# Complex Numbers

## Complex Numbers

Complex Number: If a, b are two real numbers, then a number of the form a + ib is called a complex number.

Examples:

• 7 + 3i
• -1 + i
• 3 + 2i
• 0 + 3i

etc.  are complex numbers.

Real and Imaginary Parts of a Complex Number: If z = a + ib is a complex number, then ‘a’ is called the real part of z and b is known as the imaginary part of z. The real part of z is denoted by Re(z) and the imaginary part by Im(z).

Example:

If z = 6 + 6i then Re(z) = 6 and Im (z) = 6.

Purely Real and Purely Imaginary Complex Numbers: A complex number z is purely real if it its imaginary pat is zero i.e., Im (z) = 0 and purely imaginary if its real part is zero i.e., Re(z) = 0

Set of Complex Numbers: The set of all complex numbers is denoted by C.

i.e., C = [ a+ i b | a, b ϵ R]

Since a real number ‘a’ can be written as a+i0 therefore every real number is a complex number. Hence,  where R is the set of all real numbers.

Equality of Complex Numbers: Two complex numbers z₁ = a₁ + ib₁ and z₂ = a₂ + ib₂ are equal if a₁ = a₂ and b₁ = b₂ i.e., Re(z₁) = Re(z₂) and Im(z₁) = Im(z₂).

Thus,  and Im(z₁) = Im(z₂).