# Thread: Simple intro to complex variables question

1. ## Simple intro to complex variables question

I have to show that
$\displaystyle |(2\overline{z}+5)(\sqrt{2}-i)|=\sqrt{3}|2z+5|$

So far, I have
$\displaystyle |(2\overline{z}+5)(\sqrt{2}-i)|=|2\overline{z}+5||\sqrt{2}-i)|$

My sol'n book then says this is equal to
$\displaystyle |\overline{2z+5}|\sqrt{2+1}$

I understand that for the second term they are just finding the length, but I don't understand how
$\displaystyle |2\overline{z}+5|=|\overline{2z+5}|$.

Can anyone explain the math behind that?

2. Originally Posted by paupsers
I have to show that
$\displaystyle |(2\overline{z}+5)(\sqrt{2}-i)|=\sqrt{3}|2z+5|$

So far, I have
$\displaystyle |(2\overline{z}+5)(\sqrt{2}-i)|=|2\overline{z}+5||\sqrt{2}-i)|$

My sol'n book then says this is equal to
$\displaystyle |\overline{2z+5}|\sqrt{2+1}$

I understand that for the second term they are just finding the length, but I don't understand how
$\displaystyle |2\overline{z}+5|=|\overline{2z+5}|$.

Can anyone explain the math behind that?
Suppose $\displaystyle z = a+bi$. Then $\displaystyle \overline{z} = a-bi$. So $\displaystyle \overline{2} = 2$ and $\displaystyle \overline{5} = 5$.

3. Here is a bit more.
Notice that $\displaystyle \sqrt {\left( x \right)^2 + \left( y \right)^2 } = \sqrt {\left( x \right)^2 + \left( { - y} \right)^2 }$. So that means that $\displaystyle \left| z \right| = \left| {\overline z } \right|$.