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Set of Points in complex plane

  1. Jan 15, 2015 #1
    1. The problem statement, all variables and given/known data
    Describe the set of points determined by the given condition in the complex plane:
    |z - 1 + i| = 1

    2. Relevant equations
    |z| = sqrt(x2 + y2)
    z = x + iy

    3. The attempt at a solution

    Tried to put absolute values on every thing by the Triangle inequality
    |z| - |1| + |i| = |1|
    sqrt(x2 + y2) - 1 + 1 = 1
    sqrt(x2 + y2) = 1

    Not sure if I'm approaching this question correctly...
     
  2. jcsd
  3. Jan 15, 2015 #2

    jbunniii

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    The triangle inequality ##|z + w| \leq |z| + |w|## won't help you, because it's an inequality. The set of points with ##|z - 1 + i| = 1## is not the same as the set of points with ##|z| + |1| + |i| = 1##.

    A better approach is to recognize that
    $$|z - 1 + i| = 1$$
    if and only if
    $$|z - 1 + i|^2 = 1$$
    The squared equation is easier to work with, because for any complex number ##w## we have ##|w|^2 = w\overline w##, where ##\overline w## is the complex conjugate of ##w##.
     
  4. Jan 15, 2015 #3

    Ray Vickson

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    No, you are not: ##|z - 1 + i| = 1## does NOT imply that ##|z| -|1| + |i| = |1|## or anything at all like it. In fact, it is almost always true that ##|z_1 + z_2| \neq |z_1| + |z_2|##.
     
  5. Jan 15, 2015 #4

    PeroK

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    It says "describe" the set of points. Try thinking geometrically.
     
  6. Jan 15, 2015 #5

    Mark44

    Staff: Mentor

    And along these lines, an important use of the absolute value operation is to indicate the distance between two points. So |z - 1 + i| = 1 could also be written as |z - (1 - i)| = 1.
     
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