21.3: Exercises

Exercise $\PageIndex{1}$

Plot the complex numbers in the complex plane.

1. $4+2i$
2. $-3-5i$
3. $6-2i$
4. $-5+i$
5. $-2i$
6. $\sqrt{2}-\sqrt{2}i$
7. $7$
8. $i$
9. $0$
10. $2i-\sqrt{3}$

Answer

Exercise $\PageIndex{2}$

Add, subtract, multiply, and divide, as indicated.

1. $(5-2i)+(-2+6i)$
2. $(-9-i)-(5-3i)$
3. $(3+2i)\cdot (4+3i)$
4. $(-2-i)\cdot (-1+4i)$
5. $\dfrac{2+3i}{2+i}$
6. $(5+5i)\div (2-4i)$

Answer

1. $3+4 i$
2. $-14+2 i$
3. $6+17 i$
4. $6-7 i$
5. $\dfrac{7}{5}+\dfrac{4}{5} i$
6. $-\dfrac{1}{2}+\dfrac{3}{2} i$

Exercise $\PageIndex{3}$

Find the absolute value $|a+bi|$ of the given complex number, and simplify your answer as much as possible.

1. $|4+3i|$
2. $|1-2i|$
3. $|-3i|$
4. $|-2-6i|$
5. $|\sqrt{8}-i|$
6. $|-2\sqrt{3}-2i|$
7. $|-5|$
8. $|-\sqrt{17}+4\sqrt{2}i|$

Answer

1. $5$
2. $\sqrt{5}$
3. $3$
4. $2 \sqrt{10}$
5. $3$
6. $4$
7. $5$
8. $7$

Exercise $\PageIndex{4}$

Convert the complex number into polar form $r(\cos(\theta)+i\sin(\theta))$.

1. $2+2i$
2. $4\sqrt{3}+4i$
3. $3-2i$
4. $-5+5i$
5. $4-3i$
6. $-4+3i$
7. $-\sqrt{5}-\sqrt{15}i$
8. $\sqrt{7}-\sqrt{21}i$
9. $-5-12i$
10. $6i$
11. $-10$
12. $-\sqrt{3}+3i$

Answer

1. $2 \sqrt{2}\left(\cos \left(\dfrac{\pi}{4}\right)+i \sin \left(\dfrac{\pi}{4}\right)\right)$
2. $8\left(\cos \left(\dfrac{\pi}{6}\right)+i \sin \left(\dfrac{\pi}{6}\right)\right)$
3. approximately $\sqrt{13}(\cos (-.588)+i \sin (-.588))$ or $\sqrt{13}(\cos (-.187 \pi)+i \sin (-.187 \pi))$
4. $5 \sqrt{2}\left(\cos \left(\dfrac{3 \pi}{4}\right)+i \sin \left(\dfrac{3 \pi}{4}\right)\right)$
5. approximately $5(\cos (-.644)+i \sin (-.644))$ or $5(\cos (-.205 \pi)+i \sin (-.205 \pi))$
6. approximately $5(\cos (2.498)+i \sin (2.498))$ or $5(\cos (.795 \pi)+i \sin (.795 \pi))$
7. $2 \sqrt{5}\left(\cos \left(\dfrac{4 \pi}{3}\right)+i \sin \left(\dfrac{4 \pi}{3}\right)\right)$
8. $2 \sqrt{7}\left(\cos \left(-\dfrac{\pi}{3}\right)+i \sin \left(-\dfrac{\pi}{3}\right)\right)$
9. approximately $13(\cos (4.318)+i \sin (4.318))$ or $13(\cos (1.374 \pi)+i \sin (1.374 \pi))$
10. $6\left(\cos \left(\dfrac{\pi}{2}\right)+i \sin \left(\dfrac{\pi}{2}\right)\right)$
11. $10(\cos (\pi)+i \sin (\pi))$
12. $2 \sqrt{3}\left(\cos \left(\dfrac{2 \pi}{3}\right)+ i \sin \left(\dfrac{2 \pi}{3}\right)\right)$

Exercise $\PageIndex{5}$

Convert the complex number into the standard form $a+bi$.

1. $6(\cos(134^\circ)+i\sin(134^\circ))$
2. $\dfrac 1 2 \left(\cos\left(\dfrac \pi {17}\right)+i\sin\left(\dfrac \pi {17}\right)\right)$
3. $2(\cos(270^\circ)+i\sin(270^\circ))$
4. $\cos\left(\dfrac{\pi} 6\right)+i\sin\left(\dfrac{\pi}6\right)$
5. $10\left(\cos\left(\dfrac{7\pi}{6}\right)+i\sin\left(\dfrac{7\pi}{6}\right)\right)$
6. $6 \left(\cos\left(-\dfrac{5\pi}{12}\right)+i\sin\left(-\dfrac{5\pi}{12}\right)\right)$

Answer

1. approximately $-4.168+4.316 i$
2. approximately $.491+0.0919 i$
3. $-2 i$
4. $\dfrac{\sqrt{3}}{2}+\dfrac{1}{2} i$
5. $-5 \sqrt{3}-5 i$
6. approximately $1.553-5 / 796 i$

Exercise $\PageIndex{6}$

Multiply the complex numbers and write the answer in standard form $a+bi$.

