0.9999... < 1
unc!dopey.bts
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)