Q9P - Mathematical Methods in Physical Sciences

Question

For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly find $\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{r}\mathbf{,}\mathit{\theta }$  in your head for these simple problems. Then plot the number and label it in five ways as in Figure 3.3. Also plot the complex conjugate of the number.

 2i-2. 

Step-by-Step Solution

Answer

The required values are mentioned below:

 $x=-2,y=2,r=2\sqrt{2},\theta =\frac{3\pi }{4}$

 

The graph of the number and its conjugate is shown below:

Step 1: Given Information

The complex number is 2i -2 .

Step 2: Definition of the Complex number

A complex number is a number that comprises both real and imaginary numbers.

Step 3: Find the value

The formula is mentioned below:

 $$\begin{gathered} x+iy=r\left(\cos \theta +i \sin \theta \right) \\ =r{e}^{i\theta } \end{gathered}$$

 

The complex number is:

2i -2 = x + iy

width="93" style="max-width: none; vertical-align: -4px;" $2i-2=x+iy$ 

 Compare both sides, and we get,

 $$\begin{gathered} x=-2 \\ y=2 \end{gathered}$$$$\begin{gathered} x=-2 \\ y=2 \end{gathered}$$

 

Find the value of r .

The formula is:

$r=\sqrt{x^2+y^2}$.

 

Substituting the values, and we get,

 $$\begin{gathered} r=\sqrt{4+4} \\ =2\sqrt{2} \end{gathered}$$

 

Find the value of $\theta$ .

The formula is:

 $\theta ={\cos }^{-1}\frac{x}{r}$.

 

Substituting the values, and we get,

$$\begin{gathered} \theta ={\cos }^{-1}\left(\frac{-2}{2\sqrt{2}}\right) \\ =\frac{3\pi }{4} \end{gathered}$$ 

The value of the number becomes as follows.

 $$\begin{gathered} 2i-2=2\sqrt{2}\left(\cos \left(\frac{3\pi }{4}\right)+i \sin \left(\frac{3\pi }{4}\right)\right) \\ =2\sqrt{2}\left(e^{i3\pi /4}\right) \end{gathered}$$

 

The graph of the complex number(blue) and its conjugate(red) is shown below:

The required values are mentioned below:

 

$x=2,y=2,r=2\sqrt{2},\theta =\frac{3\pi }{4}$