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The Real-Line is a fractal

  1. Jun 8, 2004 #1
    "Real line" is used to mean real axis, i.e., a line with a fixed scale so that every real number corresponds to a unique point on the line. (http://mathworld.wolfram.com/RealLine.html)

    There are two basic states that stand in the basis of the real-line, which are:

    a) = (self identity).

    b) < or > (no self identity).

    Let x be a real number.

    Any real number, which is not x cannot be but < or > than x.

    The difference between x and not_x, defines a collection of infinitely many unique real numbers.

    The magnitude of this collection can be the same in any sub collection of it, which means that we have a structure of a fractal to the collection of the real numbers.

    In short, each real number exists in at least two states:

    a) As a member of R (local state).

    b) As an operator that defines the fractal level of R (a global operator on R).

    Any fractal has two basic properties, absolute and relative.

    The absolute property:

    Can be defined in any arbitrary level of the fractal, where within the level each real number has its unique "place" on the "real-line".

    The relative property:

    Any “sub R collection” in this case is actually R collection scaled by some R member as its global operator, and this is exactly the reason why some "sub R collections" can have the same magnitude as R collection.

    We can understand it better by this picture:


    In short, R collection has properties of a fractal.

    What do you think?
    Last edited: Jun 8, 2004
  2. jcsd
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