Complex Numbers

Question:First simplify each of the following numbers to the x+iy form or to the $\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}$ form. Then plot the number in the complex plane.

$\mathbf{3}\mathbf{(}\mathit{c}\mathit{o}\mathit{s}\mathbf{28}\mathbf{°}\mathbf{+}\mathit{i}\mathit{s}\mathit{i}\mathit{n}\mathbf{28}\mathbf{°}\mathbf{)}$ .

Use a computer to find the three solutions of the equation${\mathbf{x}}^3\mathbf{-}\mathbf{3}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{=}\mathbf{0}$  . Find a way to show that the solutions can be written as . $\mathbf{2}\mathbf{cos}\mathbf{\left(}\frac{\pi }{9}\mathbf{\right)}\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{cos}\mathbf{\left(}\frac{2\pi }{9}\mathbf{\right)}\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{cos}\mathbf{\left(}\frac{4\pi }{9}\mathbf{\right)}$

Question: Describe geometrically the set of points in the complex plane satisfying the following equations.  .

For each of the following numbers, first, visualize where it is in the complete 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 numbers 1+i .

For each of the following numbers, first visualize where it is in the complex plane. With a little practice, you can quickly find $\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{r}\mathbf{,}\mathit{\theta }\mathbf{ }$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 i-1.

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 $\mathbf{1}\mathbf{-}\mathit{I}\sqrt{\mathbf{3}}$.

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.

-4i . 

For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly find x,y,r,$\theta$  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.

 -1.

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 .

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. 

 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.

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.

 $\mathbf{2}\mathbf{(}\mathbf{cos}\frac{\mathbf{\pi }}{\mathbf{6}}\mathbf{+}\mathbf{i}\mathbf{ }\mathbf{sin}\frac{\mathbf{\pi }}{\mathbf{6}}\mathbf{)}$. 

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.

 $\mathbf{4}\mathbf{(}\mathbf{cos}\frac{\mathbf{2}\mathbf{\pi }}{\mathbf{3}}\mathbf{+}\mathbf{i}\mathbf{ }\mathbf{sin}\frac{\mathbf{2}\mathbf{\pi }}{\mathbf{3}}\mathbf{)}$. 

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.

$\mathit{(}\mathit{c}\mathit{o}\mathit{s}\frac{\mathbf{3}\mathbf{\pi }}{\mathbf{2}}\mathit{+}\mathit{i}\mathit{ }\mathit{s}\mathit{i}\mathit{n}\frac{\mathbf{3}\mathbf{\pi }}{\mathbf{2}}\mathit{)}$

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.

  $\mathbf{2}\left(cos\frac{\pi }{4}+i sin\frac{\pi }{4}\right)$. 

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.

$\left(cos\mathit{\pi }\mathit{-}\mathit{i}\mathit{ }\sin \mathit{\pi }\right)$. 

For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly find  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.

 $\mathbf{5}\mathbf{(}\mathbf{cos}\mathbf{0}\mathbf{-}\mathit{i}\mathbf{sin}\mathbf{0}\mathbf{)}$. 

For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly find  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.

 $\sqrt{\mathbf{2}}{\mathit{e}}^{\mathbf{-}\mathbf{i}\frac{\mathbf{\pi }}{\mathbf{4}}}$. 

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.

$\mathbf{3}{\mathit{e}}^{{\mathbf{i}}_{\mathbf{2}}^{\mathbf{x}}}$ .

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.

$\mathbf{5}\mathbf{(}\mathit{c}\mathit{o}\mathit{s}{\mathbf{20}}^{\mathbf{°}}\mathbf{+}\mathit{i}\mathit{s}\mathit{i}\mathit{n}{\mathbf{20}}^{\mathbf{°}}\mathbf{)}$ .

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.

$\mathbf{7}\mathbf{(}\mathit{c}\mathit{o}\mathit{s}{\mathbf{110}}^{\mathbf{°}}\mathbf{-}\mathit{i}\mathit{s}\mathit{i}\mathit{n}{\mathbf{110}}^{\mathbf{°}}\mathbf{)}$ .

First simplify each of the following numbers to the  $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$ form or to the  $\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}$ form. Then plot the number in the complex plane.

${\left(\frac{1+i}{1-i}\right)}^{\mathbf{2}}$ .

Question:First simplify each of the following numbers to the x+iy form or to the $\mathit{r}{\mathit{e}}^{\mathbf{i\theta }}$ form. Then plot the number in the complex plane.

$\left(\frac{3+i}{2+i}\right)$

Question:First simplify each of the following numbers to the x+iy form or to the $\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}$ form. Then plot the number in the complex plane.

1.6-2.7i .

Question:First simplify each of the following numbers to the x+iy form or to the $\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}$ form. Then plot the number in the complex plane.

17-12i .

First simplify each of the following numbers to the$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$   form or to the   form. Then plot the number in the complex plane.  $\mathbf{5}\mathbf{(}\mathbf{c}\mathbf{o}\mathbf{s}\frac{\mathbf{2}\mathbf{\pi }}{\mathbf{5}}\mathbf{+}\mathbf{i}\mathbf{s}\mathbf{i}\mathbf{n}\frac{\mathbf{2}\mathbf{\pi }}{\mathbf{5}}\mathbf{)}$

 .

Question: First simplify each of the following numbers to the x+iy form or to the $\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}$ form. Then plot the number in the complex plane.

