Re: Problems I have with 1.999...=2

This is an old debate, and frankly it look me several years before I
firmly came to grips with it myself. My understanding came in three
parts, firstly what our motivation was for these infinite decimal
expansions, secondly what exactly we ment when we said that 'it goes on
forever' and finally dealing with the equality itself.

As best I understand it we choose to define 0.999.. = 1 (or 1.999.. =
2) because we're trying to model the real number line from classical
geometry. I assume you're familiar with the irrationality of Pi, or the
square root of two, which both come from simple geometric objects from
compass and ruler geometry. Now we know that in science we have
certain limitations on how accurate we can take measurements and in
almost all practical situations this means estimating using a rational
number. We never take our our ruler, put it against the side of a
building and then go 'yep, that's exactly Pi meters long' but at the
same time, we realize that when we're modeling such problems it's
convienent to assume that Pi is an easily constructed numbers and we
want whatever system we're working in to be able to describe that
number. After all, it's not hard to imagine a cicle of radius 1, and
we'll want a way to describe it's area.

Now, considering that context and what we're trying to describe with
our numbers, we come to the problem of what these infinite decimal
expansions mean, and how exactly they relate back to our geometric
view. Now I assume you're comfortable with the rational numbers since
we seem to use them every day to describe pretty much everything, and
in fact from a scientists point of view, the rationals are exactly the
kind of data we'd expect to be looking at when we actually take
measurements. Also, we know that rationals are dense on the real number
line, no matter what two points on the line we pick we can find a
rational snuggled up close, in fact, arbitrarily close. We could,
potentially just pick out a list of rationals that always gets closer
to the quanity we're attempting to describe until it's of sufficient
accuracy for our purposes.

Here comes the leap - what if we don't know in advance how accurate
we're going to require our estimates? What if the degree of accuracy is
arbitrary? Well in this case, our list is going to go on forever, since
we already know that no rational number exists to describe such a
simple quantity like the square root of 2. So now, to describe these
special spots on the real number line, we are forced to use infinite
sequences of rational numbers.

I assume you're also familiar with how decimal numbers are
representations of infinite sums. An infinite sum is normally defined
to be an infinite sequence of numbers with each step of the sum taking
up a place on the list. So 0.9999... would actually be a shorthand for
the sequence (0, 0.9, 0.99, 0.999, 0.9999, ....). Notice that these
infinite lists only contain rational numbers for our purposes here so
naturally we don't see any objection to the entries and we understand
that they must be infinitely long because we don't know how much
precision we're going to require. Naturally we only consider lists of
this kind that estimate a point on the real number line not some
arbitrary list. As far as I understand these special sequences are
called convergent sequences and they have the special property that as
we go along the distance between members on the list constantly
diminishes after a certain point. More formally given a sequence S
whose n-th position is called s_n, that for any any arbitrary amount of
error, say e, that their exists a k such that for all m > k that the
absolute value of s_m - s_m+1 is less then e. That is to say the
sequence is always accurate enough for our purposes no matter how
accurate we require them to be. Notice we aren't doing anything fuzzy
here by disallowing certain sequences, after all we disallow certain
types of fractions like 5/0 just because they screw things up in the
same way we disallow sequences that mess things up - after all we're
trying to model the number line.

Ok so now comes the funky part, these infinite sequences can cause us
some trouble. In fact, many of them describe exactly the same point on
the real line. Consider a simple example like (0, 0.9, 0.99, 0.999,
.....) and (0.9, 0.99, 0.999, ...) which seem to be approximating the
exact same point on the line. After all, the only difference is the
first term. We could also do something like take either of these
sequences and skip every second term or every third and still get a
list that estimates the point on the line we're looking for exactly as
the original list did. Which list should we choose?

Well another thing about rational numbers is that sometimes fractions
run into the same problem. I mean when we say 1/2 we mean the same
thing as when we say 2/4, or 4/8 right? Even though they have different
members, and aren't *exactly* identical in how we represent them, we do
realize that for practical purposes they are the *same*. Why do we say
they're the same? Well because those other ways of writing 1/2 have the
same properties as the other ways of writing it (IE we'd add, subtract,
multiply and divide them as usual) and we have a simple rule to tell
when two representations depecit the same number. For rational numbers
we say two representations, we'll call them a/b and c/d, are the same
if and only if a*d = b*c. Take a minute and convince yourself of this
if you're uncertain.

