Exploring Number Line Density: Constant or Variable? Mathematical Proof?

[Loren Booda]

Is the number density of the number line constant or variable?

Can either be proved mathematically?

 

[dodo]

Well, since no new numbers come into the line, nor any existing number jumps out, I'd call it constant... notwithstanding the fact that I have no idea on how to define 'number density' in the first place.

 

[Norwegian]

Is the number density of the number line constant or variable?

I guess Loren is free to define "number density" in any way he likes. And until he does so, it will be hard to give a sensible answer.

Can either be proved mathematically?

The wording of this question comes up quite often by non-mathemticians, and I always wonder what they mean with that last word.

 

[Loren Booda]

Is the number density of the number line constant or variable?

"Real number" lines have a variable density where they and their well-defined operations do not complete the intervals (-oo, +oo).

 

[HallsofIvy]

Okay, so please tell us what you mean by "number line density"!

 

[Loren Booda]

Thanks, Hurkyl.

Consider the real number line with number distances conserved relative to each other.

Now, for instance, distort the positive ray so that the distance between its whole numbers is doubled relative to those of the negative ray. The number line density of the positive numbers has become half that of the negative numbers.

A distortion of relative number distances on the real number line corresponds to a change in the real number line density. There exist other operations which can cause a relative change in the real number line density.

 

[SteveL27]

Thanks, Hurkyl.

Consider the real number line with number distances conserved relative to each other.

Now, for instance, distort the positive ray so that the distance between its whole numbers is doubled relative to those of the negative ray. The number line density of the positive numbers has become half that of the negative numbers.

Actually that's not true. Even if you stretch the number line, the stretched version is just as dense as the original. You can even stretch a finite interval into the entire infinitely-long real line, and both the finite interval and the entire real line are equally dense.

That's what they mean by calling it a continuum.

 

[Loren Booda]

SteveL27,

You're quite right.

However, "number density" as described next relies on infinite sets changing on similar scales:

Might different infinite cardinals vary in "number densities" (infinitely discontinuous) or are they all necessarily continuous (density one)?

Also, might discontinuous fractals describe "number densities" related to their dimensionalities D (here approximately a line, where 0 < D < 1)?

 

[dodo]

"number density" as described next [...]

Unfortunately, what follows next are two more questions, not a definition.

It seems to me that you struggle to put your finger on a concept which is hard to grasp and which you are passionate to investigate about. If that is the case, your first task should be to arrive at a definition of your new concept. Try some, even if temporarily or tentatively. Something precise, of the form "density is the cardinality of the set constructed this way..." or "density is a function from a set to the reals with the following rule...". To what objects can you apply your concept of density, besides the real line? To intervals? To sets in general, or which kind of sets? To other objects? These questions may help you complete parts of your definition. You really need to make the attempt to nail it down, as flying around with a concept between quotes will leave you nowhere (or anywhere, which is the same).

 

[Loren Booda]

Number line density can be defined generally for fractal dimensions D[>=]1.

The fractal range 0<D<1 represents singular "dust" of number density zero, and D>1 includes continuous or discontinuous, single or multiple structures.

Conventional space of whole number dimensions have number density equal to one.

A conventional, continuous line thus has number density equal to one.

Since the fractal line (or line fragments) of D=1 extend into a second dimension, their number density is diluted to less than one, but greater than zero.

Number density can be more accurately calculated with the relation N=S^D, where S is the scaling factor, D the dimension, and N the ratio between the size of a fundamental fractal fragment to a scaled-up fractal fragment.

Knowing this, the number density can be defined approximately as N/S=S^(D-1).