Ejercicios resueltos de operaciones elementales

$z_1 + z_2 = (2 + 3i) + (1 - 4i) = (2 + 1) + (3 - 4)i = 3 - i$

$z_1 - z_2 = (2 + 3i) - (1 - 4i) = (2 - 1) + (3 - (-4))i = 1 + 7i$

$z_1 \times z_2 = (2 + 3i)(1 - 4i) = 2 \cdot 1 + 2 \cdot (-4i) + 3i \cdot 1 + 3i \cdot (-4i)$

$= 2 - 8i + 3i - 12i^2 = 2 - 5i + 12 = 14 - 5i$

$z_1 \div z_2 = \frac{2 + 3i}{1 - 4i} \cdot \frac{1 + 4i}{1 + 4i} = \frac{(2 + 3i)(1 + 4i)}{(1 - 4i)(1 + 4i)}$

$= \frac{2 + 8i + 3i + 12i^2}{1 + 16} = \frac{2 + 11i - 12}{17} = \frac{-10 + 11i}{17} = -\frac{10}{17} + \frac{11}{17}i$

$z_1 + z_2 + z_3 = (-1 + 2i) + (3 - i) + (2 + 5i) = (-1 + 3 + 2) + (2 - 1 + 5)i = 4 + 6i$

$z_1z_2 = (-1 + 2i)(3 - i) = -3 + i + 6i - 2i^2 = -3 + 7i + 2 = -1 + 7i$

$z_1z_2 + z_3 = (-1 + 7i) + (2 + 5i) = 1 + 12i$

$(z_1 + z_2) = (-1 + 2i) + (3 - i) = 2 + i$

$(z_1 + z_2) \times z_3 = (2 + i)(2 + 5i) = 4 + 10i + 2i + 5i^2 = 4 + 12i - 5 = -1 + 12i$

$\frac{z_1z_2}{z_3} = \frac{-1 + 7i}{2 + 5i} \cdot \frac{2 - 5i}{2 - 5i} = \frac{(-1 + 7i)(2 - 5i)}{4 + 25}$

$= \frac{-2 + 5i + 14i - 35i^2}{29} = \frac{-2 + 19i + 35}{29} = \frac{33 + 19i}{29} = \frac{33}{29} + \frac{19}{29}i$

$z_1z_2 = (1 - 3i)(-2 + 4i) = -2 + 4i + 6i - 12i^2 = -2 + 10i + 12 = 10 + 10i$

$z_3z_4 = (3 + 2i)(1 - i) = 3 - 3i + 2i - 2i^2 = 3 - i + 2 = 5 - i$

$z_1z_2 - z_3z_4 = (10 + 10i) - (5 - i) = 5 + 11i$

$z_1 + z_3 = (1 - 3i) + (3 + 2i) = 4 - i$

$z_2 - z_4 = (-2 + 4i) - (1 - i) = -3 + 5i$

$\frac{z_1 + z_3}{z_2 - z_4} = \frac{4 - i}{-3 + 5i} \cdot \frac{-3 - 5i}{-3 - 5i} = \frac{(4 - i)(-3 - 5i)}{9 + 25} = \frac{-12 - 20i + 3i + 5i^2}{34} = \frac{-17 - 17i}{34} = -\frac{1}{2} - \frac{1}{2}i$

$z_1z_3 = (1 - 3i)(3 + 2i) = 3 + 2i - 9i - 6i^2 = 3 - 7i + 6 = 9 - 7i$

$z_2z_4 = (-2 + 4i)(1 - i) = -2 + 2i + 4i - 4i^2 = -2 + 6i + 4 = 2 + 6i$

$(z_1z_3)(z_2z_4) = (9 - 7i)(2 + 6i) = 18 + 54i - 14i - 42i^2 = 18 + 40i + 42 = 60 + 40i$

$\frac{z_1z_2}{z_3z_4} = \frac{10 + 10i}{5 - i} \cdot \frac{5 + i}{5 + i} = \frac{(10 + 10i)(5 + i)}{25 + 1} = \frac{50 + 10i + 50i + 10i^2}{26} = \frac{40 + 60i}{26} = \frac{20 + 30i}{13}$

$\left|\frac{z_1z_2}{z_3z_4}\right| = \left|\frac{20 + 30i}{13}\right| = \frac{\sqrt{20^2 + 30^2}}{13} = \frac{\sqrt{400 + 900}}{13} = \frac{\sqrt{1300}}{13} = \frac{10\sqrt{13}}{13}$

$wu = (2 - i)(-1 + 3i) = -2 + 6i + i - 3i^2 = -2 + 7i + 3 = 1 + 7i$

$vs = (4 + 2i)(3 - 4i) = 12 - 16i + 6i - 8i^2 = 12 - 10i + 8 = 20 - 10i$

$(wu)(vs) = (1 + 7i)(20 - 10i) = 20 - 10i + 140i - 70i^2 = 20 + 130i + 70 = 90 + 130i$

$wv = (2 - i)(4 + 2i) = 8 + 4i - 4i - 2i^2 = 8 + 2 = 10$

$us = (-1 + 3i)(3 - 4i) = -3 + 4i + 9i - 12i^2 = -3 + 13i + 12 = 9 + 13i$

$(wv)(us) = 10(9 + 13i) = 90 + 130i$ ✓

$wuv = (1 + 7i)(4 + 2i) = 4 + 2i + 28i + 14i^2 = 4 + 30i - 14 = -10 + 30i$

$|wuv| = |-10 + 30i| = \sqrt{100 + 900} = \sqrt{1000} = 10\sqrt{10}$

$|w||u||v| = |2 - i||-1 + 3i||4 + 2i| = \sqrt{5} \cdot \sqrt{10} \cdot \sqrt{20} = \sqrt{1000} = 10\sqrt{10}$ ✓

$wu + vs = (1 + 7i) + (20 - 10i) = 21 - 3i$

$\overline{wu + vs} = \overline{21 - 3i} = 21 + 3i$

$\overline{wu} + \overline{vs} = \overline{1 + 7i} + \overline{20 - 10i} = (1 - 7i) + (20 + 10i) = 21 + 3i$ ✓

$s^2 = (3 - 4i)^2 = 9 - 24i + 16i^2 = 9 - 24i - 16 = -7 - 24i$

$\frac{wuv}{s^2} = \frac{-10 + 30i}{-7 - 24i} \cdot \frac{-7 + 24i}{-7 + 24i} = \frac{(-10 + 30i)(-7 + 24i)}{49 + 576}$

$= \frac{70 - 240i - 210i + 720i^2}{625} = \frac{70 - 450i - 720}{625} = \frac{-650 - 450i}{625} = -\frac{26}{25} - \frac{18}{25}i$