Re: Real numbers , what are they good for ?

On Dec 22, 4:26 am, Kira Yamato <kira...@xxxxxxxxxxxxx> wrote:
On 2007-12-21 04:49:53 -0500, simple.pop...@xxxxxxxxx said:



On Dec 17, 4:13 am, "porky_pig...@xxxxxxxxxxx" <porky_pig_jr@my-
deja.com> wrote:
On Dec 16, 5:37 pm, jane <jane1...@xxxxxxxxxx> wrote:

A bit shaky question :

A friend of mine was asked the following question:

What are real numbers good for ?

How would you answer this question, stressing properties which, differ
exactly real numbers and which are useful in analysis ?

I can give my own answer as well, but i would like to see other
opinions, perhaps written shorter and having more information.

Thanks.

Analysis formalizes the intuitive notion of 'continuum', that is,
continuous time-space fabric, 'without holes'. On one (spatial)
dimension that would be a continuous line, which analysis models as a
'real line'. The basic requirement for such line is that if some
sequence converges to some point, that point must be on that line,
otherwise it's a 'hole in a line'. It can be shown that the rational
numbers can't serve such purpose (e.g., we can think of some sequence
whose limit is sqrt(2), but it can be shown that sqrt(2) is not a
rational number). So we enhance (complete) rational numbers, but
plugging up all the potential holes (the formal operation is known as
'completion', there are basically two ways to do so, Cauchy-Cantor
limit based approach, and Dedekind cut. the bottom line that it is
doable in a rigorous way). So, informally, we ending up constricting
some number system such that if some sequence converges to some
number, that number is a real number, on a real line, no holes in real
line. A formal definition (if set of real numbers is bounded above, it
has the least upper bound, or supremum) serves as a starting point of
the real analysis.

(Completing the rational numbers is rather messy procedure. We want to
show that the resulting system has all the properties of rational
numbers we got used to: associativity, commutativity, distributive law
etc. Lots of gory details, but in the end everything gets proven).

Excellent informal summary. May be you could have done more justice
for Dedekind cut also by giving it also a line or two. or is it
points?

In my view, the real number to which a Cauchy sequence converges to is
a a point having infinitesimal width and not a Dedekind cut. And any
two different Dedekind cuts are separated by a finite or infinitesimal
interval, but not a zero interval. I might change/correct of my views
later - but at this moment I should tell what I think ...

What's an infinitesimal interval?

I knew it is not in standard reals, and I'm swimming against the tide
here.

The infinitesimal is an infinitely small number, as opposed to that of
infinitely large number. Note that an "infinitely large" is a relative
term, and not some absolute largest (roof). So, zero, being an
absolute smallest positive number (floor), can't be the reciprocal to
"infinitely large" number. The "infinitely small" or infinitesimal
number takes this role of being reciprocal to "infinitely large"
number.

Obviously all this can be shown invalid in standard real analysis. But
I think nature's symmetry needs this.
.

Relevant Pages

  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... construction of the reals is designed to patch all of the holes. ... Then that proves that "these holes exist", ... Can a general expression ... example of such a polynomial equation. ...
    (sci.math)
  • Re: Real numbers , what are they good for ?
    ... Analysis formalizes the intuitive notion of 'continuum', that is, ... numbers can't serve such purpose (e.g., we can think of some sequence ... for Dedekind cut also by giving it also a line or two. ... at least in the standard model of the reals. ...
    (sci.math)
  • Re: Peanos space-filling curve
    ... > because if these 'holes' cause me a problem, ... > assures us that sets of reals can divide into two without ... Without Dedekind cuts (or Cauchy sequences, ...
    (sci.fractals)
  • Re: Peanos space-filling curve
    ... > because if these 'holes' cause me a problem, ... > assures us that sets of reals can divide into two without ... Without Dedekind cuts (or Cauchy sequences, ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... "is continuum completely filled up". ... reals e.g. the 2D fractal curve filling all space, ... mapping of points from P1 to P2 via linear interpolation, ... response was that, if we are concerned about holes amongst the reals, ...
    (sci.math)