Lösung für Aufgabe 6.5.10
Bestimmen Sie für die komplexen Zahlen $$z_{1}=3+2i,\ z_{2}=2-4i,\ z_{3}=-i,\ z_{4}=1-i,\ z_{5}=5-3i$$ $\bar z_{i}$, $|z_{i}|$, $\arg z_{i}$ und $1/z_{i}$ für $i=1,\dots,5$. Berechnen Sie weiters $z_{i}+z_{j}$, $z_{i}-z_{j}$, $z_{i}z_{j}$ und $z_{i}/z_{j}$ für $i,j=1,\dots,5$. Stellen Sie das Resultat jeweils in der Form $a+ib$ mit $a,b\in\R$ dar.\begin{gather*} \bar z_1 = 3-2i,\quad \bar z_2 = 2+4i,\quad \bar z_3 = i,\quad \bar z_4 = 1+i,\quad \bar z_5 = 5+3i,\\ |z_1| = \sqrt{13}\quad |z_2| = 2\sqrt5,\quad |z_3| = 1,\quad |z_4| = \sqrt2,\quad |z_5| = \sqrt{34},\\ \arg z_1 = \arctan{\tfrac23} \approx 0.588003,\quad \arg z_2 = -\arctan{2} \approx -1.10715,\quad \arg z_3 = -\tfrac{\pi}2,\\ \arg z_4 = -\tfrac{\pi}4,\quad \arg z_5 = -\arctan{\tfrac35} \approx -0.54042,\\ 1/z_1 = \tfrac3{13}-\tfrac2{13}i,\quad 1/z_2 = \tfrac1{10}+\tfrac15i,\quad 1/z_3 = i,\quad 1/z_4 = \tfrac12+\tfrac12i,\quad 1/z_5 = \tfrac5{34}+\tfrac3{34}i,\\ \end{gather*} Außerdem haben wir $$ \begin{array}{lllll} z_1+z_1 = 6+4i,& z_1+z_2 = 5-2i,& z_1+z_3 = 3+i,& z_1+z_4 = 4+i,& z_1+z_5 = 8-i,\\ z_2+z_2 = 4-8i,& z_2+z_3 = 2-5i,& z_2+z_4 = 3-5i,& z_2+z_5 = 7-7i,& z_3+z_3 = -2i,\\ z_3+z_4 = 1-2i,& z_3+z_5 = 5-4i,& z_4+z_4 = 2-2i,& z_4+z_5 = 6-4i,& z_5+z_5 = 10-6i,\\ z_1-z_1 = 0,& z_1-z_2 = 1+6i,& z_1-z_3 = 3+3i,& z_1-z_4 = 2+3i,& z_1-z_5 = -2+5i,\\ z_2-z_1 = -1-6i,& z_2-z_2 = 0,& z_2-z_3 = 2-3i,& z_2-z_4 = 1-3i,& z_2-z_5 = -3-i,\\ z_3-z_1 = -3-3i,& z_3-z_2 = -2+3i,& z_3-z_3 = 0,& z_3-z_4 = -1,& z_3-z_5 = -5+2i,\\ z_4-z_1 = -2-3i,& z_4-z_2 = -1+3i,& z_4-z_3 = 1,& z_4-z_4 = 0,& z_4-z_5 = -4+2i,\\ z_5-z_1 = 2-5i,& z_5-z_2 = 3+i,& z_5-z_3 = 5-2i,& z_5-z_4 = 4-2i,& z_5-z_5 = 0,\\ z_1z_1 = 5+12i,& z_1z_2 = 14-8i,& z_1z_3 = 2-3i,& z_1z_4 = 5-i,& z_1z_5 = 21+i,\\ z_2z_2 = -12-16i,& z_2z_3 = -4-2i,& z_2z_4 = -2-6i,& z_2z_5 = -2-26i,& z_3z_3 = -1,\\ z_3z_4 = -1-i,& z_3z_5 = -3-5i,& z_4z_4 = -2i,& z_4z_5 = 2-8i,& z_5z_5 = 16-30i,\\ z_1/z_1 = 1,& z_1/z_2 = -\tfrac{1}{10}+\tfrac{4}{5}i,& z_1/z_3 = -2+3i,& z_1/z_4 = \tfrac{1}{2}+\tfrac{5}{2}i,& z_1/z_5 = \tfrac{9}{34}+\tfrac{19}{34}i,\\ z_2/z_1 = -\tfrac{2}{13}-\tfrac{16}{13}i,& z_2/z_2 = 1,& z_2/z_3 = 4+2i,& z_2/z_4 = 3-i,& z_2/z_5 = \tfrac{11}{17}-\tfrac{7}{17}i,\\ z_3/z_1 = -\tfrac{2}{13}-\tfrac{3}{13}i,& z_3/z_2 = \tfrac{1}{5}-\tfrac{i}{10},& z_3/z_3 = 1,& z_3/z_4 = \tfrac{1}{2}-\tfrac{i}{2},& z_3/z_5 = \tfrac{3}{34}-\tfrac{5}{34}i,\\ z_4/z_1 = \tfrac{1}{13}-\tfrac{5}{13}i,& z_4/z_2 = \tfrac{3}{10}+\tfrac{i}{10},& z_4/z_3 = 1+i,& z_4/z_4 = 1,& z_4/z_5 = \tfrac{4}{17}-\tfrac{i}{17},\\ z_5/z_1 = \tfrac{9}{13}-\tfrac{19}{13}i,& z_5/z_2 = \tfrac{11}{10}+\tfrac{7}{10}i,& z_5/z_3 = 3+5i,& z_5/z_4 = 4+i,& z_5/z_5 = 1,\\ \end{array} $$