# 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 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}).