1. **Problem Statement:** Show the following properties of complex conjugates for two complex numbers $z_1$ and $z_2$: (i) $\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$ (ii) $\overline{z_1 \cdot z_2} = \overline{z_1} \cdot \overline{z_2}$ (iii) $\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\overline{z_1}}{\overline{z_2}}$ --- 2. **Recall the definition:** If $z = a + bi$ where $a,b \in \mathbb{R}$ and $i^2 = -1$, then the conjugate is $\overline{z} = a - bi$. --- 3. **Proof of (i):** Start with the left side: $$\overline{z_1 + z_2} = \overline{(a_1 + b_1 i) + (a_2 + b_2 i)} = \overline{(a_1 + a_2) + (b_1 + b_2) i} = (a_1 + a_2) - (b_1 + b_2) i$$ Right side: $$\overline{z_1} + \overline{z_2} = (a_1 - b_1 i) + (a_2 - b_2 i) = (a_1 + a_2) - (b_1 + b_2) i$$ Since both sides are equal, property (i) holds. --- 4. **Proof of (ii):** Calculate the left side: $$\overline{z_1 z_2} = \overline{(a_1 + b_1 i)(a_2 + b_2 i)} = \overline{a_1 a_2 + a_1 b_2 i + b_1 i a_2 + b_1 b_2 i^2}$$ $$= \overline{a_1 a_2 + (a_1 b_2 + b_1 a_2) i + b_1 b_2 (-1)} = \overline{(a_1 a_2 - b_1 b_2) + (a_1 b_2 + b_1 a_2) i}$$ $$= (a_1 a_2 - b_1 b_2) - (a_1 b_2 + b_1 a_2) i$$ Calculate the right side: $$\overline{z_1} \cdot \overline{z_2} = (a_1 - b_1 i)(a_2 - b_2 i) = a_1 a_2 - a_1 b_2 i - b_1 i a_2 + b_1 b_2 i^2$$ $$= a_1 a_2 - (a_1 b_2 + b_1 a_2) i + b_1 b_2 (-1) = (a_1 a_2 - b_1 b_2) - (a_1 b_2 + b_1 a_2) i$$ Both sides are equal, so property (ii) holds. --- 5. **Proof of (iii):** Start with the left side: $$\overline{\left(\frac{z_1}{z_2}\right)} = \overline{\frac{a_1 + b_1 i}{a_2 + b_2 i}}$$ Multiply numerator and denominator by the conjugate of denominator: $$= \overline{\frac{(a_1 + b_1 i)(a_2 - b_2 i)}{(a_2 + b_2 i)(a_2 - b_2 i)}} = \overline{\frac{(a_1 a_2 + b_1 b_2) + (b_1 a_2 - a_1 b_2) i}{a_2^2 + b_2^2}}$$ Taking conjugate: $$= \frac{(a_1 a_2 + b_1 b_2) - (b_1 a_2 - a_1 b_2) i}{a_2^2 + b_2^2}$$ Right side: $$\frac{\overline{z_1}}{\overline{z_2}} = \frac{a_1 - b_1 i}{a_2 - b_2 i}$$ Multiply numerator and denominator by $a_2 + b_2 i$: $$= \frac{(a_1 - b_1 i)(a_2 + b_2 i)}{(a_2 - b_2 i)(a_2 + b_2 i)} = \frac{(a_1 a_2 + b_1 b_2) + (b_2 a_1 - a_2 b_1) i}{a_2^2 + b_2^2}$$ Note that $b_2 a_1 - a_2 b_1 = -(b_1 a_2 - a_1 b_2)$, so the numerator matches the conjugate form. Hence, $$\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\overline{z_1}}{\overline{z_2}}$$ --- **Final answer:** All three properties hold true for complex conjugates.