SOLUTION: Suppose z=a+bi is a complex number, and w=x+yi is another complex number such that z+w is a real number and zw is also a real number. Show that w must be the conjugate of z.

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Suppose z=a+bi is a complex number, and w=x+yi is another complex number such that z+w is a real number and zw is also a real number. Show that w must be the conjugate of z.      Log On


   

Question 539691: Suppose z=a+bi is a complex number, and w=x+yi is another complex number such that z+w is a real number and zw is also a real number. Show that w must be the conjugate of z.
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
If z+w is real then (here, Im(z) stands for the imaginary part of z), so b = -y. Also, if zw is real, then

. We can replace y with -b.



Since a = x and y = -b, z and w are conjugates of each other.