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).

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