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November 16th, 2017, 02:31 AM  #1 
Newbie Joined: Nov 2017 From: Poland Posts: 1 Thanks: 0  Proof that real part of complex number is equal 0
Given that $\displaystyle w = 1 + \sqrt{3}$ and $\displaystyle z = 1 + i $ show that $\displaystyle \Re(\frac{\sqrt{2}z + w}{\sqrt{2}z  w}) = 0$ I only found that $\displaystyle \Re(w) = \frac{1}{2} (w + \bar{w})$ but still I have no idea how to solve it. 
November 16th, 2017, 03:09 AM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 635 Thanks: 401 Math Focus: Dynamical systems, analytic function theory, numerics 
I suspect you should have $w  \bar{w}$ in your simplification. The solution then follows by thinking about the real part of any complex number minus its own conjugate.

November 16th, 2017, 08:07 AM  #3 
Senior Member Joined: Dec 2015 From: holland Posts: 162 Thanks: 37 Math Focus: tetration 
Why use the conjugate? Just fill in w and z in the given expression.

November 16th, 2017, 10:10 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,828 Thanks: 2160 
I suggest, aga150, that you correct the mistake in the statement of the problem.

January 9th, 2018, 04:20 AM  #5 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
I suspect that should be $w= 1+ i\sqrt{3}$ NOT "$1+ \sqrt{3}$".

January 9th, 2018, 11:29 AM  #6 
Member Joined: Aug 2011 From: Nouakchott, Mauritania Posts: 85 Thanks: 14 Math Focus: Algebra, Cryptography 
Good afternoon ! As Country Boy has said, we suspect that it should be $\displaystyle w=1+i\sqrt3$. In this case , let $\displaystyle A=\frac{\sqrt{2}z + w}{\sqrt{2}z  w}$. Then : $\displaystyle \overline A=\frac{\sqrt{2}\bar z + \bar w}{\sqrt{2}\bar z  \bar w}$. We have : $\displaystyle \begin{align*} A\times\frac1{\overline A}&=\frac{\sqrt{2}z + w}{\sqrt{2}z  w}\times\frac{\sqrt{2}\bar z  \bar w}{\sqrt{2}\bar z + \bar w}=\frac{2z^2w^2+\sqrt2(w\bar zz\bar w)}{2z^2w^2\sqrt2(w\bar zz\bar w)}\\& =\frac{2\times24+\sqrt2(w\bar z\overline{w\bar z})}{2\times24\sqrt2(w\bar z\overline{w\bar z})}\\& =\frac{\sqrt2\times 2i\Im(w\bar z)}{\sqrt2\times 2i\Im(w\bar z)}\\&=1 \end{align*} $ Thus : $\displaystyle A=\overline A$ and this means that : $\displaystyle \Re(A)=0$ 

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complex, equal, number, part, proof, real 
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