Chapter 2: Complex Numbers

Show that ${\mathbf{\left(}{\mathbf{a}}^{\mathbf{b}}\mathbf{\right)}}^{\mathbf{c}}$ can have more values than.${{\mathit{a}}^{\mathbf{b}}}^{\mathbf{c}}$As examples compare

1. ${\mathbf{\left[}{\mathbf{(}\mathbf{-}\mathbf{i}\mathbf{)}}^{\mathbf{2}\mathbf{+}\mathbf{i}}\mathbf{\right]}}^{\mathbf{2}\mathbf{-}\mathbf{i}}\mathbf{\text{and}}{{\mathbf{(}\mathbf{-}\mathbf{i}\mathbf{)}}^{\mathbf{(}\mathbf{2}\mathbf{+}\mathbf{i}\mathbf{)}}}^{\mathbf{(}\mathbf{2}\mathbf{-}\mathbf{i}\mathbf{)}}\mathbf{=}{\mathbf{(}\mathbf{-}\mathbf{i}\mathbf{)}}^{\mathbf{5}}$;
2. ${\mathbf{\left(}{\mathbf{i}}^{\mathbf{i}}\mathbf{\right)}}^{\mathbf{i}}\mathbf{\text{and}}{\mathbf{(}\mathbf{-}\mathbf{i}\mathbf{)}}^{\mathbf{-}\mathbf{1}}$.

Find the values of the indicated roots.

$\sqrt[\mathbf{8}]{\frac{\mathbf{-}\mathbf{1}\mathbf{-}\mathbf{I}\sqrt{\mathbf{3}}}{\mathbf{2}}}$

Find the real part, the imaginary part, and the absolute value of

COS(ix)

Find the values of the indicated roots:

$\sqrt[\mathbf{5}]{\mathbf{-}\mathbf{1}\mathbf{-}\mathbf{i}}$

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 rectangularform .$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$

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 writtenas .$\mathbf{2}\mathbf{cos}\mathbf{\left(}\frac{\pi }{9}\mathbf{\right)}\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{cos}\mathbf{\left(}\frac{2\pi }{9}\mathbf{\right)}\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{cos}\mathbf{\left(}\frac{4\pi }{9}\mathbf{\right)}$

Question: Find the real part, the imaginary part, and the absolute value of

$\mathbf{sin}\mathbf{(}\mathit{x}\mathbf{-}\mathit{y}\mathbf{i}\mathbf{)}$.

Question: Find the real part, the imaginary part, and the absolute value of

cosh(2 - 3i) .

Find the values of the indicated roots.

$\sqrt[\mathbf{5}]{\mathbf{i}}$

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.

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