Q27P - 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.

 $\frac{\mathbf{2}\mathbf{+}\mathbf{3}\mathbf{i}}{\mathbf{1}\mathbf{-}\mathbf{i}}$.

Step-by-Step Solution

Answer

The absolute value is $\left|z\right|=\frac{\sqrt{13}}{\sqrt{2}}$ .

Step 1: Given Information

The complex number is $\frac{2+3i}{1-i}$.

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 $\frac{2+3i}{1-i}$ .

 

The absolute value can be calculated as below:

$$\begin{gathered} \left|z\right|=\left|\frac{2+3i}{1-i}\right| \\ =\left|\frac{2+3i}{1-i}\right| \\ =\frac{\sqrt{4+9}}{\sqrt{1+1}} \\ \left|z\right|=\frac{\sqrt{13}}{\sqrt{2}} \end{gathered}$$ 

Thus, the final answer is obtained as:

$\left|z\right|=\frac{\sqrt{13}}{\sqrt{2}}$ .