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Straight line?

  1. Mar 20, 2007 #1
    Hi
    My thought might be a bit weird or maybe childish, but I can not find anything to object it or to prove that it is wrong.
    As a basic definition of a straight line that it is an infinite number of points that are connected together.
    Regardless the length of that straight line, it consists of infinite number of points.
    I've thought about breaking that line into two halves, so I would get one half of infinity(starting from zero and going to +ve infinity and omitting the other half from -ve infinity to zero), what if I keep breaking the line for infinite times? the conclusion would be that the straight line is nothing but a point?
    If it was an infinite number of points, I would always get infinity even if I break it to an infinite number of times, right?
    I am really confused and would like someone to help me how to think it out.
     
  2. jcsd
  3. Mar 20, 2007 #2
    My 2-cents worth:

    A single, non-moving point can not define a line because the concept of a vector would not exist(with respect to a "line")

    You need at least 2 points.
     
  4. Mar 20, 2007 #3
    it will approach one point, but never actually be one point, it will just be an infinitesimal small line, which can be halved and halved and halved infinitly, the world could be a very small place, thats my thought at least. if you look at it as a specific size point, it may eventually get to a point where the line is so small it looks like one point, but it wont be
     
  5. Mar 20, 2007 #4
    From that point of view, there is no real point? because no matter how small that point is, it can be seen as a straight line from other point of view?
     
  6. Mar 20, 2007 #5
    on the contrary, i would say that there is a real point, but you cannot turn a line into a real point, and likewise you cannot turn a point into a line. Visually, yes, what you might say is the case, but eyes are deceiving. In reality, I would look at a point as a dot, zoom in a billion times, its still a dot, etc, but with a line, you zoom in enough, eventually you will see it is a line
     
  7. Mar 20, 2007 #6
    What does it mean to have an infinite number of points connected together? What does it mean to "see" a straight line? In analysis, a straight line is a function which rate of change is constant. The word straight is not relevant; it is just a, how would you say, visual definition.
     
  8. Mar 21, 2007 #7

    Gib Z

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    You're all thinking of a point as a little dot. A point is not a little dot, it is what it is called, a point. It doesn't matter how small our little dot is.

    Say we want it to represent co ordinate (0,0), with a dot of radius r. It doesn't matter how tiny out dot is, it excompasses all points which satisfy [tex]x^2+y^2 < r^2[/tex]. Since the reals are dense in the sense between any 2 non-equal reals in another real, it encompasses an infinite number of values.

    I sure as hell hope you don't think of fundamental particles like you do points, if you do physics..

    Basically, NO DOTS.
     
  9. Mar 21, 2007 #8

    Gib Z

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    Did you mean, with a curve, zoom in enough and see it becomes linear? Because that is only true of differentiable functions. Try fractals.
     
  10. Mar 21, 2007 #9
    That was helpful, thanks a lot all
     
  11. Mar 21, 2007 #10
    Trying to grasp the concept infinity mathematically can be difficult at first. Well done for trying. Think about what you wrote in the OP

    What do you mean by connected together? What connects the points?

    What is one half of infinity?

    For example take the positive integers

    1, 2, 3, 4, 5, 6, 7, ...................................................................

    an infinite set.

    These split into two halves the odds and the evens


    Take the half with the evens

    2, 4, 6, 8, 10, .......................................................................

    now start counting them. To do this put the count above the number

    1, 2, 3, 4, 5, .....................
    2, 4, 6, 8, 10, ................

    We may have split the numbers into two halves but there is as many in each half as we started with.

    You may find this site interesting

    http://www.mattababy.org/~belmonte/Publications/Books/CSaW/5_infinity.html
     
  12. Mar 21, 2007 #11
    line can't be a point

    How can one consider a line as a point I coulden't understood from which point of view u can consider it as a point
     
    Last edited: Mar 21, 2007
  13. Mar 21, 2007 #12
    It needs to be remembered that there's a difference between analysis and the physical. A point is a singularity and has thus no physical equivalent; you can't represent it with a dot. Seeing a straight line is just seeing the more or less accurate physical representation of a certain function. The word "dot" and "connected" are not relevant when it comes to analysis.
     
