|
|
Re: Proof 0.999... is not equal to one.
Posted:
Jun 1, 2007 6:20 AM
|
|
On May 31, 7:07 am, William Hughes <wpihug...@hotmail.com> wrote:
> On May 31, 8:01 am, chaja...@mail.com wrote:
>
> > > You do however have to give some sort of definition for 0.999...
> > > Whatever you define it to be it will either be equal to 1
> > > or it will not be a real number.
>
> > My definition of 0.999... is the sum of all elements of an infinite
> > set defined by 0.9*(1/10)^n for all n in the set of wholes numbers
> > including 0 and is included in my proof.
>
> You are defining 0.999... in terms of something else
> you have not defined, the sum of all elements of an infinite
> set. The standard definition uses a limit and produces
> a real number. So you are not using the standard definition.
> If you do not use the standard definition you need to
> supply a definition (and it would be good practice to
> use another name).
>
>
>
>
>
> > I do not attribute to this
> > entity any other characteristics, need not for it to be real or non-
> > real. I allow the consequence of the infinite contibutive values alone
> > to dictate all else.
>
> > > No. If you use the standard limit definition, 10x-x = 9.
> > > You are only correct if you use some other definition for
> > > 0.999... in which case the "subtraction" is not in the
> > > real numbers. You need to define a new set of "numbers".
> > > When you do so you will not get a field.
>
> > > - William Hughes
>
> > I avoid all limit definitions. To me limits are their own area of
> > study that tell you potentially more about what bounds an entity that
> > about the entity itself.
>
> > The limit of 1/x as x approaches infinity is 0, but this is not
> > representative of the function at all which will never yield a zero
> > value. I do not reject ideas that discuss and define 0.999... as a
> > limit - I only reject ideas that attempt use a limit to ~equate~ to
> > the entity described by that limit when this is not justified.
> > I leave
> > this disclaimer only to take into account a constant limit, such as
> > the limit of 3 as x approaches infinity - where the limit is exactly
> > equal to the number that yields it.
>
> Unfortunately, there are sequences like (.9,.99,.999,...) that do
> not reach anything and do not have an element equal to the limit.
> You want to talk about such sequences but you are on the horns of a
> dilemma. You
> do not wish to use the limit, but you have nothing else to
> replace it with.
>
> -William Hughes- Hide quoted text -
>
> - Show quoted text -
It didn't feel it was a long stretch. If you have a set defined by {2,
3, 6} the sum of its elements is 11. If you have a set defined by
9/10^n for all n in the set of whole numbers not including zero, if
you summed all of its elements you would get 0.999...
I originally wrote a substanially larger proof that defined everything
from scratch but I didn't want to insult anyone's intelligence or bore
those who could tell what I meant off the bat.
I have come increasingly close to seeing that ~trained~ mathematicians
have lost all ability to generalize. It seems nearly any student at
the local community college would read what I have wrote and taken it
for what it is. A trained mathematician is flustered at paragraph one.
Yet trained mathematician are the ones who teach kids that 0.999... is
exactly in everyway equal to one without mentioning limits once, and
completely ignoring what students will originally take 0.999... to be.
Then, when one of these students, namely me, tries to talk about the
entity on its own terms, suddenly everyone refuses to allow me, trying
to corral me into having implied limits everywhere.
This has been educational, I simply realize that when communicating to
a chinese person I shouldn't use english, and when communicating to a
mathematician I have to know the entire body of mathematics and be
explicit to the nth degree. Sounds good to me - fun even.
--charlie
|
|
