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"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)

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

a)= (self identity).

b)< or > (no self identity).

Letbe a real number.x

Any real number, which is notcannot be but < or > thanx.x

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

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 inat leasttwo states:

a)As a member ofR(local state).

b)As an operator that defines the fractal level ofR(a global operator onR).

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 “subRcollection” in this case is actuallyRcollection scaled by someRmember as its global operator, and this is exactly the reason why some "subRcollections" can have the same magnitude asRcollection.

We can understand it better by this picture:

http://www.geocities.com/complementarytheory/Real-Line.pdf

In short,Rcollection has properties of a fractal.

What do you think?

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# The Real-Line is a fractal

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