October 24th, 2016, 09:14 AM  #1 
Newbie Joined: Oct 2016 From: netherlands Posts: 1 Thanks: 0  Complex numbers question
Hey everyone, I'm currently making assignments about complex numbers (Calculus, Appendix A7) I've been stuck at some question however, and I can't seem to figure it out, or find any helpful information about it online. Graph the points z=x+iy that satisfy the following conditions: a. z+i=z1 b. z+1≥z Let Re(z) denote the real part of z, and Im(z) denote the imaginary part of z. Show that the following relations hold for any complex numbers z, z1, and z2. a. Re(z)≤ z b. z1+z2≤ z1+z2 Any help would be very, very much appreciated. Thank you in advance! Sincerely, Karim 
October 24th, 2016, 10:52 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,160 Thanks: 866 
We can interpret "a b" as "the distance between complex number a and complex number b in the complex plane. Saying that z a= z b means that z is equidistant between a and b. That is the line that perpendicularly bisects the line segment from a to b. We can write any complex number, z, as z= a+ ib for a and b real numbers. In terms of a and b. Re(z)= a, and . Do you see why it is true that tex]a\le \sqrt{a^2+ b^2}[/tex]? Let z1= a+ bi, z2= c +di. Then [tex]z1+ z2= \sqrt{(a+ c)^2+ (b+ d)^2}[tex] and . Look the square of both of those. 
October 24th, 2016, 11:08 AM  #3  
Senior Member Joined: Sep 2015 From: USA Posts: 1,938 Thanks: 1003  Quote:
$z = \sqrt{(Re(z))^2 + (Im(z))^2} \geq \sqrt{(Re(z))^2} = Re(z)$ b) is just the triangle inequality  

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