$\mathbf{2}\mathbf{.}\mathbf{8}{\mathit{e}}^{\mathbf{-}\mathbf{i}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{1}\mathbf{)}}$ .

Question: First simplify each of the following numbers to the x+iy form or to the $\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}$ form. Then plot the number in the complex plane.

$\frac{\mathbf{5}\mathbf{-}\mathbf{2}\mathbf{i}}{\mathbf{5}\mathbf{+}\mathbf{2}\mathbf{i}}$ .

Question:First simplify each of the following numbers to the x+iy form or to the $\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}$ form. Then plot the number in the complex plane.

$\frac{\mathbf{1}}{\mathbf{0}\mathbf{.}\mathbf{5}\mathbf{(}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{40}\mathbf{°}\mathbf{+}\mathbf{i}\mathbf{ }\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{40}\mathbf{°}}$ .

Question:First simplify each of the following numbers to the x+iy form or to the $\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}$ form. Then plot the number in the complex plane.

${\left(1.7-3.2i\right)}^{\mathbf{2}}$ .

First simplify each of the following numbers to the $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$  form or to the $\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}$   form. Then plot the number in the complex plane.

${\mathbf{(}\mathbf{0}\mathbf{.64}\mathbf{+}\mathbf{0}\mathbf{.77}\mathbf{i}\mathbf{)}}^{\mathbf{4}}$ .

Find each of the following in rectangular $\mathit{a}\mathbf{+}{}_{\mathbf{z}}\mathit{b}$ form $\mathit{i}\mathit{f}\mathbf{ }\mathit{z}\mathbf{=}\mathbf{2}$if ; if $\mathit{z}\mathbf{=}\mathit{x}\mathbf{+}{}_{\mathbf{z}}\mathit{y}$ .

  $\frac{\mathbf{1}}{\mathbf{z}\mathbf{+}\mathbf{1}}$.

Find each of the following in rectangular $\mathit{a}\mathbf{+}\mathit{z}\mathit{b}$  form if  $\mathit{z}\mathbf{=}\mathbf{2}\mathbf{-}\mathit{z}\mathbf{3}$ ; if $\mathit{z}\mathbf{=}\mathit{x}\mathbf{+}\mathit{z}\mathit{y}$ .

$\frac{\mathbf{1}}{\mathbf{z}\mathbf{-}\mathbf{1}}$

Find each of the following in rectangular $\mathit{a}\mathbf{+}\mathit{z}\mathit{b}$ form if  $\mathit{z}\mathbf{=}\mathbf{2}\mathbf{-}{}_{\mathbf{t}}\mathbf{3}$ ; if $\mathit{z}\mathbf{=}\mathit{x}\mathbf{+}{}_{\mathbf{z}}\mathit{y}$ .

$\frac{\mathbf{1}\mathbf{+}\mathbf{z}}{\mathbf{1}\mathbf{-}\mathbf{z}}$.

Find each of the following in rectangular $\mathit{a}\mathbf{+}{}_{\mathbf{z}}\mathit{b}$  form if  $\mathit{z}\mathbf{=}\mathbf{2}\mathbf{-}{}_{\mathbf{t}}\mathbf{3}$ ; if $\mathit{z}\mathbf{=}\mathit{x}\mathbf{+}{}_{\mathbf{z}}\mathit{y}$ .

 $\frac{\mathbf{z}}{\mathbf{z}}$.

Prove that the conjugate of the quotient of two complex numbers is the quotient of the conjugates. Also prove the corresponding statements for difference and product. Hint: It is easier to prove the statements about product and quotient using the polar coordinate form; for the difference, $\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}$it is easier to use the rectangular form .$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$

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.

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

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.

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

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.

 $\frac{\mathbf{z}}{\overline{\mathbf{z}}}$.

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.

 ${\left(1+2\mathit{i}\right)}^{\mathbf{3}}$.

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.

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

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.

 $\frac{\mathbf{5}\mathbf{-}\mathbf{2}\mathbf{i}}{\mathbf{5}\mathbf{+}\mathbf{2}\mathbf{i}}$.

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.

 ${\left(2-3i\right)}^{\mathbf{4}}$.

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.

 $\frac{\mathbf{25}}{\mathbf{3}\mathbf{+}\mathbf{4}\mathbf{i}}$.

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.

${\mathbf{\left(}\frac{\mathbf{1}\mathbf{+}\mathbf{i}}{\mathbf{1}\mathbf{-}\mathbf{i}}\mathbf{\right)}}^{\mathbf{5}}$

Solve for all possible values of the real numbers $\mathit{x}$  and  $\mathit{y}$ in the following equations.

 $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}\mathbf{=}\mathbf{3}\mathit{i}\mathbf{-}\mathbf{4}$

Solve for all possible values of the real numbers  and in the following equations.

$\mathbf{2}\mathit{i}\mathit{x}\mathbf{+}\mathbf{3}\mathbf{=}\mathit{y}\mathbf{-}\mathit{i}$

Solve for all possible values of the real numbers x and y in the following equations.

$\mathbf{2}\mathit{i}\mathit{x}\mathbf{+}\mathbf{3}\mathbf{=}\mathit{y}\mathbf{-}\mathit{i}$

Solve for all possible values of the real numbers $\mathit{x}$ and $\mathit{y}$ in the following equations.

$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}\mathbf{=}\mathbf{0}$

Solve for all possible values of the real numbers  and in the following equations.

$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}\mathbf{=}\mathbf{2}\mathit{i}\mathbf{-}\mathbf{7}$