My Math Forum About argument of complex numbers
 User Name Remember Me? Password

 Algebra Pre-Algebra and Basic Algebra Math Forum

 June 3rd, 2018, 10:28 AM #1 Senior Member     Joined: Jan 2012 Posts: 140 Thanks: 2 About argument of complex numbers Hi, Let, z, $\displaystyle z{_{1}}$, $\displaystyle z{_{2}}$ be three complex numbers. How to prove the following two properties: (i) arg(z) - arg(-z) = $\displaystyle \pm \pi$ (ii) $\displaystyle \mid z{_{1}}+z{_{2 }}\mid=\mid z{_{1}}-z{_{2 }}\mid \Leftrightarrow arg(z{_{1}}) - arg(z{_{2}})=(\pi /2)$ Thx.
 June 3rd, 2018, 10:45 AM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,529 Thanks: 1389 Well the first step is to spend at least an hour trying to tackle it yourself before giving up and asking for help. You're never going to learn anything if you just keep asking for help here and never try to do the problems yourself. Are they really teaching complex numbers in high school algebra these days? (i) is trivial (ii) I suggest you scale and rotate the coordinate axes so that $z_1$ is transformed to the complex number $1+i0 = 1$ $z_2$ is transformed to some complex number $x + i y$ and you can easily determine the magnitudes of the sum and difference of $z_1,~z_2$ The relation you need to prove is preserved by such coordinate transformations and it reduces the algebra of the problem considerably. Last edited by romsek; June 3rd, 2018 at 11:04 AM.
 June 3rd, 2018, 07:52 PM #3 Senior Member     Joined: Jan 2012 Posts: 140 Thanks: 2 Thx bt without transforming axes, can we prove in general? Yes..actually I tried it asuming z1 & z2 as general complex numbers, but did not come to conclusions. Well I teach, so any topic should be known even when it is not being taught. Nothing is like that we can ask only those topics which are being currently taught. Its more of a general type of doubt. But I will request you to elaborate the second query assuming complex numbers in general, if possible without rotation. Last edited by happy21; June 3rd, 2018 at 07:54 PM.
 June 3rd, 2018, 08:01 PM #4 Senior Member   Joined: Aug 2012 Posts: 2,355 Thanks: 737 What is the geometrical relationship between $z$ and $- z$? Therefore what is the difference of their arguments?
June 3rd, 2018, 09:04 PM   #5
Senior Member

Joined: Sep 2015
From: USA

Posts: 2,529
Thanks: 1389

Quote:
 Originally Posted by happy21 Thx bt without transforming axes, can we prove in general?
The proof is no less general because we changed the coordinate system.

 Tags argument, complex, numbers

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post Laplace Complex Analysis 3 June 19th, 2015 05:57 AM jiasyuen Algebra 2 May 24th, 2015 10:08 AM Arbes Greca Complex Analysis 5 October 20th, 2014 05:04 AM sachinrajsharma Algebra 2 March 26th, 2013 02:26 PM maximus101 Number Theory 2 November 9th, 2012 07:43 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top

Copyright © 2019 My Math Forum. All rights reserved.