Chapter I, Problem 6E

To determine

To evaluate: The expression $\overline{2i\left(\frac{1}{2}-i\right)}$ in the form of $a+bi$.

Answer to Problem 6E

The value of the expression $\overline{2i\left(\frac{1}{2}-i\right)}$ is $2-i$.

Explanation of Solution

Use the property of conjugates $\overline{z+w}=\overline{z}+\overline{w}$ and $\overline{zw}=\overline{z}\overline{w}$ to simplify the expression as follows.

$\begin{array}{c}\overline{2i\left(\frac{1}{2}-i\right)}=\overline{2i}\overline{\left(\frac{1}{2}-i\right)}\\ =\overline{2}\left(\overline{i}\right)\left[\overline{\left(\frac{1}{2}\right)}-\left(\overline{i}\right)\right]\\ =-2i\left[\frac{1}{2}-\left(-i\right)\right]\\ =-2i\left(\frac{1}{2}\right)+\left(-2i\right)\left(i\right)\\ =2-i\end{array}$

Therefore, the value of the expression $\overline{2i\left(\frac{1}{2}-i\right)}$ is $2-i$.