Complex Number

What is Complex Number?

Complex numbers are defined as expressions of the form x = iy

where x, y ∈ R, and i = √-1 . It is denoted by z  i.e. z= x + iy

The numbers x and y are called respectively real and imaginary parts of complex number z.

I.e.     x = Re (z) and y = Im (z)

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.

Note:

• The Set R of real numbers is a proper subset of the Complex Numbers. Hence the Complex Number system is N ⊂ W ⊂ I ⊂ Q ⊂ R ⊂ C.
• Zero is both purely real as well as purely imaginary but not imaginary.
• √x √y = √xy  Only if atleast one of either x or y is non negative.
• i = √-1 is called the imaginary unit. Also i² = -1; i³ = -i; i⁴ = 1 etc.

Equality of Complex Numbers

Two complex numbers Z₁ = a₁ + ib₁ and z₂ = ac₂ + ib₂ are equal, iff a₁ = a₂ and b₁ = b₂ i.e. Re (Z₁) = Re (z₂) and Im (Z₁) = Im (z₂).

Addition and subtraction of complex numbers

Addition of complex numbers is defined by separately adding real
and imaginary parts; so if

z = a + bi, w = c + di
then
z + w = (a + c) + (b + d)i .
Similarly for subtraction.

Example complex numbers

QUESTION 1. Express each of the following in the form
x + yi.
(a) 3 + 5i + =2 − 3i
(b)3 + 5i + 6
(c) 7i − = 4 + 5i

Solution
(a) 3 + 5i + ( ) 2 − 3i = 3 + 2 + ( ) 5 − 3 i = 5 + 2i
(b)3 + 5i + 6 = 9 + 5i
(c)7i − ( ) 4 + 5i = 7i − 4 − 5i = −4 + 2i

Multiplication of complex numbers

Multiplication is straightforward provided you remember that

i² = −1.

Example

Simplify in the form x + yi :
(a) 3( 2 + 4i)
(b) (5 + 3i)i
(c) (2 − 7i ) (3 + 4i)

Solution
(a) 3 (2+ 4i)

= 3 ( 2) + 3 (4i)

= 6 +12i
(b) (5 + 3i) i

= (5 i +3 (i) i

= 5i + 3 (i²)

= 5i + −(1 )  3

= − 3 + 5i
(c) (2 − 7i) (3 + 4i )

= ( 2)  (3 )  − (7i )  ( 3)  + ( 2)  (4i )  − ( 7i) ( 4i)
= 6 − 21i + 8i − −( 28)
= 6 − 21i + 8i + 28
= 34 −13i
In general, if

z = a + bi ,
w = c + di,
then
z w = (a + bi)(c + di)
= a c − b d + (a d + b c)i

Division of complex numbers

The complex conjugate of a complex number is obtained by
changing the sign of the imaginary part. So if
z = a + bi, its
complex conjugate, z , is defined by

z = a − bi
Any complex number
a + bi has a complex conjugate
a − bi
and from Activity 5 it can be seen that
( a + bi ) (a − bi )  is a real
number. This fact is used in simplifying expressions where the
denominator of a quotient is complex.

Example

Simplify the expressions:
(a) 1/i

Solution

To simplify these expressions you multiply the numerator and
denominator of the quotient by the complex conjugate of the
denominator.
(a) The complex conjugate of i is −i, therefore