Chapter 3.5, Problem 80E

To determine

To prove: the provided statement.

Explanation of Solution

Given information: it is provided that z = a + bi and w = c + di , the statement provided in the question is,

  $z=\overline{z}$ if and only if a real number.

Proof:

Rewrite the left-hand side using a and b,

  $z=a+bi$

Since, z is real, b = 0,

  $z=a+0i$

  $z=a$ ……(1)

Now, rewrite the right-hand side using a and b

  $\overline{z}=\overline{\left(a+bi\right)}$

Complex conjugate of $\overline{\left(a+bi\right)}=a-bi$

Take the conjugate as,

  $\begin{array}{l}\overline{\left(a+bi\right)}=\left(a-bi\right)\\ \overline{z}=a-bi\end{array}$

Since, z is real, b = 0,

  $\overline{z}=a-0i$

  $\overline{z}=a$ ……(2)

Compare (1) and (2),

  $z=\overline{z}$

Hence, proved that $z=\overline{z}$ if and only if z is a real number.