From: Guillaume Marcais <guslist@...>
Date: 2004-10-29T04:46:51+09:00
Subject: Re: Rounding error, (100.0 2000 2001 Desktop Money Projects	Protege.png Protege_2.1 Q3.DIR Quicken RCS Screenshot-1.png	Screenshot-2.png Screenshot-3.png Screenshot.png afs-backup afshome	amixer.works b basic.edl bin bruin-woods c4isr.owl c4isr.pprj	cellphones.sxc comedi comedilib config config-n6uni config.austin	conv-factors.sxc debian drvinfo.txt dry.xml ds-tools emachines evolution	float.sxc fnfix foomatic-db-current.tar.gz grp-backup iFog-Aqua20-1.tar.gz	id_dsa.pub income jimo-core-sjenkins job-search ken-cole latex2docbook	letters lfront log mbse modules n6uni-etc.tar.gz n6uni-home.tar.gz	n6uni.tar.gz nasa nisotr old-ruby-tools openafs photos pickaxe2.pdf	rdtmerge resume rexml.sxc ruby-bug ruby-oscon.sxi ruby-tidal ruby-tools	smbmount.3.0.7 software themes tidal-3-11 tidal-3-5-patches	tidal-cvsroot-archive tidal-data tidal-head tidal-poster-corrected.ppt	tidal-poster.ppt tidal-web tidal-xdemo tidalsim.tar.gz uuid	vim-ruby-snapshot-2003-10-12 wet.xml yepp 9.95).to_i == 994

On Thu, 2004-10-28 at 14:29, Steven Jenkins wrote:
> Guillaume Marcais wrote:
> > Another funny failing of the notation is that not everything that looks
> > like a number actually represents a number. The following is quite
> > intriging:
> > 
> > x = 0.99999... (infinite serie of 9s)
> > 10*x = 9.99999...
> > 10*x - x = 9*x = 9
> > So x = 1!
> > 
> > IIRC, any 'word' that ends with an infinite serie of 9s does not
> > represent an actual real number.
> 
> Sure it does. As you just showed, .9999... == 1.0.
> 
> Decimal notation is just convention. We take it for granted, but it's 
> convention. Any infinite decimal expanson is interpreted as a power 
> series. Such series always converge, and you can show (as you just did) 
> that any repeating decimal expansion converges to a rational number.

True, the power serie somme of 9*10^(-n) does converge to 1. But the
point I was making is that we usually take for granted that the
representaion of real numbers as decimals is a one to one mapping to the
set of real numbers, but it is not. The set of real numbers is 'smaller'
than the set of possible representations.

The set of all words of finite size with characters in 0-9 has a one to
one mapping to the positive integers (0, 1, ..., 10, 11, ...). The
fraction notation n/m, with n and m two integers (positive or negative)
is NOT a one to one mapping to the rational numbers (-2/3 == 4/-6), but
it is if you add some constraints on n and m (remove all common factors,
have m positive not null). We just easily forget that to get a one to
one mapping to the real numbers you must add some constraint on the
suite of decimals.

Guillaume.