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Topic: Proof 0.999... is not equal to one.
Replies: 194   Last Post: Feb 16, 2017 5:56 PM

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Re: Proof 0.999... is not equal to one.
Posted: May 31, 2007 5:12 AM

> So what about sqrt(2)? or Pi?
> Are those real numbers?

It depends on what we mean by real number. And yes, I'm painfully
aware that many consider there to be one exact specific definition and
nearly disallow all introspection into the idea itself due to this....
I regard them as real numbers because I have been taught to.
If by "are they real" you mean can every other real number be
expressed as strictly less or greater, than I would say quite likely
and I currently hold it to be so. Yet I find this to be seemingly true
about 0.999... as well and consider it in need of a different label
than real number.

The issue gets conceptually very complex for me. In a sense, it seems
no real number can intervene between 0.999... and one since decimal
cannot express one. But decimal cannot express a whole range of real
numbers that exist between 0 and anything it could ever hope to
express.

If we accept ideas we have been taught, sqrt(2) and pi will never be
able to explicitly located with a decimal representation. I do not
believe an infinity of positions would explicitly locate them either.
If you asked infinity itself to string digits together in decimal
notation until it had an exact location of an irrational number, I
doubt it would be able to comply in infinite completion. Much as we
just can't divide 100 stones into 3 piles no matter how we go about
it. This is something I have come to believe true. For the same reason
0.999... does not equal one, an infinite decimal for any irrational
number would be strictly less as well - even if astronomically beyond
our needs for accuracy. At no position within pi have we explicitly
located pi, and should we have an infinity of digits that accurately
represent the next needed increment in our system, I am unable to
attribue an ability to the structure of infinity to allow the final
infinite set to be any more explicit in its location.

>
So why do you accept those infinite
> decimals as real but not .999... ?

I do not accept infinite numbers to be real in my current
understanding. They seem more a relation. For any real number you
point out they will be more or less, but never specifying an exact
location themselves. This is an interesting idea to point if you just
step back and look at what it means to have an exact location on the
number line, as between any two real numbers lie an infinity of real
numbers infinitely nested.

In my current personal definition of real number, a real number
requires you stop at some point and say it's exactly in one knowable
position. Many may say it is itself a real number, but I do not feel
we can know other non-real relations such as 0.999... to have
properties we would expect of real numbers.

You'll have to excuse the implications of the ideas I am sharing, as
they are new understandings that I feel in no way relate to my proof
or are required by my proof. I share them only out of a sense of
openess and exploration with you.

>
> For that matter, what about the fraction 1/3?
> I'm sure you accept 1/3 as a real number.
> Do you accept 1/3 = .333... ?

I believe 1/3 is exactly a real number. I believe any division of a
span of the number line will yield a real number.

I do not accept 1/3 = 0.333.... I regard 0.333... as strictly less. In
my personal understanding, I could never hope to say 0.999... does not
equal one while simultaneously saying 1/3 = 0.333... so I am surprised
you would expect I might accept that as literally valid.

> Recall that the sum of an
> infinite series means, by definition, the limit of the partial sums,
> providing the limit exists.

The sum, a limit, is a definition of convention as it yields a value
usable by us. To me a limit expresses a bound on such an entity. It
does not dicate the nature of the entity itself. I believe the limit
as the number of places grows in 0.999... is 1 in terms of decimal
notation, but I believe an infinity of real numbers must exist between
0.999... and 1.

I feel it important people undertstand the first paragraph in my proof/
analysis. Decimal notation has a huge range of real numbers it cannot
ever hope to express.

--charlie

--charlie

Date Subject Author
5/31/07 karl
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6/1/07 Richard Tobin
5/31/07 Glen Wheeler
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5/31/07 Glen Wheeler
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5/31/07 David W. Cantrell
6/5/07 Michael Press
5/31/07 Dr. David Kirkby
5/31/07 mensanator
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5/31/07 Jesse F. Hughes
5/31/07 Dik T. Winter
5/31/07 Rupert
2/2/17 wolfgang.mueckenheim@hs-augsburg.de
2/2/17 JÃÂ¼rgen R.
2/2/17 abu.kuanysh05@gmail.com
5/31/07 William Hughes
5/31/07 Virgil
5/31/07 quasi
5/31/07 quasi
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5/31/07 William Hughes
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6/1/07 hagman
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5/31/07 T.H. Ray
5/31/07 Jesse F. Hughes
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6/3/07 T.H. Ray
2/2/17 wolfgang.mueckenheim@hs-augsburg.de
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2/2/17 bassam king karzeddin
2/2/17 wolfgang.mueckenheim@hs-augsburg.de
2/2/17 JÃÂ¼rgen R.
5/31/07 William Hughes
5/31/07 Dave Seaman
6/1/07 Glen Wheeler
5/31/07 William Hughes
6/1/07 William Hughes
2/2/17 wolfgang.mueckenheim@hs-augsburg.de
5/31/07 Glen Wheeler
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6/1/07 Richard Tobin
6/1/07 Jesse F. Hughes
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6/1/07 Jesse F. Hughes
6/1/07 Brian Quincy Hutchings
5/31/07 Dr. David Kirkby
5/31/07 quasi
5/31/07 quasi
5/31/07 quasi
6/1/07 Dr. David Kirkby
6/1/07 Virgil
6/1/07 Dr. David Kirkby
6/1/07 Dr. David Kirkby
6/1/07 Dik T. Winter
6/1/07 bassam king karzeddin
6/1/07 Dr. David Kirkby
3/22/13 John Gabriel
3/22/13 John Gabriel
6/1/07 Dr. David Kirkby
6/1/07 Denis Feldmann
2/7/13 Brian Q. Hutchings
2/8/13 JT
2/8/13 Virgil
2/8/13 JT
2/8/13 Virgil
2/8/13 Virgil
2/8/13 JT
2/8/13 Virgil
2/21/13 John Gabriel
6/1/07 JEMebius
6/1/07 bassam king karzeddin
2/2/17 bassam king karzeddin
6/1/07 mike3
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9/27/07 Brian Quincy Hutchings
6/2/07 OwlHoot
6/3/07 jsavard@ecn.ab.ca
6/5/07 zuhair
6/10/07 Brian Quincy Hutchings
2/2/17 wolfgang.mueckenheim@hs-augsburg.de
2/2/17 JÃÂ¼rgen R.
2/2/17 Robin Chapman
2/2/17 JÃÂ¼rgen R.
2/2/17 R.J.Chapman
2/2/17 JÃÂ¼rgen R.
2/2/17 JÃÂ¼rgen R.
2/3/17 R.J.Chapman
2/8/17 George Cornelius
2/8/17 abu.kuanysh05@gmail.com
2/13/17 Dan Christensen
2/13/17 bassam king karzeddin
2/13/17 bursejan@gmail.com
2/15/17 William Hughes
2/15/17 netzweltler
2/15/17 William Hughes
2/15/17 William Hughes
2/15/17 netzweltler
2/15/17 William Hughes
2/15/17 netzweltler
2/15/17 Peter Percival
2/16/17 bassam king karzeddin
2/16/17 Peter Percival
2/15/17 William Hughes