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

Explanation

we have to let`z_1 =x_1 +iy_1` and `z_2 =x_2+iy_2` be two complex numbers.

Suppose `z_1,z_2` are conjugate of each other. Then,`z_2=bar{z_1} =x_1+iy_1`

so, `z_1+z_2=z_1+ bar(z_1) =2x_1` , which is real and `z_1z_2=z_1bar(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_1z_2`are real when `z_1` and `z_2` are conjugate complex.

Now let the sum`z_1+z_2` and product `z_1z_2`be real. We have `z_1+z_2=(x_1+x_2)+i(y_1+y_2 )`

and `z_1z_2=x_1x_2−y_1y_2+i(x_1y_2+x_2y_1)`

Since `z_1+z_2` and` z_1z_2` are both real, we must have `y_1+y_2=0` and `x_1y_2+x_2+y_1=0`

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