Chapter 2: Complex Numbers

Show that the sum of the n nth roots of any complex number is zero.

Find each of the following in the x + iyform and check your answers by computer.

$\mathbf{sinh}\left(\ln 2+\frac{i\pi }{3}\right)$

Use Problems 27 and 28 to find the following absolute values. If you understand Problems 27 and 28 and equation (5.1), you should be able to do these in your head.

$\left|5e^{2\mathrm{\pi i}/3}\right|$

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 each of the following in the x + iyform and check your answers by computer.

$\mathbf{cosh}\left(\frac{\mathrm{i\pi }}{2}-\ln 3\right)$

The three cube roots of +1 are often called 1,$\mathbf{\omega }$, and ${\mathbf{\omega }}^{\mathbf{2}}$. Show that this is reasonable, that is, show that the cube roots of +1 are +1 and two other numbers, each of which is the square of the other.

Use Problems 27 and 28 to find the following absolute values. If you understand Problems 27 and 28 and equation (5.1), you should be able to do these in your head.

$\left|3e^{2+4i}\right|$

Use Problems 27 and 28 to find the following absolute values. If you understand Problems 27 and 28 and equation (5.1), you should be able to do these in your head.$\mathbf{\left|}\mathbf{2}{\mathbf{e}}^{\mathbf{3}\mathbf{+}\mathbf{i\pi }}\mathbf{\right|}$

Find each of the following in the x + iyform and check your answers by computer.

tani

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.

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