« Section 6.7
Section 7.2 »

Section 7.1

Please post comments if you find any errors, or if you wish to contribute answers for additional problems. You may find information on using \LaTeX here.

1. (a) w + z = 1 + 3i

(c) 3w - 2z = 13 - 26i

(e) zw = 22 + 29i

(f) \displaystyle{\frac{z}{w} = -\frac{34}{25} + \frac{13}{25}i}

(g) |z| = \sqrt{53}

(i) \Re(z - w) = -5

(j) \Im(3z + w) = 17

2. (a) \displaystyle{\Re\left(\frac{1}{i}\right) = 0}, \displaystyle{\Im\left(\frac{1}{i}\right) = -1}

(c) \displaystyle{\Re\left(\frac{3-4i}{-2+3i}\right) = -\frac{18}{13}}, \displaystyle{\Im\left(\frac{3-4i}{-2-3i}\right) = -\frac{1}{13}}

3. (a) z = 3i

(c) \displaystyle{z = -\frac{1}{2\sqrt{2}} - \frac{1}{2\sqrt{2}}i}

4. (a) |z| = 1, \mathrm{Arg}(z) = -\dfrac{\pi}{2}

(c) |z| = \sqrt{2} , \mathrm{Arg}(z) = \dfrac{\pi}{4}

(e) |z| = 4 , \mathrm{Arg}(z) = \dfrac{\pi}{3}

5. (a) \left|w^2\right| = 9 , \mathrm{Arg}\left(w^2\right) = \dfrac{\pi}{3} , \displaystyle{w^2 = \frac{9}{2} + \frac{9\sqrt{3}}{2}i}

(c) |wz| = 6 , \mathrm{Arg}(wz) = -\dfrac{\pi}{6} , \displaystyle{wz = 3\sqrt{3} - 3i}

(e) \left|\dfrac{z}{w^2}\right| = \dfrac{2}{9} , \mathrm{Arg}\left(\dfrac{z}{w^2}\right) = -\dfrac{2\pi}{3} , \displaystyle{\frac{z}{w^2} = -\frac{1}{9} - \frac{1}{3\sqrt{3}}i}

(f) \left|w^5\right| = 243 , \mathrm{Arg}\left(w^5\right) = \dfrac{5\pi}{6} , \displaystyle{w^5 = -\frac{243\sqrt{3}}{2} + \frac{243}{2}i}

6. If z = \dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i , then the roots are 1, z, z^2, z^3, z^4 , and z^5 .

8. (c) The square roots of 1 + \sqrt{3}i are \sqrt{\dfrac{3}{2}} + \dfrac{1}{\sqrt{2}}i and -\sqrt{\dfrac{3}{2}} - \dfrac{1}{\sqrt{2}}i .
The square roots of -9 are 3i and -3i .

This entry was posted on 12 August 12 2007 at 9:19 pm and is filed under Section 7.1. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

Leave a Reply