If the product and the sum of two complex numbers are real, what can we say about the numbers? If the product and the sum of two complex numbers are real, what can we say about the numbers? Prove it.


Explanation will be very helpful. Actually the question asks for a proof. Hence, a proof is all that we need. I actually am quite confused with it. I have proved a lot more complex questions in complex numbers and this little proof blocked my brain. Please stop this block.

Thanks in advance!
Let the two complex numbers be $z = x + iy$ and $w = u + iv$.
Then, calculate $zw$ and $z+w$ and set the imaginary parts to zero.
If $r$ and $s$ are your complex numbers then the polynomial $(z-r)(z-s) = z^2-(r+s)z +rs$.  The coefficients are the sum and product and must be real.  So the polynomial is of the form $z^2+az+b$ with $a$ and $b$ real.   If you set it equal to zero, and solve, the roots (which are $r$ and $s$) must be conjugates.