Adopey.1985
net.math
utzoo!decvax!duke!unc!dopey.bts
Fri Mar 19 10:46:33 1982
0.9999... < 1
     Just because we've all learned that .9999...  =  1,  we
should  not pretend that there's nothing upsetting about the
result.  We accept it because it's true within the system we
commonly  use,  but  this  is not the only system, nor do we
know that it provides the  best  model  of  nature.   It  is
possible--  if you will accept standard axioms of set theory
with the axiom  of  choice--  to  make  up  a  system  where
.9999... -- given a particular interpretation-- is different
from 1.  The  difference  between  them  is  an  "infinitely
small" number, an infinitesimal.  It's not easy to come up a
notation which gives names to particular infinitesimals, but
even  in the standard real numbers there are lots of numbers
without names. (Either consider the usual cardinality  argu-
ment,  or  consider a fraction with a non-computable decimal
expansion.)
     We all know-- if not, any text on analysis will provide
details--  how  to construct the real numbers from sequences
of rationals.  If we use function notation  for  sequences--
our  terminals  do  not in general allow subscripts-- we say
that  two  sequences  of  rational  numbers  s  and  t   are
equivalent, s ~ t, if

		Lim   | s(n) - t(n) |  =  0
	     n -> infinity

Then, the real numbers are the equivalence classes of  these
rational  sequences.   The  rational numbers we started with
are "embedded" in the reals by identifying  them  with  con-
stant   sequences.   Hence,  the  real  number  1/2  is  the
equivalence    class    of     the     rational     sequence
1/2,1/2,1/2,1/2,...
     To construct a  simple  "non-standard"  number  system,
start  with  sequences  of reals and an ultrafilter U on the
natural numbers, the indices for our sequences. (Here's  all
you  need  to  know about the ultrafilter we'll use.  It's a
collection of sets of indices.  Every set  of  indices  with
finite  complement is in U.  Every set of indices is in U or
its complement is in U.  And, if a set of indices is  in  U,
then  any super-set of indices of that set is in U as well.)
Now we say that two sequences s and t are equivalent, s ~ t,
if
		{ n : s(n) = t(n) } is in U.

Now, as you'd expect, the non-standard reals  are  just  the
equivalence classes of the sequences of reals.  The standard
real numbers are embedded in the  non-standard  universe  by
identifying them with constant sequences.  For instance, the
real number pi is associated with the equivalence  class  of
the sequence pi,pi,pi,pi,... , a typical infinitesimal might
be the equivalence class of the sequence

	0.1, 0.01, 0.001, 0.0001, 0.00001, ...

and a typical infinite  element  might  be  the  equivalence
class of the sequence

	1, 10, 100, 1000, 10000, 100000, ...

     Finally-- and this is the  first  deviation  from  what
you'll find in a text on non-standard analysis-- let's agree
on the following interpretation of  non-terminating  decimal
fractions.   If x is a non-terminating decimal fraction, let
x(n) be the n-th symbol of  x,  read  from  left  to  right.
Associate  x  with the equivalence class of the sequence sx,
where

		sx(n) = x(1)x(2) ... x(n)

This  means that  0.9999...  will  be  associated  with  the
equivalence class of the sequence

	0, 0., 0.9, 0.99, 0.999, 0.9999, ...

     There's one more technical detail, then I'll  get  back
to  0.9999...  and  1.  In general, a formula about two ele-
ments of the non-standard universe is true if, when you take
a  sequence  from  each equivalence class the set of indices
for which the reals in the sequences satisfy the formula  is
in  U.   Let  s be the sequence of associated with 0.9999...
and t the constant sequence 1.  Then 0.9999... < 1, since

	{ n : s(n) < t(n) }

	= { n : 0.999... n chars ...9 < 1 }

	= { all natural numbers } is in U

The difference between 0.9999... and 1, is infinitely small,
hence  smaller  that  any  standard  real number.  It is, in
fact, the equivalence class of the sequence

	1, 1, 0.1, 0.01, 0.001, 0.0001, ...

     I'm not trying to say that this is  a  useful   way  to
interpret  non-terminating  fractions.   I will claim that a
naive intuition can lead to  interesting  mathematics,  how-
ever.  Here's a way to assign a value to 0.9999... in a con-
sistent way, so that it is less than  1,  though  infinitely
close.
     If anyone out there in net-land is interested  in  non-
standard analysis after reading this, I'll be glad to supply
references.  And, since I'm  far  from  an  expert  in  non-
standard  analysis,  I'd  be  very happy to what anyone else
might be using it for.

		Bruce T. Smith, UNC-CH
		(...!duke!unc!bts)