On Sep 26, 8:54 am, JEMebius <jemeb...@xs4all.nl> wrote: <snip> ======================================================================================= > > > I'd bet $1,000,000 that yourproofcontains 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 => quotient0.9remainder 1) so as to keep the ball rolling? > > Johan E. Mebius >
With money though it quantizes at the cent mark, so it never gets to an infinite string of 9s.
But with the decimal system, an infinite repeating decimal like 0.9999.... is *defined* as the *LIMIT* of the sequence
0.a, 0.aa, 0.aaa, ... (0 <= a <= 9)
So here's a question: What is the LIMIT of
0.9, 0.99, 0.999, ...
By *definition*, that is the value of 0.9999..., OK?
It would take an infinite amount of time then to get $1,000,000 if paid according to your scheme. Just as to get 1/3 of that amount, paid off in installments of $300,000, $30,000, $3,000, $300, $30, $3, $0.3, $0.03, $0.003, etc. would.