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Homework Help: 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?

  2. jcsd
  3. Oct 1, 2014 #2


    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

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    Interpret ##a^2 -2a + b^2 \geq 0##.
  5. Oct 1, 2014 #4
    [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


    Staff: Mentor

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


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


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    [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


    Staff: Mentor

    The inequality represents the set of points on the circle or outside it.
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