Complex no & conjugate

ramble

I would like to prove [itex]\mid z + \overline{z} \mid \leq 2 \mid z \mid[/itex]

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?

matt grime

Nothing is wrong with it. what is z-z*? It's not a real number, is it.....

fargoth

okay got it..
lets say b is Im(z).
then b^2 is always positive, and (ib)^2 is always negative...

you just got the signs wrong to start with.

ramble

according to my proof, instead of [itex]\mid z + \overline{z} \mid \leq 2 \mid z \mid[/itex], it is [itex]\mid z + \overline{z} \mid \geq 2 \mid z \mid[/itex]...

z is a complex no. and [itex]\overline{z}[/itex] is its conjugate. Have I mixed up some basic rules in complex no. with those in real no.?

ramble

Sorry forgoth I haven't noticed your reply when i post mine... but I do not understand... do you mean that I cannot square a complex no?

fargoth

[tex]-(z-z^*)^2=-(a+ib-a+ib)^2=4b^2>0[/tex]
your last line had the wrong signs... or i just missed something.

matt grime

When ever you do something like this, always step back and think: what happens in a simple example. For instance why not put z=i in and see what happens?

But you are confusing some rules of real numbers with complex ones. Sure, if you square a real number it is positive, but the whole raison d'etre of complex number is that you have things that square to negative numbers.

ramble

I see... thx~
Too much used to real no... have never thought that there would be a problem on that line
(never thought of posting a question in a forum could get immediate response too~ ^^)