I would like to prove [itex]\mid z + \overline{z} \mid \leq 2 \mid z \mid[/itex](adsbygoogle = window.adsbygoogle || []).push({});

The first way I could think of:

[tex]

\begin{multline}

RHS^2 - LHS^2\\

=4\mid z \mid^2 - \mid z + \overline{z} \mid ^ 2\\

=4z\overline{z} - (z + \overline{z})(\overline{z}+z)\\

=4z\overline{z} - z\overline{z}-z^2-\overline{z}^2-z\overline{z}\\

=2z\overline{z}-z^2-\overline{z}^2\\

=-(z^2-2z\overline{z}+\overline{z}^2)\\

=-(z-\overline{z})^2\\

\leq 0 ???

\end{multline}

[/tex]

I now know the correct proof is as follow:

[tex]

\begin{multline}

\mid z + \overline{z} \mid\\

\leq \mid z \mid + \mid \overline{z} \mid\\

= \mid z \mid + \mid z \mid\\

= 2 \mid z \mid\\

\end{multline}\\

[/tex]

But what is wrong with my original proof?

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# Homework Help: Complex no & conjugate

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