So notice, even though the fractions we use to describe the rationals
sometimes overlap we didn't throw them out, we just made a simple rule
to tell when two are the same. The real numbers and our infinitely long
lists are no different. Our problem is to make sure that we have a
simple rule telling us when two of these sequences represent the same
point on the line. Mathamaticians do this by defining what's called an
'Equivilence Relation' on the set of convergent sequences. An
equivilence relation is an abstraction of what we mean when we say two
things are the same. For example, you might only be interested in how
long something is, or what colour it is, or what species. Each of these
ideas captures to some degree the idea of 'sameness' depending on the
context, but would still fail to be *exactly* the same in every
concievable way. But as we can see with the rational numbers we have a
useful motivation for letting some objects that aren't exactly the same
be considered the same in our context.

Equivilence relations have some rules, namely x = x, x = y implies y =
x and finally if x = y and y = z then x = z. Lets try to build your
intuition by replacing the = sign with the phrase "is the same colour
as". Naturally if x is red, then x is the same colour as x. Moreso, if
x is the same colour as y, then y is the same colour as x. On top of
that, if x is the same colour as y, and y is the same colour as z, then
x must naturally be the same colour as z. It's a simple example, but we
can see how this captures the idea of sameness and moreso, these are
the three rules we use in algebra to make substitutions, prove
equality, and otherwise exploit all the advantages that sameness gives
us.

These three simple rules also do something else very clever. We could
say that Partition a group up into catagories of similar things (based
on context) and say anything in the same little group is 'the same' as
everything else in the group. So we could organize our wardrobe by
colour for example. Now considering these groups and our three rules,
we can see clearly that if x is in a group, then x is in that group,
and if x is in the same group as y, then y is in the same group as x.
Morso, if x is in the same group as y, and y is in the same group as z,
then of course x is in the same group as z.

Lets apply this idea to the rational numbers. For the rationals we had
fractions, and sometimes they would be the same number like 1/2 and
2/4. We notice these are just pairs of numbers, split by a / sign with
the special property that in the integer number system 1*4 = 2*2. So we
take all of the numbers written in the form p/q with q not 0 and
given any two of these objects we group a/b and c/d together if and
only if a*d = b*c. So we've taken all of the objects of the form p/q
and grouped them into bunches based on this rule.

Now for the real numbers we have a whole bunch of convergent sequences
some of which get arbitrarily close to the same point on the number
line. What rule should we use to bunch them up? We would say that given
some arbitrary error, say e, which for convinence we always say is
greater then 0, that the distance between the terms of our two
sequences eventually is lower then e. So we have two convergent
sequences Q (with terms of the form q_n) and R (with terms of the form
r_n), we would say that Q = R if and only if their exists a positive
interger k such that for all m > k the absolute value of Q_m - r_m < e.
Remember that e is arbitrary so no matter how small the error is, we'll
always be able to find terms of Q and R that satisfy these properties.
We now partition the set of convergent sequences into groups based on
this rule, and say that everything inside these groups is the 'same' in
our context, which is the real number line we've been trying to model
all along. Since this forms a nice partition, they follow the three
rules of an equivilence relation, and we can suddenly do all the
algebra we're familiar with with a few more simple explainations of
what we mean by 'adding' and 'multiplying' these sequences together.
I'll let you discover those on your own.

So finally we can see how we've come full circle, we've defined what we
mean by these infinite decimal expansions and from the context of
geometry slowly built up the properties we wanted them to have using
the rational numbers and our own reasoning. Since the above defines
precisely what we mean by two of these infinite sequences being the
same, we now turn to the question of 1.999... =2. In this case we have
two sequences, the first being (1, 1.9, 1.99, 1.999, 1.9999, ....) and
the other being (2, 2, 2, 2, 2, 2, 2, 2, ...). Both of these are
convergent sequences, so we know they're allowable, the only question
left is do they both represent the same point on the real number line?
So we turn to our equivilence relation that says if we subract these
two term by term, given some arbitrarily small number e, can we find a
spot where this difference is less then e.

Lets consider what happens when we subtract these two sequences term by
term. We get a new list, that goes like this (1, 0.1, 0.01, 0.001,
0.0001, ...) Now no matter what e I chose to start with, I could always
find a point on that new list that is less then e since e is greater
then 0 and fixed. This means we would bundle (2, 2, 2, 2,...) and (1,
1.9, 1.99,...) into the same group and would indeed call them equal in
the context we're discussing. I hope this answers you're question.

.

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