Chapter 6.8, Problem 17E

To determine

If $\left(w\right)\text{ and }\left(z\right)$ are complex numbers then show that $\overline{w-z}=\overline{w}-\overline{z}$

Explanation of Solution

Given information: The given complex numbers are $\left(w\right)\text{ and }\left(z\right)$ .

Proof: Consider that $w=w_1+i{w}_2$ and $z=z_1+i{z}_2$ where $w_1,w_2,z_1\text{ and }{z}_2$ belongs to real numbers. Now evaluate $\overline{w-z}$ with the help of $w=w_1+i{w}_2$ and $z=z_1+i{z}_2$ .

  $\begin{array}{c}w-z=\left(w_1+i{w}_2\right)-\left(z_1+i{z}_2\right)\\ w-z=\left(w_1-z_1\right)+i\left(w_2-z_2\right)\\ \overline{w-z}=\overline{\left(w_1-z_1\right)+i\left(w_2-z_2\right)}\mathrm{....}\left(\text{Conjugate}\right)\\ \overline{w-z}=\left(w_1-z_1\right)-i\left(w_2-z_2\right)\mathrm{....}\left(1\right)\end{array}$

Similarly evaluate $\overline{w}+\overline{z}$ ,

  $\begin{array}{c}\overline{w}-\overline{z}=\overline{\left(w_1+i{w}_2\right)}-\overline{\left(z_1+i{z}_2\right)}\\ \overline{w}-\overline{z}=\left(w_1-i{w}_2\right)-\left(z_1-i{z}_2\right)\\ \overline{w}-\overline{z}=\left(w_1-z_1\right)-i\left(w_2-z_2\right)\mathrm{....}\left(2\right)\end{array}$

Hence combined (1) and (2) it is proved that $\overline{w-z}=\overline{w}-\overline{z}$ .