Chapter 6.8, Problem 14E

To determine

Prove that $z+\overline{z}$ is twice the real part of $\left(z\right)$ .

Explanation of Solution

Given information: The given complex number is $\left(z\right)$ .

Proof: Consider that $z=x+iy$ where $x,y$ belongs to real numbers.

  $\begin{array}{c}z=x+iy\mathrm{....}\left(\text{Assumption}\right)\mathrm{....}\left(1\right)\\ \text{such that, }\overline{z}=\overline{x+iy}\mathrm{....}\left(\text{Conjugate}\right)\\ \overline{z}=x-iy\mathrm{....}\left(2\right)\end{array}$

Adding (1) and (2) gives,

  $\begin{array}{c}z+\overline{z}=\left(x+iy\right)+\left(x-iy\right)\\ z+\overline{z}=2x\mathrm{....}\left(x\in R\right)\mathrm{....}\left(3\right)\end{array}$

From (3) $z+\overline{z}=2x$ where $\left(x\right)$ belongs to real number. Hence proved that $\left(z+\overline{z}\right)$ is twice the real $\left(x\right)$ part of $\left(z\right)$ .