Decide |z|, arg ??¯,Re z, Im ??¯ for z = 1 - i

Question

Decide |z|, arg ??¯,Re z, Im ??¯    for    z = 1 − i

Answer 1

Since   z = 1 - i      |z| = sqrt( (1- i)(1+i)) = √2

 

  With this result    (1- i) = √2 exp(i θ) = √2( cos θ + i sin θ)         where θ = arg z.     

   This is satisfied with  θ = -pi/4

 

     Obviously   Re z = 1,        Im z = -i