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I Complex Numbers Moduli Problem

  1. Apr 15, 2016 #1
    This may be a simple thing but due to some reason I am not able to understand.
    I am not able to understand an example from Brown-Churchill book. I have provided the screenshot in the attached picture. Request help.
     

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  3. Apr 15, 2016 #2

    andrewkirk

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    I think you may have copied the wrong equation for (9). The triangle inequality says that ##|a+b|\leq|a|+|b|##. Applying that to your equation we get:

    $$|z^3+3z^2-2z+1|\leq|z^3+3z^2-2z|+|1|\leq|z^3+3z^2|+|-2z|+|1|\leq|z^3|+|3z^2|+|-2z|+|1|$$
    [applying the triangle inequality three times in succession]
    $$=|z|^3+3|z|^2+|-2||z|+1$$
    [applying (8) ]
    $$=|z|^3+3|z|^2+2|z|+1<2^3+3\cdot 2^2+2\cdot 2+1=25$$

    EDIT: Just saw Samy's post. I don't have the book but, based on that picture, it looks more likely a typo in that the ref to (9) should be to (10), rather than you miscopying it.
     
  4. Apr 15, 2016 #3

    Samy_A

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    May well be a typo. He is referring to the following:
    complex.jpg
     
  5. Apr 16, 2016 #4
    Thank you very much.
     
  6. Apr 16, 2016 #5
    Thank you very much.
     
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