1. $4(\cos(27^\circ)+i\sin(27^\circ)) \cdot 10(\cos(33^\circ)+i\sin(33^\circ))$
2. $7\left(\cos\left(\dfrac{2\pi}{9}\right)+i\sin\left(\dfrac{2\pi}{9}\right)\right) \cdot 6\left(\cos\left(\dfrac{\pi}{9}\right)+i\sin\left(\dfrac{\pi}{9}\right)\right)$
3. $\left(\cos\left(\dfrac{13\pi}{12}\right)+i\sin\left(\dfrac{13\pi}{12}\right)\right) \cdot \left(\cos\left(\dfrac{-11\pi}{12}\right)+i\sin\left(\dfrac{-11\pi}{12}\right)\right)$
4. $8\left(\cos\left(\dfrac{3\pi}{7}\right)+i\sin\left(\dfrac{3\pi}{7}\right)\right) \cdot 1.5\left(\cos\left(\dfrac{4\pi}{7}\right)+i\sin\left(\dfrac{4\pi}{7}\right)\right)$
5. $0.2(\cos(196^\circ)+i\sin(196^\circ)) \cdot 0.5(\cos(88^\circ)+i\sin(88^\circ))$
6. $4\left(\cos\left(\dfrac{7\pi}{8}\right)+i\sin\left(\dfrac{7\pi}{8}\right)\right) \cdot 0.25\left(\cos\left(\dfrac{-5\pi}{24}\right)+i\sin\left(\dfrac{-5\pi}{24}\right)\right)$

Answer

1. $40\left(\cos \left(60^{\circ}\right)+i \sin \left(60^{\circ}\right)\right)=20+20 \sqrt{3} i$
2. $42\left(\cos \left(\dfrac{\pi}{3}\right)+i \sin \left(\dfrac{\pi}{3}\right) \right)=21+21 \sqrt{3} i$
3. $\cos \left(\dfrac{\pi}{6}\right)+i \sin \left(\dfrac{\pi}{6}\right)=\dfrac{\sqrt{3}}{2}+\dfrac{1}{2} i$
4. $12(\cos (\pi)+i \sin (\pi))=-12$
5. $.1\left(\cos \left(284^{\circ}\right)+i \sin \left(284^{\circ}\right)\right) \approx .0242-.0970 i$
6. $\cos \left(\dfrac{2 \pi}{3}\right)+i \sin \left(\dfrac{2 \pi}{3}\right)=-\dfrac{1}{2}+\dfrac{\sqrt{3}}{2} i$

Exercise $\PageIndex{7}$

Divide the complex numbers and write the answer in standard form $a+bi$.

1. $\displaystyle\frac{18(\cos(\frac{\pi}{2})+i\sin(\frac{\pi}{2}))}{3(\cos(\frac{\pi}{6})+i\sin(\frac{\pi}{6}))}$
2. $\displaystyle\frac{10(\cos(254^\circ)+i\sin(254^\circ))}{15(\cos(164^\circ)+i\sin(164^\circ))}$
3. $\displaystyle\frac{\sqrt{24}(\cos(\frac{11\pi}{14})+i\sin(\frac{11\pi}{14}))}{\sqrt{6}(\cos(\frac{2\pi}{7})+i\sin(\frac{2\pi}{7}))}$
4. $\displaystyle\frac{\cos(\frac{8\pi}{5})+i\sin(\frac{8\pi}{5})}{2(\cos(\frac{\pi}{10})+i\sin(\frac{\pi}{10}))}$
5. $\displaystyle\frac{42(\cos(\frac{7\pi}{4})+i\sin(\frac{7\pi}{4}))}{7(\cos(\frac{5\pi}{12})+i\sin(\frac{5\pi}{12}))}$
6. $\displaystyle\frac{30(\cos(-175^\circ)+i\sin(-175^\circ))}{18(\cos(144^\circ)+i\sin(144^\circ))}$

Answer

1. $6(\cos (\pi / 3)+i \sin (\pi / 3))=3+3 \sqrt{3} i$
2. $\dfrac{2}{3}\left(\cos \left(90^{\circ}\right)+i \sin \left(90^{\circ}\right)\right)=\dfrac{2}{3} i$
3. $2(\cos (\pi / 2)+i \sin (\pi / 2))=2 i$
4. $\dfrac{1}{2}\left(\cos \left(\dfrac{3 \pi}{2}\right)+i \sin \left(\dfrac{3 \pi}{2}\right)\right)=-\dfrac{1}{2} i$
5. $6\left(\cos \left(\dfrac{4 \pi}{3}\right)+i \sin \left(\dfrac{4 \pi}{3}\right)\right)=-3-3 \sqrt{3} i$
6. $\dfrac{5}{3}\left(\cos \left(-319^{\circ}\right)+i \sin \left(-319^{\circ}\right)\right) \approx 1.258+1.093 i$