Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 ross.finlayson@gmail.com Posts: 2,720 Registered: 2/15/09
Re: Simple analytical properties of n/d
Posted: Mar 6, 2013 12:18 AM

On Mar 5, 7:40 am, "Ross A. Finlayson" <ross.finlay...@gmail.com>
wrote:
> On Mar 4, 8:08 pm, William Elliot <ma...@panix.com> wrote:
>

> > On Mon, 4 Mar 2013, Ross A. Finlayson wrote:
> > > Does n/d have these simple analytical properties?
>
> > Yes, that's a verbose characteristic of f_d which I [think]
> > you're talking about and not some arbitrary function f.

>
> Thank you, yes, that's so.
>

Thanks,

Rather leading questions, obviously enough anyone familiar with my
extended writings knows this is Socratic about the "natural/unit
equivalency function."

http://math.stackexchange.com/questions/283192/what-are-the-features-of-n-d-n-rightarrow-d-d-rightarrow-infty-n-d-in

Is there any neighborhood in [0,1] not containing infinitely many
points in the range of f? No, the points in the range of f are dense
in [0,1].

Then this simple construct of a function modeled by a family of
functions, "ranges" from zero to one, with that over the domain the
"linear" function sees values in the range between and including zero
and one and corresponding to the natural order of the domain, that
"ranges" doesn't need quotes, f ranges from zero to one.

Then, those are some simple properties of this construct. Then, the
free mathematical mind simply enough considers conditions as of other
analytical properties of this function. With n e R, or x e R+ for
f_d(x), Int_0->d f_d(x) dx = d/2. This is simply enough the area
under the line connecting the origin and (d,1). How about for n e R?
It, for r e ran(f_d(n)), f_d(rd), looks like f(x) = x, from zero to
one, the simple ray of a line with unit slope, from zero to one. The
area under that is 1/2. But, the "area" under f_oo(n) = 1. Consider
lim_d->oo Int_0->d 1/d dx, that equals one. This looks like a flat
line infinitesimally greater than zero, the area under which sums to
one. So, there are some _interesting_ properties, of f_d(n) = n/d.