Re: Need some help with digits...

In article <1150004577.941631.271150@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
ytyourclothes@xxxxxxxxxxx wrote:

Caveat: I am not a mathematician. I had a couple semester of math on
the college level. (They never appeared to deal with anything
interesting ;)

I'm trying to understand the concept of a "digit" as it is being used
in math. The context is cantor's diagonal argument. Let me lay it out
as I understand it, maybe there's already something wrong with my
understanding.

Suppose there were some interesting list of real numbers that purports
to contain all rael numbers. You could write that list down and then
construct a new number by taking the n'th digit of the n'th number and
increasing it by one or setting it to zero if it is nine.

Actually, one wants a rule which will not ever replace a digit with
either a 0 or a 9, as these might lead to numbers with more than one
decimal representation. An acceptable rule would be to replace any digit
except 1 by 1 and replace 1 by 2.

This newly constructed number would have a first digit which is
different from the first digit of the first number on the list, its
second
digit would differ from the second digit of the second number, its
third digit would differ from the third digit of the third number and
so
forth. And therefore the new number can't be on this list as it is
different from each of the numbers by at least one digit.

(Someone correct me if I have mangled that).

Now I have always taken digits to be something ... dunno
... profane. Something to do with writing down numbers which in
themselves exist in any representation we can imagine (including none
at all). In particular, it is utterly obvious that the same number can
be expressed in a multitude of ways ( 1/4 vs 0.25 vs. "a quarter" etc).

My confusion stems from the fact that it is not clear to me that two
numbers are necessary unequal, just because they don't use the same
digits. For example there's the classical case of 0.999...=1.000...
where I'm expressing the same number using two different decimal
expressions.

As noted above, one can assure that this situation does not arise.

Most numbers have unique decimal representations.

Those that have more than one representation gain that ambiguity by
either having all but finitely many digits being 0 or all but finitely
many being 9.

Using a rule like that above, one can assure that the newly created
number contains neither of these digits at all.

Pushing this to an extreme: imagine Cantor's list was really very
boring and consisted only of the numbers {1.000..., 1.000..., 1.000...}
then I could claim that the number 0.999... is the "diagonal number"
of this list and proclaim to have proven that 0.999... is in fact NOT
equal to 1.000... because its first digit is different from that of
the first number on the list, its second digit is different from the
second digit of the second number etc.

Not if you use a rule which prohibits the use of either 0 or 9 in the
"diagonal" number.

Obviously there's something wrong here, but I'm kinda stuck with
this. So I figured I'd throw this out here and see if someone can
whack me over the head with the (undoubtedly obvious) mistake I'm
making here somewhere...

The problem is real if one is careless about how one constructs the new
number, but is easily solved with a little forethought.
.

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