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Topic: Simple analytical properties of n/d
Replies: 20   Last Post: Mar 11, 2013 11:01 PM

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 ross.finlayson@gmail.com Posts: 2,688 Registered: 2/15/09
Re: Simple analytical properties of n/d
Posted: Mar 6, 2013 11:15 PM

On Wednesday, March 6, 2013 12:22:52 AM UTC-8, William Elliot wrote:
> So far, skipping the stuff for the undefined f,
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> we have the definition f_d:R -> R, x -> x/d for d /= 0.
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> > But, the "area" under f_oo(n) = 1.
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> You haven't defined f_oo
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> > Consider lim_d->oo Int_0->d 1/d dx, that equals one.
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> lim(d->oo) integeral(0,d) 1/d dx = lim(d->oo) x/d|_0^d = 1
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> > This looks like a flat line infinitesimally greater than zero, the area
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> > under which sums to one.
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> Meaningless.
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> > So, there are some _interesting_ properties, of f_d(n) = n/d.
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> Exercise. Graph f:(R^2 - Rx{0}) -> R, (x,y) -> x/y.

Mapping R^2 \ (0,0) the pointed disc to x/y, it is a many-to-one function where for each x =/= 0, for each r =/= 0 there is y s.t. for each x/y = r. So its image is the four quadrants minus the axes.

What then of each r/y or x/r, as r ranges?

In this context f_d(n) = n/d sees that there is a goal in the determination of what is continuity. The points on a line are not necessarily continuous (bear with me here, functions are continuous or the reals are a continuum etc) but the continuous line is points on a line, and the points in the line.

A _very_ simple notion of what is continuity is the evolution of points in a space, from the origin and ever expanding in space from the origin, a spiral space-filling curve as natural continuum and substrate of space. Then the strata or layers of the space, here casually, would be to defining integer points and regular tilings. Here as above with the consideration that there is one variable, the spiral space-filling curve forms the the unit disc (and unit hyperball) or might be surmised to do so, given geometric properties, of the circle, as maximizing area for a maximum distance from the origin. Then the triangle or trilateral and square also have properties, of regular polygons with a minimum area, for a maximum distance from the origin, and greatest ratio between rays through corners and bisectors of sides.

Here then as to why that is relevant to f_d(n) = n/d, it is that the shells of the spiral that fill the unit disc cross the x axis at the values of the range of f_oo, f_\infty or lim d->oo f_d, in exactly the same order as for n = 0, 1, 2, .... This polydimensional perspective of a thing that varies, as little as it can to vary in all ways, for simple principles of conservation and symmetry toward the definition of mathematical quantities, this polydimensional perspective restricted to the line is of the establishment of the continuum of the unit line segment, from the natural continuum of the natural integers. (And variously the naturals may have a point at infinity and be compact with regards to the closed and clopen unit line segment.)

On the line, the points have only one side. In the line (as of a set) they have two sides. So, when all the point-widths of the region under the points of f_oo are are scaled symmetrically back from infinity: the resulting tri-lateral shape, has twice the area of the shape with its bounds.

This is a remarkable feature of n/d.

https://duckduckgo.com/?q=%22spiral+space-filling+curve%22
https://groups.google.com/forum/?fromgroups#!search/%22spiral$20space-filling$20curve%22
https://duckduckgo.com/?q=%22polydimensional+perspective%22
https://groups.google.com/forum/?fromgroups#!search/%22points$20on$20a\$20line%22

The continuum is a central feature of mathematics.

Regards,

Ross Finlayson