Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 hagman Posts: 1,923 Registered: 1/29/05
Re: Proof 0.999... is not equal to one.
Posted: Jun 1, 2007 5:31 AM

On 1 Jun., 08:55, chaja...@mail.com wrote:
> > If you insist, here is the first obvious mistake in the paper:
>
> > -------------------------------------------
> > Let S be the set of all real numbers in the interval [0,1]
> > Let T be the set of all real numbers in the interval (-1,0]

>
> > If you applied an operator '+' to these two sets that sums the elements
> > of both sets, you would get: S + T = 1
> > -------------------------------------------

>
> > You are trying to sum an uncountable set of numbers. Please define what
> > this means. Also consider: if you pair the numbers differently, can you
> > make the sum come out to something else?

>
> > FInally, I didn't notice anything looking like a proof in your article,

>
> > --
> > Eric Schmidt

>
> > --
> > Posted via a free Usenet account fromhttp://www.teranews.com

>
> Hi Eric,
>
> what I am attempting to point out in the S + T result is that any
> element of the infinite set T corresponds to exactly one element in
> the infinite set S of equal value and opposite sign, except for the
> positive 1 in set S. Should you hold the infinity of values in your
> awareness, all cancel except for one that cannot.
>
> You would not be able to have a final result upon summing the entire
> infinity of elements in S union T other than 1.
>
> --charlie

Oh, so you want to define sum(X) for certain subsets X of the reals?
This should be easy if X is finite: just sum the elements (fortunately
It should also be easy to define sum(X) if sum(Y) is defined
where Y = X\(-X): just set sum(X)=sum(Y).
One should investigate any further definitions as to whether all that
stuff yields a consistent definition of sum.

Your example amounts to obtaining sum( (-1,1] ).
With X = (-1,1], I would calculate Y = X\(-X) = {1} and obtain
sum(X)=sum(Y)=1.

Nothing has been said yet about sum(S) if S\(-S) is infinite, esp.
this has no relevance yet if S is the set of all rationals of the form
9/10^n.

However, we can form the set union of -S and 10 times S, i.e.
let C = (-S) u (10*S).
It turns out that C\(-C) is {9}.
We are forced to conclude sum(C)=9.

If t is a non-zero number and sum(S) is defined, can sum(t*S) be
anything
but defined and equal to t*sum(S)?

If A and B are disjoint and both sum(A) and sum(B) are defined,
can sum(A u B) be anything but defined and equal to sum(A)+sum(B) ?

In fact, the last two issues are what motivates the way to
calculate sum(X) from sum( X\(-X) ) used above and introduced by you.

Thus sum(-S) = -sum(S), sum(10*S)=10 *sum(S) and finally
sum(-S u 10*S) = 9*sum(S).
Hence sum(S) = sum(C)/9 = 1.

hagman

Date Subject Author
5/31/07 karl
5/31/07 karl
5/31/07 karl
5/31/07 Virgil
6/1/07 Richard Tobin
5/31/07 Glen Wheeler
5/31/07 The Ghost In The Machine
2/2/17 wolfgang.mueckenheim@hs-augsburg.de
5/31/07 Glen Wheeler
5/31/07 Glen Wheeler
5/31/07 David W. Cantrell
6/5/07 Michael Press
5/31/07 Dr. David Kirkby
5/31/07 mensanator
5/31/07 mensanator
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
5/31/07 quasi
5/31/07 William Hughes
5/31/07 William Hughes
6/1/07 hagman
6/1/07 William Hughes
5/31/07 T.H. Ray
5/31/07 Jesse F. Hughes
5/31/07 T.H. Ray
5/31/07 Jesse F. Hughes
5/31/07 T.H. Ray
5/31/07 Jesse F. Hughes
5/31/07 Denis Feldmann
5/31/07 T.H. Ray
5/31/07 T.H. Ray
5/31/07 Dave Seaman
5/31/07 T.H. Ray
5/31/07 William Hughes
5/31/07 Jesse F. Hughes
6/1/07 Eric Schmidt
6/3/07 T.H. Ray
2/2/17 wolfgang.mueckenheim@hs-augsburg.de
2/2/17 wolfgang.mueckenheim@hs-augsburg.de
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
5/31/07 Marshall
6/5/07 Michael Press
5/31/07 bassam king karzeddin
5/31/07 Glen Wheeler
5/31/07 bassam king karzeddin
5/31/07 bassam king karzeddin
5/31/07 neilist
5/31/07 tommy1729
5/31/07 neilist
5/31/07 tommy1729
5/31/07 neilist
5/31/07 tommy1729
5/31/07 Dave Seaman
5/31/07 quasi
5/31/07 quasi
6/1/07 Dr. David Kirkby
6/1/07 quasi
6/1/07 hagman
5/31/07 hagman
6/1/07 Dr. David Kirkby
6/1/07 hagman
6/1/07 Eric Schmidt
6/1/07 hagman
6/2/07 hagman
5/31/07 Richard Tobin
5/31/07 mathedman@hotmail.com.CUT
5/31/07 Richard Tobin
5/31/07 William Hughes
5/31/07 Jesse F. Hughes
5/31/07 Brian Quincy Hutchings
5/31/07 Brian Quincy Hutchings
6/1/07 Richard Tobin
6/1/07 Jesse F. Hughes
6/1/07 Richard Tobin
6/1/07 Dik T. Winter
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
9/26/07 JEMebius
9/26/07 mike3
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