JEMebius
Posts:
913
Registered:
12/13/04


Re: Proof 0.999... is not equal to one  is a joy for ever!
Posted:
Sep 26, 2007 9:59 AM


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? >
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> 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 nonzero 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 noninfinite 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 baseB 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!!! > > >

