Question

Prove that the sum and product of two complex numbers are real if and only if they are conjugate of each other.

Answer 1

Let $z_1=x_1+{iy}_1$ and $z_2=x_2+{iy}_2$ be two complex numbers.
First suppose $z_1,z_2$ are conjugate of each other. Then,
$z_2={\overline{z}}_1=x_1+{iy}_1$
Hence $z_1+z_2={z_1+\overline{z}}_1=2x_1$, which is real and $z_1{z}_2={z_1\overline{z}}_1=(x_1+{iy}_1)(x_1-{iy}_1)={x_1}^2+{y_1}^2$, which is also real.
Thus sum $z_1+z_2$ and product $z_1{z}_2$ are real when $z_1$ and $z_2$ are conjugate complex.
Now let the sum $z_1+z_2$ and product $z_1{z}_2$ be real. We have $z_1+z_2=(x_1+x_2)+i(y_1+y_2)$
and $z_1{z}_2=x_1{x}_2-y_1{y}_2+i(x_1{y}_2+x_2{y}_1)$
Since $z_1+z_2$ and $z_1{z}_2$ are both real, we must have $y_1+y_2=0$ and $x_1{y}_2+x_2+y_1=0$
These give $y_2=-y_1$ and $x_2=x_1.$
Hence $z_2=x_2+{iy}_2=x_1-{iy}_1={\overline{z}}_1$, that is, $z_1$ and $z_2$ are conjugate complex.