Geometrical Representation of Fundamental Operations

(i) Geometrical representation of addition

If two points P and Q represents complex z₁ and z₂ respectively in the Argard Plane, then the sum z₁+z₂ is represented by the extremely R of the diagonal OR of parallelogram OPRQ having OP and OQ as two adjacent sides.

(ii) Geometrical representation of subtraction -

(iii) Modules and argument of multiplication of two complex numbers -

Theorem: For any two complex numbers z₁, z₂

We have, |z₁, z₂| = |z₁| |z₂| and arg(z₁, z₂) = arg(z₁) + arg(z₂)

Proof: z₁ = r₁ e^iθ, z₂ = r₂ e^iθ₂

        z₁z₂ = r₁ r₂ e^i(θ1+θ2)

        => |z₁ z₂|=|z₁|=|z₂|

arg (z₁z₂) = arg(z1) + arg(z2)

i.e., to multiply two complex numbers, we multiply their absolute values and add their arguments.