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

 May 3, 2021
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Suppose z and w are complex numbers.  
Prove that \(z + \overline{z}\) and \(w \overline{w}\) are real.

 

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

 

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 May 3, 2021

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