Question

If $z_1$ and $z_2$ are complex numbers, such that $z_1+z_2$ is a real number, then

• A. $z_1=-\overline{z_2}$
• B. $z_2=\overline{z_1}$
• C. $z_1$ and $z_1$ are any two complex numbers
• D. $z_1=\overline{z_1},z_2=\overline{z_2}$

Answer 1

The correct option is B $z_2=\overline{z_1}$

Let $z_1=x_1+i{y}_1$

$\therefore \overline{z_1}=x_1-i{y}_1$

Now $z_2=\overline{z_1}\Rightarrow z_1+z_2=z_1+\overline{z_1}$

$=x_1+i{y}_1+x_1-i{y}_1=2x_1$ which is real

Hence, results holds if $z_2=\overline{z_1}$.