# What is a Complex Number?

If we denote A complex number is of the form*a*+

*ib*where

*a*and

*b*are real numbers.

If

*z*=

*a*+

*ib*,

*a*is the real part of

*z*, denoted by Re(

*Z*) and

*b*is the imaginary part of

*Z*, denoted by Im(

*Z*)

**Note:-**

- If the real part of a complex number is zero, then the number is purely imaginary

- If the imaginary part of a complex number is zero, the number is purely real

- 0 + 0
*i*represents number 0 in complex number system.

**â€‹Example**

**1**. Write the following as complex numbers.

**Example 2**. Find the real and imaginary parts of the following numbers.

**Example 3.**Find the real and imaginary parts of

**Solution:**

*i*

^{20}= (

*i*

^{4})

^{5}= (1)

^{5}= 1 [Since

*i*

^{2}= -1 and

*i*

^{4}= (

*i*

^{2})

^{2 }= (âˆ’1)

^{2}= 1]

*i*

^{19}=

*i*

^{16}.

*i*

^{3}= 1.(âˆ’

*i*) = âˆ’

*i*[Since

*i*

^{2}= âˆ’1 and

*i*

^{16}= (

*i*

^{2})

^{8 }= (âˆ’1)

^{8}= 1,

*i*

^{3}

*= i*

^{2}

*.i =*(âˆ’1)

*.i =*âˆ’

*i*]