Question

For any two complex numbers z₁ and z₂, prove that Re (z₁z₂) = Re z₁ Re z₂ – Imz₁ Imz₂

Answer 1

Let z₁ = a + ib, z₂ = c + id

∴ z₁z₂ = (a + ib)(c + id)

= ac + adi + bci + i²bd

= (ac – ba) + (ad + bc) i [∴ i² = – 1]

Real part of Re(z₁z₂) = ac – bd

= Rez₁ Rez₂ – Imz₁ Imz₂

Here real part of Rez₁ = a, similarly Rez₂ = c

Imz₁ = imaginary part of z₁ = b

Similarly Imz₂ = d.