  14. Mar 21, 2007 #13
    well, I need two known points to draw a straight line, this is clear, but when this straight line is drawn, there in an infinite number of points connecting the two points I have connected together.
    for example
    f(x) = 2x + 5
    f(0) = 5
    f(1) = 7
    now, connect these two points on the graph paper, you get a straight line, right?
    how many points lay on that line? the answer is infinity
    now, resubstitute in the equation
    f(0) = 5
    f(0.5) = 6
    now the line is broken into half, but still having infinite number of points
    if and I say if I substitute this way
    f(0) = 5
    f(0+s) = 5
    where s a very small fraction that tends to be zero.
    so, what I've got here?
    two points, but no straight line?
     
  15. Mar 21, 2007 #14

    JasonRox

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    How does Analysis come into play here?

    What he means that all the points connected is that if you cross the line with another line, it will intersect at some point. Of course he didn't say that, but it's kind of implied. It's all semantics. He's not a mathematician.

    For example, if the line were filled with just rational points, it is not "connected" in the OP's definition. Why?

    Anyways, let's move on with the OP's question.

    The answer is no it never becomes a point.

    Say for example the length of the line is 64. The distance between one end point and the other. (You'll understand in a second why I picked this nice number.

    Divide that line in half and we get 32.

    Again, and we get 16.

    Again, 8.

    Again, 4.

    Again, 2.

    Again, 1.

    Again, 1/2.

    Again, 1/4.

    Again, 1/8.

    Again, 1/16.

    Again 1/32.

    See what's going on here? The line ALWAYS has length. It does converge to 0 though, but it is never 0. So, the line always has endpoints, call them a and b. They certainly are different points because b = a + e, where e is the length of the line. But how many points are there between a and b? There are infinitely many of course!

    If this line contained only the rationals, then it still has infinitely many points and that's not even fully "connected" in the OP's definition!
     
  16. Mar 21, 2007 #15

    JasonRox

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    Really?

    The real line is connected, and Analysis does in fact take note of this. And connectedness is a big property and not something irrelevant. Also, "dots" are also relevant because "dots" are limit points to sequences.
     
  17. Mar 21, 2007 #16

    JasonRox

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    Straight or linear, it's the same thing. Like I said, it's all semantics.
     
  18. Mar 21, 2007 #17
    So, if I take the limit of x when x tends to zero, what do I get?
     
  19. Mar 21, 2007 #18

    JasonRox

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    Um... 0. :surprised
     
  20. Mar 21, 2007 #19
    But the way he used connected pertains to the visual. Certainly if connectivity is analogous to continuity, analysis does take this into account. Because of continuity you see the visual representation as being "connected". However, if you see a "connected" line, it doesn't necessarily mean that it represents a continuous function; it could very well be discreet and defined on very close x's. And by dots, I meant big dots made with pencils. Not singular points. What I'm trying to point out is that the visual interpretation of things in the physical world are not necessarily translated to analysis. Simply saying "connected" and "dots" is not true to the definitions of analysis. But this debate is kind of useless; it's more philosophical than mathematical really.
     
    Last edited: Mar 21, 2007
  21. Mar 21, 2007 #20
    My 2 cents, and i don't know how qualified I am to say this (probably not very!) but a line is not an infinite number of dots. To me, a dot is a point, it has 0 dimensions, no length shape or size, nothing - in that sense it doesn't really physically exist, except as a mathematical tool to define a position, I suppose it fits into that other mathematical realm which is order. If you have infinite many dots, you just have infinite points in space, because each of these points have no length then they cannot make up the entire line, because the line must have a length > 0, and the sum of the lengths of all the points would be 0.
     
    Last edited: Mar 21, 2007
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