Re: rational or irrational?
- From: "Ross A. Finlayson" <raf@xxxxxxxxxxxxxxx>
- Date: 14 Jun 2005 16:01:51 -0700
Timothy Little wrote:
> Nurxgwkk wrote:
> > is 0.00000000.....0000001 with and infinite amount of zeros
> > in-between an irrantional number or not?
>
> Not. It is not even a real number.
>
>
> - Tim
You'll notice that the variety of replies are quick to note that that
kind of construction is not a "real" number. This is where there are a
variety of small, but positive, numbers in various forms of what they
call the "nonstandard" real numbers, that are smaller than .00...001
for any finite number of zeros but greater than zero.
These infinitesimals do not have a decimal representation. They're
numbers that can not be represented by decimals.
Luckily, there is a thing called the transfer principle, so most any
thing that you can show true about the nonstandard real numbers can
also be shown true about the real numbers. For some, that means
they're one and the same thing. For example, if infinitesimals exist
in the hyperreals, and the reals are complete, then infinitesimals
exist in the reals.
That's arguable, just because there are everywhere reals between zero
and one, that there are infinitesimals between zero and one, if there
exist any infinitesimals, as there are, because via induction there is
no largest integer.
I think there are ways and ways to talk about, say, a least positive
real. It's easy to see why you can take any positive number, and halve
it, so it's not the least positive real number.
Ross F.
.
- References:
- rational or irrational?
- From: Nurxgwkk
- Re: rational or irrational?
- From: Timothy Little
- rational or irrational?
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