Question

If z₁, z₂ and z₃, z₄ are two pairs of conjugate complex numbers, then find arg`(z_1/z_4)` + arg`(z_2/z_3)`.

Answer 1

Let the polar form of z₁ = r₁(cosθ₁ + isinθ₁)

∴ z₂ = `barz_1` 

= r₁(cosθ₁ + isinθ₁)

= r₁[cos(–θ₁) + isin(–θ₁)]

Similarly, z₃ = r₂(cosθ₂ + isinθ₂)

∴ z₄ = `barz_3`

= r₂(cosθ₂ + isinθ₂)

= r₂[cos(–θ₂) + isin(–θ₂)]

arg`(z_1/z_4)` + arg`(z_2/z_3)` = arg(z₁) – arg(z₄) + arg(z₂) – arg(z₃)

= θ₁ – (–θ₂) + (–θ₁) – θ₂

= θ₁ + θ₂ – θ₁ – θ₂

= 0

Hence, arg`(z_1/z_4)` + arg`(z_2/z_3)` = 0.