Q8P - 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.

3 .

Step-by-Step Solution

Answer

The required values are mentioned below:

$x=3,y=0,r=3,\theta =0$ 

 

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

Step 1: Given Information

The complex number is 3 .

Step 2: Definition of the Complex number

Imaginary and real numbers are both required to represent a complex number.

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:

 3 = x +iy

 

Compare both sides, and we get,

 x = 3

 y = 0

Find the value of r .

The formula is:

 $r=\sqrt{x^2+y^2}$

 

Substituting the values, and we get,

 $$\begin{gathered} r=\sqrt{9+0} \\ =3 \end{gathered}$$

 

Find the value of  $\theta$.

The formula is:

$\theta ={\cos }^{-1}\frac{x}{r}$ 

 

Substituting the values, and we get,

$$\begin{gathered} \theta ={\cos }^{-1}\left(\frac{3}{3}\right) \\ =0 \end{gathered}$$

The value of the number becomes as follows:

$$\begin{gathered} 3=3\left(\cos \left(0\right)+i \sin (0\right) \\ =3e^{i0} \end{gathered}$$ 

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

 

The required values are mentioned below:

 $x=3,y=0,r=3,\theta =0$