Re: abundance of irrationals
From: Virgil (ITSnetNOTcom#virgil_at_COMCAST.com)
Date: 01/11/05
- Next message: John Baez: "Re: too much information!"
- Previous message: Will Twentyman: "Re: How many numbers in N have property is_positive?"
- In reply to: mueckenh_at_rz.fh-augsburg.de: "Re: abundance of irrationals"
- Next in thread: mueckenh_at_rz.fh-augsburg.de: "Re: abundance of irrationals"
- Reply: mueckenh_at_rz.fh-augsburg.de: "Re: abundance of irrationals"
- Messages sorted by: [ date ] [ thread ]
Date: Tue, 11 Jan 2005 13:53:35 -0700
In article <1105458371.912296.225750@f14g2000cwb.googlegroups.com>,
mueckenh@rz.fh-augsburg.de wrote:
> Virgil wrote:
> > In article <1105368373.075488.304720@c13g2000cwb.googlegroups.com>,
> > mueckenh@rz.fh-augsburg.de wrote:
>
> > > No, not for any positive epsilon. That is the big mistake. x^2 - 2
> = 0
> > > cannot be solved to a precision of 1/10^10^100 in x.
> >
> > X = sqrt(2) and x = -sqrt(2) are solutions with better precision than
>
> > that. They are just not expressed as decimal expansions.
>
> That are names for desired solutions, not numbers. If you compare
> Cantor's work you will find that this was his position too.
Decimal expansions are also "just names", not numbers, too. There is
always a distinction between the number and any of its representations.
That some representations are more useful thatn others for certain
purposes does not mean that they are anything more than representations.
Depending on one's model, a real number is either a certain type of
partitioning of the rationals (Dedekind) or a collection of infinite
sequences of rationals (Cauchy sequences whose differences are null
sequences). It is not, in any truly mathematical model, merely a decimal
expansion.
>
> Could you decide whether sqlmn < sqrlp, unless you have rational
> approximations for those names? That would be the minimum requirement
> for a number. For sqrt(2) you know a lot of rational approximations
> which enable you to decide the <, =, or > -question for a lot of other
> rational numbers. But you can't leave the rational domain.
> Regards, WM
Given either the partition form or the family of sequences form, as
referred to above, of both numbers, such questions are trivial.
- Next message: John Baez: "Re: too much information!"
- Previous message: Will Twentyman: "Re: How many numbers in N have property is_positive?"
- In reply to: mueckenh_at_rz.fh-augsburg.de: "Re: abundance of irrationals"
- Next in thread: mueckenh_at_rz.fh-augsburg.de: "Re: abundance of irrationals"
- Reply: mueckenh_at_rz.fh-augsburg.de: "Re: abundance of irrationals"
- Messages sorted by: [ date ] [ thread ]
Relevant Pages
|
