Q28P - Mathematical Methods in Physical Sciences

Question

Find the absolute value of each of the following using the discussion above. Try to do simple problems like these in your head-it saves time.

 $\frac{\mathbf{z}}{\overline{\mathbf{z}}}$.

Step-by-Step Solution

Answer

The absolute value is 1.

Step 1: Given Information

The complex number is $\frac{z}{\overline{z}}$.

Step 2: Absolute value of the Complex number

Let z be a complex number consisting of two real numbers as:

$\mathit{z}\mathbf{=}\mathit{x}\mathbf{+}{}_{\mathbf{z}}\mathit{y}$

 

Here x and y are the real numbers.

 

The absolute value of z can be calculated as:

$\mathbf{\left|}\mathbf{z}\mathbf{\right|}\mathbf{=}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{=}{\mathbf{y}}^{\mathbf{2}}}$

Step 3: Find the value

The complex number is $\frac{z}{\overline{z}}$ .

 

The absolute value can be calculated as below:

$$\begin{gathered} \left|z\right|=\left|\frac{x+yi}{x-yi}\right| \\ =\left|\frac{x+yi}{x-yi}\right| \\ =\frac{\sqrt{x^2+y^2}}{\sqrt{x^2=y^2}} \\ \left| z\right|=1 \end{gathered}$$

  

Thus, the final answer is obtained as:

$\left|z\right|=1$