Question

Recall that the symbol $\overline{z}$represents the complex conjugate of z. If $z=a+bi\text{and}w=c+di$, show that each statement is true.

$z+\overline{z}\text{isarealnumber.}$

Short answer

It is shown that $z+\overline{z}$is a real number.

Step-by-step solution

Step 1. Concept of complex conjugate.

The complex conjugate of the complex number$a+bi$ is represented by a horizontal bar over it, that is, the complex conjugate of$a+bi$ is denoted by $\overline{a+bi}$.

The complex conjugate of any complex number can be obtained by changing the sign of the imaginary part of the complex number. The complex conjugate of $a+bi$is $a-bi$.

The product of a complex number and its complex conjugate is given by

$\left(a+bi\right)\left(a-bi\right)=a^2+b^2$

The imaginary number i is defined as the square root of negative one. That is, $i^2=-1$.

Step 2. Concept of addition of complex numbers.

To add or subtract two complex numbers, simply add/subtract the real parts together and the imaginary parts together. That is,

$\left(a+bi\right)+\left(c+di\right)=\left(a+c\right)+\left(b+d\right)i$

To multiply two complex numbers, multiply them as binomial expressions and use the result of imaginary number, $i^2=-1$.

Step 3. Evaluate the expressions.

In order to show that the given statement is true, find the complex conjugate of z and add it with the complex number z as shown below

$$\begin{gathered} z+\overline{z}=\left(a+bi\right)+\left(\overline{a+bi}\right) \\ =\left(a+bi\right)+\left(a-bi\right) \\ =\left(a+a\right)+\left(b+\left(-b\right)\right)i \\ =\left(2a\right)+\left(0\right)i \\ =2a \end{gathered}$$

Thus, the sum of$z+\overline{z}$ is a real number.