linear algebra

Question

For the complex numbers w and z, is Im(w+z)=Im(w) + Im(z). Prove

Answer 1

Let z = z₁ + iz₂ where z₁ = Re(z) and z₂ = Im(z)

 

Let w = w₁ + iw₂ where w₁ = Re(w) and w₂ = Im(w)

 

Then

 

w + z = (w₁ + iw₂) + (z₁ + iz₂) = (w₁ + z₁) + i(w₂ + z₂)

 

Then

 

Im(w + z) = w₂ + z₂ = Im(w) + Im(z)