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.Com
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) (Show Source): You can put this solution on YOUR website!
If z+w is real then  = 0 \Rightarrow b+y = 0)
(here, Im(z) stands for the imaginary part of z), so b = -y. Also, if zw is real, then
(x+yi)) = 0)
. We can replace y with -b.
(x-bi)) = 0 \Rightarrow -abi + bxi = 0 \Rightarrow a = x)
Since a = x and y = -b, z and w are conjugates of each other.
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