complex numbers

Suppose z and w are complex numbers.  
Prove that $z + \overline{z}$ and $w \overline{w}$ are real.

$\begin{array}{|rcll|} \hline \text{Let $z=a+bi$} \\ \text{Let $\overline{z}=a-bi$} \\ \begin{array}{|rcll|} \hline z + \overline{z} &=& a+bi +a-bi \\ \mathbf{z + \overline{z}} &=& \mathbf{2a} \\ \hline \end{array}\\ \\ \text{Let $w=a+bi$} \\ \text{Let $\overline{w}=a-bi$} \\ \begin{array}{|rcll|} \hline w \overline{w} &=& (a+bi)(a-bi) \\ w \overline{w} &=& a^2-(bi)^2 \\ w \overline{w} &=& a^2-b^2i^2 \quad | \quad \mathbf{i^2 = -1} \\ \mathbf{w \overline{w}} &=& \mathbf{a^2+b^2} \\ \hline \end{array}\\ \hline \end{array}$