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.

$\overline{z}+\overline{w}=\overline{z+w}$

Short answer

It is shown that $\overline{z}+\overline{w}=\overline{z+w}$.

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,

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

Step 3. Evaluate the expression.

In order to show that the given statement is true, we will solve the left hand side expression and right hand expression and show that both are equal.

To find the value of the left hand side expression, first find the complex conjugates of the given complex numbers and then add them as shown below

$$\begin{gathered} \overline{z}+\overline{w}=\left(\overline{a+bi}\right)+\left(\overline{c+di}\right) \\ =\left(a-bi\right)+\left(c-di\right) \\ =\left(a+c\right)+\left(-b+\left(-d\right)\right)i \\ =\left(a+c\right)-\left(b+d\right)i \end{gathered}$$

Now, to solve the right hand side expression, first add the given numbers and then find the complex conjugate of the resulting sum as shown below

$$\begin{gathered} \overline{z+w}=\overline{\left(a+bi\right)+\left(c+di\right)} \\ =\overline{\left(a+c\right)+\left(b+d\right)i} \\ =\left(a+c\right)-\left(b+d\right)i \end{gathered}$$

Since the left hand side expression is equal to the right hand side expression, so we can conclude that $\overline{z}+\overline{w}=\overline{z+w}$.