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complex numbers

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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

#1
+26213
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Suppose z and w are complex numbers.
Prove that $$z + \overline{z}$$ and $$w \overline{w}$$ are real.

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

May 3, 2021