1. Problem: Divide complex numbers and find real and imaginary parts. 2. Formula: To divide complex numbers $\frac{a+bi}{c+di}$, multiply numerator and denominator by the conjugate of the denominator $c-di$: $$\frac{a+bi}{c+di} \times \frac{c-di}{c-di} = \frac{(a+bi)(c-di)}{c^2 + d^2}$$ 3. i. Divide $5 - 2i$ by $4 + 3i$: $$\frac{5 - 2i}{4 + 3i} \times \frac{4 - 3i}{4 - 3i} = \frac{(5 - 2i)(4 - 3i)}{4^2 + 3^2} = \frac{20 - 15i - 8i + 6i^2}{16 + 9}$$ Since $i^2 = -1$: $$= \frac{20 - 23i + 6(-1)}{25} = \frac{20 - 23i - 6}{25} = \frac{14 - 23i}{25} = \frac{14}{25} - \frac{23}{25}i$$ 4. ii. Given $z_1 = 2 - 3i$, $z_2 = 3 - 4i$, find $\frac{z_1}{z_2}$: $$\frac{2 - 3i}{3 - 4i} \times \frac{3 + 4i}{3 + 4i} = \frac{(2 - 3i)(3 + 4i)}{3^2 + 4^2} = \frac{6 + 8i - 9i - 12i^2}{9 + 16}$$ $$= \frac{6 - i + 12}{25} = \frac{18 - i}{25} = \frac{18}{25} - \frac{1}{25}i$$ 5. iii. Find $\operatorname{Re}\left(\frac{3 - 2i}{1 + 4i}\right)$ and $\operatorname{Im}\left(\frac{3 - 2i}{1 + 4i}\right)$: $$\frac{3 - 2i}{1 + 4i} \times \frac{1 - 4i}{1 - 4i} = \frac{(3 - 2i)(1 - 4i)}{1^2 + 4^2} = \frac{3 - 12i - 2i + 8i^2}{1 + 16}$$ $$= \frac{3 - 14i + 8(-1)}{17} = \frac{3 - 14i - 8}{17} = \frac{-5 - 14i}{17} = -\frac{5}{17} - \frac{14}{17}i$$ Real part: $-\frac{5}{17}$ Imaginary part: $-\frac{14}{17}$ 6. Given $z_1 = 1 - 3i$, $z_2 = -2 + 5i$, $z_3 = -3 - 4i$: 7. i. Find $\frac{z_1}{z_3}$: $$\frac{1 - 3i}{-3 - 4i} \times \frac{-3 + 4i}{-3 + 4i} = \frac{(1 - 3i)(-3 + 4i)}{(-3)^2 + 4^2} = \frac{-3 + 4i + 9i - 12i^2}{9 + 16}$$ $$= \frac{-3 + 13i + 12}{25} = \frac{9 + 13i}{25} = \frac{9}{25} + \frac{13}{25}i$$ 8. ii. Find $\frac{z_1 z_2}{z_1 + z_2}$: First, calculate numerator: $$z_1 z_2 = (1 - 3i)(-2 + 5i) = -2 + 5i + 6i - 15i^2 = -2 + 11i + 15 = 13 + 11i$$ Calculate denominator: $$z_1 + z_2 = (1 - 3i) + (-2 + 5i) = (1 - 2) + (-3i + 5i) = -1 + 2i$$ Divide: $$\frac{13 + 11i}{-1 + 2i} \times \frac{-1 - 2i}{-1 - 2i} = \frac{(13 + 11i)(-1 - 2i)}{(-1)^2 + 2^2} = \frac{-13 - 26i - 11i - 22i^2}{1 + 4}$$ $$= \frac{-13 - 37i + 22}{5} = \frac{9 - 37i}{5} = \frac{9}{5} - \frac{37}{5}i$$