## 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 i
^{2}= -1; i^{3}= -i; i^{4}= 1 etc.

## Equality of Complex Numbers

Two complex numbers Z_{1} = a_{1} + ib_{1} and z_{2} = ac_{2} + ib_{2} are equal, iff a_{1} = a_{2} and b_{1} = b_{2} i.e. Re (Z_{1}) = Re (z_{2}) and Im (Z_{1}) = Im (z_{2}).

## 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

i^{2} = −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 (i^{2})

= 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