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Question about groups and limit points?

  1. Feb 9, 2012 #1
    1. The problem statement, all variables and given/known data
    We are supposed to say how many limit points the set A={sin(n)} where n is a positive integer.
    My teacher said to use a theorem by Kronecker to help with it.
    His theorem says from wiki, that an infinite cyclic subgroup of the unit circle group is a dense
    subset.
    3. The attempt at a solution
    Im not sure if sin(n) is a cyclic group, don't know a lot about group theory. But if that theorem says that its a dense set, then I would think that all the numbers in [-1,1] would be limit points because eventually the sin(n) would eventually get close to all the points in that interval. And certainly all the points in the set are limit points. Any help would be appreciated.
     
  2. jcsd
  3. Feb 9, 2012 #2

    micromass

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    sin(n) is not on the circle. Do you know how to make an element with it that IS on the circle??
     
  4. Feb 9, 2012 #3
    make it some multiple of pi like [itex] sin(n\pi) [/itex]
    or maybe since sin(n) is irrational I could possibly multiply something by it to get pi.
     
  5. Feb 9, 2012 #4

    micromass

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    Sin(n) is a NUMBER, it can't be an element of a circle.

    The number 4 is not on the circle. An element of the circle would be (0,-1) or (1,0) or something.
     
  6. Feb 9, 2012 #5
    so your saying to be an element of a circle in need a point labeled in x y coordinates.
    so once I found sin(n) I could use like Pythagorean theorem to find the x coordinate
    so I would have (x,sin(n) )
     
  7. Feb 9, 2012 #6

    micromass

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    Uuh, yes, you could do that. Or you could use the very definition of the sine. Surely a sine has something to do with a circle??
     
  8. Feb 9, 2012 #7
    sin(n)= L/M
    L=opposite M=hypotenuse.
     
  9. Feb 9, 2012 #8

    micromass

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    That's really your definition of the sine?? Wow, I hope they fire your trig teacher.

    Anyway, just find the x such that (x,sin(n)) is on the circle. Use the definition of the circle or Pythagoras theorem or whatever...
     
  10. Feb 9, 2012 #9
    how would you define sine
     
  11. Feb 9, 2012 #10

    micromass

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    Like this:

    Trig_functions_on_unit_circle.PNG

    But your definition is also ok.
     
  12. Feb 9, 2012 #11
    When we say a set is dense do we mean that between any two rationals there is an irrational number?
     
  13. Feb 9, 2012 #12

    micromass

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    No, dense has another meaning. In a metric space, a set A is dense in X iff every point in X is the limit of a sequence of points in A. Or equivalently, that every open set contains a point of A. Your definition of dense specializes to my definition in [itex]\mathbb{R}[/itex].

    However, your definition doesn't make sense for the circle since there is no thing as "between" in the circle.
     
  14. Feb 9, 2012 #13
    We know through kroneckers approximation theorem that given an irrational θ, then the sequence of numbers {nθ}, is dense in the unit interval.

    Given an [itex]\alpha[/itex], 0≤[itex]\alpha[/itex]≤1, and any ε>0, then there exists a positive integer k, such that -ε<{kθ}-[itex]\alpha[/itex]<ε.

    That is the kronecker approximation theorem you want to use right?
     
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