#math/linear-algebra Suppose $w,z\in \mathbf{C}$. Then the following equalities and inequalities hold. 1. $z+\bar{z}=2\text{Re }z$ 2. $z-\bar{z}=2(\text{Im }z)i$ 3. $z\bar{z}=\lvert z \rvert^{2}$ 4. $\overline{w+z}=\bar{w}+\bar{z}$ and $\overline{wz}=\bar{w}\bar{z}$ 5. $\large{\bar{\bar{z}}}=z$ 6. $\lvert \text{Re }z \rvert \leq{} \lvert z \rvert$ and $\lvert \text{Im } z \rvert \leq{}\lvert z \rvert$ 7. $\lvert \bar{z} \rvert=\lvert z \rvert$ 8. $\lvert wz \rvert=\lvert w \rvert\lvert z \rvert$ 9. $\lvert w+z \rvert\leq{}\lvert w \rvert+\lvert z \rvert$