0.99999, infinity

[yale-com!nglasser]

It is true, and well known, that 0.9999... is equal to 1. But what does it
mean to say 0.000...1? At what decimal place does the 1 occur? 0.9999...
is well defined, but 0.000..1 does not make any sense. It is like saying,
"Take an infinite list of numbers, and at the 'end' add another number."
This is obviously obsurd. We could add another number at any given point in the
list, but there is no end to the list, so we cannot add a number to the end
of it.

Another point about 0.9999 = 1. The proofs of the form

x = 0.99999...
10x = 9.99999...
9x = 9
x = 1

are not valid. Otherwise you could use a similar argument to show that the
series ln2 = 1 - 1/2 + 1/3 - 1/4 + ... converges to any number you
like, or even diverges. You must add some more details to the proof, such
as the fact that the series in question converges absolutely, that addition
of series term by term is valid, etc. to make it valid. The easiest proof is
the one already given in this newsgroup evaluating this number as the sum of
an infinite geometric series.

As to infinity, most books on set theory will be able to give you a good
picture of cardinalities of infinite sets, and whether sets are countable
or uncountable.

- Nathan Glasser
..decvax!yale-comix!nglasser