Q31P - 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.

 $\frac{\mathbf{5}\mathbf{-}\mathbf{2}\mathbf{i}}{\mathbf{5}\mathbf{+}\mathbf{2}\mathbf{i}}$.

Step-by-Step Solution

Answer

The absolute value is $\left|z\right|=1$ .

Step 1: Given Information

The complex number is $\frac{5-2i}{5+2i}$.

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 $\frac{5-2i}{5+2i}$ .

 

The absolute value can be calculated as below:

$$\begin{gathered} \left|z\right|=\left|\frac{5-2i}{5+2i}\right| \\ =\left|\frac{5-2i}{5+2i}\right| \\ =\frac{\sqrt{25+4}}{\sqrt{25+4}} \\ \left|z\right|=1 \end{gathered}$$

 

Thus, the final answer is obtained as:

 $\left|z\right|=1$.