JEMebius
Posts:
913
Registered:
12/13/04
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Re: Proof 0.999... is not equal to one - is a joy for ever!
Posted:
Sep 26, 2007 9:59 AM
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mike3 wrote:
> On May 31, 12:16 am, chaja...@mail.com wrote:
>> I have written a proof that 0.999... cannot be equal to one in the
>> system of real numbers.
>>
>> While at the end of it all you may not fully agree with my proof, much
>> I as have never seen a proof asserting they were equal that I was able
>> to consider valid, I'm sure you will agree that the
>> ideas I present are not a simply rehashing of basic objections of
>> others before me.
>>
>> It is available in several formats:http://www17.brinkster.com/chajadan/Math/Proofs/Proof1.dochttp://www17.brinkster.com/chajadan/Math/Proofs/Proof1.odthttp://www17.brinkster.com/chajadan/Math/Proofs/Proof1.txt
>>
>> --Charles J. Daniels
>> chaja...@mail.com
>
> _
> 0.9 must equal 1, however, at least in the real numbers. It may
> not in a system with infinitesimal/differential numbers but in
> _
> the reals 0.9 = 1, OK?
>
=======================================================================================
> I'd bet $1,000,000 that your proof contains some sort of logical
> fallacy. Based on what bit of reading of it I did it seems like
> you are assuming a system where infinitesimals exist, but such
> a system is NOT the real numbers no matter how much you'd like it
> to be.
A somewhat belated reply.
Proposal: The $1,000,000.00 will be paid off not in a single installment, but rather as
$900,000 + $90,000 + $9,000 + $900 + $90 + $9 + $0,90 + $0,09 + $0,09 +...,
often denoted as $999,999.999,999,....., quite in the vein of this newsgroup.
Got the relation to a manner of long division in which one always leaves a non-zero
remainder (10 div 10 => quotient 0.9 remainder 1) so as to keep the ball rolling?
Johan E. Mebius
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>
> "Again, the problem is introduced because decimal representation
> is not able to express or perceive a distance between a number
> that is infinite in position beyond the decimal point relative to
> other non-infinite decimals."
>
> Such a number is an infinitesimal, and thuse do not exist in the
> _
> reals. So 0.9, as a representation of a _real number_, *IS* equal
> to one.
>
> The other problem is that you seem to thin the definition of
> decimal system must be different. The sum
>
> sum_{n=-infinity...infinity} d_n B^n, 0 <= d_n < B
>
> is the _definition_ of the base-B repesentation of a real
> number!
> _
> For 0.9, d_n = 0 for n >= 0 and 9 for n < 0. The DEFINITION
> of infinite summation is
>
> sum_{n=-infinity...infinity} a_n =
> lim_{k->infinity} sum_{n=-k...k} a_n
>
> by DEFINITION, folks.
>
> This LIMIT equals ONE when a_n = 9 * 10^n when n < 0 and
> 0 when n >= 0. Since this Limit ***DEFINES*** the infinite
> sum and the infinite sum ***DEFINES*** the decimal then
> _
> 0.9 = 1, by DEFINITION.
>
> QED!!!
>
>
>
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