nor associative, like in real numbers. Also,
there does not exist any identity element for this operation and so inverse
element also does not exists.
Identities Related to Complex Numbers
For any complex numbers z1 , z 2, we have
(i) ( z1 + z 2 )2 = z12 + 2 z1z 2 + z 22
(ii) ( z1 − z 2 )2 = z12 − 2z1z 2 + z 22
(iii) ( z1 + z 2 )3 = z13 + 3z12z 2 + 3z1z 22 + z32
(iv) ( z1 − z 2 )3 = z13 − 3z12z 2 + 3z1z 22 − z32
(v) z12 − z 22 = ( z1 + z 2 ) ( z1 − z 2 )
These identities are similar as the algebraic identities in real numbers.
Conjugate of a Complex Number
If z = x + iy is a complex number, then conjugate of z is denoted by z,
i.e. z = x − iy
