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Question about a cirlce

  1. Oct 23, 2012 #1
    Lets say I take the natural numbers from 1 to infinity and I wrap the positive real line into a circle.
    So I have a circle with an infinite circumference and the natural numbers are spaced 1 unit away from each other at the edge of the circle. Now I draw a line from each natural number
    to the center of the circle. Now is there a point from the center of the circle at a finite distance such that there are no gaps in a second test circle drawn at this finite distance.
    And then can I construct a circle with infinite circumference and have an uncountable number of points and each point is separated from the other by a positive finite distance.
    And then draw a line from each point to the center of my circle. will I have another circle inside this that has no gaps in it.
     
  2. jcsd
  3. Oct 23, 2012 #2
    If the circle has an infinite circumference how can it have a center?

    How can a circle have an infinite circumference? What does that mean?

    Say such a circle exists, and say you have your points 1, 2, 3 on the circle, now say that your point 1 lies on the origin, and the center of your circle lies on the positive x axis. Then what is the angle with respect to the x-axis of the vector passing through 1 and 2? Is it not 0? If it is 0, then you do not have a circle, if it is not 0 then your circumference is not "infinite".

    I think you are playing with things that can not exist. I think your circle is not well-defined. Thus you can draw conclusions that make no sense.

    Edit: spelling and I made an error.
     
  4. Oct 23, 2012 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    The difficulty is that you can't do any of the things you say! In particular, you cannot, geometrically, "wrap the positive real line into a circle." And, after that, nothing you say is correct.
     
  5. Oct 23, 2012 #4
    A simple proof that you can't do the wrap is: (I claim copyright on this dance)

    Consider any circle.
    Start anywhere on the circumference and mark zero
    Continue marking unit distance for points 1, 2,3,.....

    But length of the circumference is non integral so your last (joining) marked interval will be of different size from the rest ie non unit.

    Thus the original premise is false.
     
  6. Oct 23, 2012 #5
    you guys are good. Why cant the circumference be integral
    why cant my circle exist in the metaphysical realm of the ideals? Just kidding
     
    Last edited: Oct 23, 2012
  7. Oct 23, 2012 #6
    The real numbers are [STRIKE]countably[/STRIKE] uncountably infinite. That is there are more real numbers than there are natural numbers.

    http://en.wikipedia.org/wiki/Countable_set

    Be careful about the use of the word 'ideal' it has a special meaning in algebra and number theory.
     
    Last edited: Oct 23, 2012
  8. Oct 23, 2012 #7

    Mark44

    Staff: Mentor

    Edit inline, in blue.
    The natural numbers, integers, and rational numbers are countably infinite; the reals are uncountably infinite. That's probably what you meant to say...
     
  9. Oct 23, 2012 #8
    Thank you for the correction, Mark.

    I'm glad somebody knew what I meant, because I don't always myself.
     
    Last edited: Oct 23, 2012
  10. Nov 14, 2012 #9
    ok so maybe my circle doesn't exist in the euclidean plane. Why cant I define a new mathematical object and maybe we need to use non-standard analysis
     
  11. Nov 14, 2012 #10
    Like someone else said, if you put the center of this "circle" on the x axis, the center occurs at (infinity,0). The radius is not a finite distance

    By the way, what is your question?
     
  12. Nov 15, 2012 #11
    Why cant we construct the circle by taking the limit.
    we start with the circle centered at (0,0) with radius 1 and then take the limit as
    r goes to infinity. now out at infinity we start by drawing a line from the outer edge back to the center, and then we go like 1 unit over and do the same thing and draw a line back to the center and we do this countably infinite times. Now we make another circle at a finite distance from the center, now where these lines cross the circle of finite radius
    will there be gaps in this circle.
     
  13. Nov 15, 2012 #12
    Hello Cragar.

    Your construction is still not well defined because you can't draw the lines after you blow your circle up out to infinity. However, I will take the liberty of reformulating your question and you can tell me if it matches the idea in your head.

    Start with a circle of radius N centered at the origin, where N is a positive integer. The circumference has length 2piN, so we divide it up into N equal arcs of length 2pi each. (I am modifying the spacing to 2pi so that we divide the circle into an even number of pieces). Then we draw a ray from the origin to each vertex creating a system of N rays spaced by equal angles from the origin.

    Now we fix another circle of a definite radius R (also centered at the origin), and we mark the points of intersection of this circle with the rays. That set of intersection points depends on the radius N of the big circle. Now we consider that set of points as N grows larger.

    Question: As N goes to infinity, are there any gaps left in the circle of radius R?
     
  14. Nov 16, 2012 #13
    ya thats about it
     
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