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Math Help - Complex number

  1. #1
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    Complex number

    a) modulus of Z-3i<3. Find the greatest possible value of modulus of z. Modulus means square bracket.

    I have no idea about finding the greatest possible value.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by kingkaisai2
    a) modulus of Z-3i<3. Find the greatest possible value of modulus of z. Modulus means square bracket.

    I have no idea about finding the greatest possible value.
    The region of the complex plain such that |z-3i|<3 is the interior of the circle of
    radius 3 centred at 3i. If the region was in fact closed (that is |z-3i|\le 3)
    then z=6i would give the maximum modulus of |z|=6, but the region is open,
    so there is no z with maximum modulus in the region.

    RonL
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  3. #3
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    Quote Originally Posted by kingkaisai2
    a) modulus of Z-3i<3. Find the greatest possible value of modulus of z. Modulus means square bracket.

    I have no idea about finding the greatest possible value.
    I will explain differently.
    Let,
    z=x+iy
    Then,
    |z-3i|=|x+iy-3i|=|x+i(y-3)|=\sqrt{x^2+(y-3)^2}
    But,
    |z-3i|<3
    Thus,
    \sqrt{x^2+(y-3)^2}<3
    Square,
    x^2+(y-3)^2<9
    This is a circle center at (0,3)
    Now look upon CaptainBlank's post.
    He said that is the region is closed meaning,
    x^2+(y-3)^2\leq 9 then you can find a maximum value, namely, 9. Which would give 6.
    If the region is open meaning,
    x^2+(y-3)^2<9 then you cannot find a maximum value.
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