My Math Forum Proof that real part of complex number is equal 0

 Complex Analysis Complex Analysis Math Forum

 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{2|z|^2-|w|^2+\sqrt2(w\bar z-z\bar w)}{2|z|^2-|w|^2-\sqrt2(w\bar z-z\bar w)}\\& =\frac{2\times2-4+\sqrt2(w\bar z-\overline{w\bar z})}{2\times2-4-\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 : $\displaystyle \Re(A)=0$

 Tags complex, equal, number, part, proof, real

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post Bogdano Complex Analysis 2 September 3rd, 2014 02:19 AM archer18 Complex Analysis 3 March 2nd, 2014 12:46 PM rayman Complex Analysis 4 February 22nd, 2013 09:21 PM Punch Complex Analysis 4 April 1st, 2012 09:23 AM TsAmE Complex Analysis 1 October 18th, 2010 04:38 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top