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Section 7.1

By cssp

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.

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