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.

2-2i.

Short answer

The required values are mentioned below:

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

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

Step-by-step solution

Given Information

The complex number is 2-2i .

Definition of the Complex number

A complex number has both real and imaginary numbers.

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:

2-2i = x + iy

Compare both sides, and we get,

x = 2

y = -2

Find the value of .

The formula is:

localid="1658474350436" $r=\sqrt{x^2+y^2}$

Substituting the values, and we get,

localid="1658474354105" $$\begin{gathered} r=\sqrt{4+4} \\ =2\sqrt{2} \end{gathered}$$

Find the value of $\theta$.

The formula is:

localid="1658474357878" $\theta ={\cos }^{-1}\frac{x}{r}$

Substituting the values, and we get,

localid="1658474361473" $$\begin{gathered} \theta ={\sin }^{-1}\left(\frac{-2}{2\sqrt{2}}\right) \\ =\frac{\pi }{4} \end{gathered}$$

The value of the number becomes as follows:

localid="1658474365280" $$\begin{gathered} 2-2i=2\sqrt{2}\left(\cos \left(\frac{\pi }{4}\right)-i\sin \left(\frac{3\pi }{4}\right)\right) \\ =2\sqrt{2}\left(e^{-i\pi /4}\right) \end{gathered}$$t

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

The required values are mentioned below:

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