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June 3rd, 2018, 10:28 AM  #1 
Senior Member Joined: Jan 2012 Posts: 113 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: 2,124 Thanks: 1103 
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: 113 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: 2,047 Thanks: 585 
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: 2,124 Thanks: 1103  

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