Q30P - 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.

 $\frac{\mathbf{3}\mathbf{i}}{\mathbf{i}\mathbf{-}\sqrt{\mathbf{3}}}$.

Step-by-Step Solution

Answer

The absolute value is $\left|z\right|=\frac{3}{2}$ .

The absolute value is .

The complex number is $\frac{3i}{i-\sqrt{3}}$.

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:

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

Step 3: Find the value

The complex number is $\frac{3i}{i-\sqrt{3}}$ .

$$\begin{gathered} \left|z\right|=\left|\frac{3i}{i-\sqrt{3}}\right| \\ \left|z\right|=\left|\frac{z_1}{z_2}\right| \end{gathered}$$

The absolute value can be calculated as below:

$$\begin{gathered} \left|z_1\right|=\sqrt{0^2+3^2} \\ \left|z_1\right|=3 \\ \left|z_1\right|=\sqrt{1+{\left(\sqrt{3}\right)}^2} \\ \left|z_1\right|=2 \end{gathered}$$

 

Thus, the final answer is obtained as:

 $\left|z\right|=\frac{3}{2}$.