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Complex Plane

  1. Oct 10, 2007 #1

    danago

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    Given that [tex]|z-1+i| \le 1[/tex], find the maximum and minimum value of |z| and Arg(z).

    I realise that the equation given defines the interior of a circle of radius 1 centered at (1,-1), which includes the circumference.

    For the first part of the question, i am able to represent the equation graphically. From what i understand, |z| is the distance from the origin to any point lying on or within the circle. If this is the case, i can see the minimum and maximum points, but im not too sure on how to calculate their locations.

    For the next part, finding the extreme values of Arg(z), i just read straight from my graph and said that the minimum is [tex]-\pi/2[/tex] and the maximum is 0. Is that right?

    Thanks in advance,
    Dan.
     
  2. jcsd
  3. Oct 10, 2007 #2

    mjsd

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    from description, it appears that this is your task:
    given the set of points defined by [tex]\{z:|z-1+i|\leq 1\}[/tex] find the complex numbers z such that the vector going from origin to z has max/min length. likewise for angle (so -p1/2 and 0 seem wrong). but then again you said you read straight from your graph, how does your graph look like, or how you derived it? (to help with pin-pointing potential mistakes)


    EDIT: sorry my mistake
     
    Last edited: Oct 10, 2007
  4. Oct 10, 2007 #3

    danago

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    I simply drew a circle centered at (1,-1) with radius 1. Both the x and y axis are tangental to the circle. That pretty much explains what ive drawn.

    My book says the argument of a complex number should be defined within -pi to pi, which is how i got 0 and -pi/2, since the circle touches the positive x axis and the negative y axis.
     
  5. Oct 10, 2007 #4

    Dick

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    Ok, seems like your picture is fine. For |z|, how far is the center of the circle from the origin? Now how far are the closest and farthest points from the center of the circle?
     
  6. Oct 10, 2007 #5

    danago

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    Ah ok thats a good way to think about it. The distance from the origin to (1,-1) is [tex]\sqrt{2}[/tex], plus another 1 unit (the circles radius) gives a maximum distance of [tex]\sqrt{2}+1[/tex]. The minimum distance will just then be [tex]\sqrt{2}-1[/tex]. Am i right?
     
  7. Oct 10, 2007 #6

    Dick

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    Absolutely.
     
  8. Oct 10, 2007 #7

    danago

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    Alright thanks for the help :smile: What about the argument of z? Was i right with that?
     
  9. Oct 10, 2007 #8

    HallsofIvy

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    Yes, you were.
     
  10. Oct 10, 2007 #9

    danago

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    Alright, thanks alot :smile:
     
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