1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Complex Number: What's the set?

  1. Oct 1, 2014 #1
    What's the set [tex]\{ z \in \mathbb{C}| |z|^2 \geq z+ \bar{z} \}[/tex]?

    I've set z=a+ib and found [tex]a^2 + b^2 \geq 2a \Rightarrow b^2 \geq a(2-a)[/tex]

    I'm not sure how to interpret this geometrically ie what it looks like?

    I suppose it is the set of vectors whose length is bigger than twice their real part. I guess if I take the square root then I find [tex]\{ b \geq \pm \sqrt{a} \} \cap \{ b \geq \pm \sqrt{2-a} \}[/tex]

    How do we draw this?

    Thanks.
     
  2. jcsd
  3. Oct 1, 2014 #2

    Mark44

    Staff: Mentor

    ##a^2 - 2a + b^2 \geq 0##
    Complete the square in the a terms, and see what you get.
     
  4. Oct 1, 2014 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Interpret ##a^2 -2a + b^2 \geq 0##.
     
  5. Oct 1, 2014 #4
    so
    [tex](a-1)^2+(b-0)^2-1 \geq 0 [/tex]

    [tex](a-1)^2+(b-0)^2 \geq 1 [/tex]

    so circle radius 1 centre (1,0)?
     
  6. Oct 1, 2014 #5

    Mark44

    Staff: Mentor

    Yes.
     
  7. Oct 1, 2014 #6
    what was wrong with saying the length was bigger than twice the real part?
     
  8. Oct 2, 2014 #7

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Nothing, but which description is easier to visualize?
     
  9. Oct 2, 2014 #8

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    [tex](a- 1)^2+ b^2= 1[/tex]
    is a circle of radius 1 with center at (1, 0).

    [tex](a- 1)^2+ b^2\ge 1[/tex]
    is the set of points outside that circle.
     
  10. Oct 2, 2014 #9

    Mark44

    Staff: Mentor

    The inequality represents the set of points on the circle or outside it.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted