Re: Demonstrating that 0.999... = 1

From: PRL (prl_at_cyclops.com)
Date: 09/24/04 

Date: Fri, 24 Sep 2004 10:40:03 GMT

On Thu, 16 Sep 2004 17:06:02 -0500, David C. Ullrich <ullrich@math.okstate.edu>
wrote:

Having read all the posts on this now rather long thread, I have some comments
to make. Please excuse the tetchiness in places, but I consider some of the
responses to be unnecessarily snide and self-satisfied - not to mention
downright hostile. I get the impression that quite a few people are not here to
help others, but rather to impress with their knowledge; rather like contrasting
the class knowall to a good teacher.

>>Secondly, I apologise for not being aware that 1/9 does not mean 1 divided by 9,
>>or one ninth, but rather "a number such that when multiplied by 9 it yields
>>one". Whilst I can see that the later defition is true, I was not aware that it
>>differed in any way from the more commonly taught "one divided by 9".
>
>It doesn't - it's the _definition_ of what "1 divided by 9" _means_.

Firstly, I'm not aware of a mathematical distinction between what something is,
and what it means. I'd be grateful for a pointer.

>Yes, the problem is that you purport to prove something about .999...
>without realizing that you first need to get straight what a symbol
>_denotes_ before trying to prove something about it.

Hmm, denotes. A third word.

How does what it _denotes_ differ from its _definition_, differ from what it
_means_ ?

>
>No, ".9 recurring" or "a decimal point with an unterminating string
>of digit '9's following" are not the definition of .999... .
>They are definitions of the _string_ ".999...". That string is
>a name for a certain number, namely .999...; the number is not
>the string.

My understanding is that they define a string which is a representation of a
certain number.
>
>Since you've been kind enough to tell us how to explain why
>.999... = 1 I just assumed that you knew what .999... was,

I think I do.

>else you wouldn't be talking about it.

Actually, that is a wholly illogical statement. If I _knew_ that I didn't know
what it meant, then I _would_ be talking about it: I'd be trying to find out
what it meant. If I _didn't_ know (that I didn't know), then why wouldn't I be
talking about it. I'm sure there are many things that you _think_ you know, but
that others might think you _don't_ know, but I doubt that stops you talking
about them.

> In case you missed
>the post where I explained what .999... is, the following
>is an informal definition. (It's a little more formal-looking
>than what appears in that other post, because there I said
>what the sentence "0.999... = 1" means, which is one level
>simpler logically than saying what 0.999... is):
>
>(*) 0.999... is the number x with this property: you can make
>the finite decimal .999...9 as close to x as you want, just
>by taking enough digits.

I didn't miss it, but I can't quite see the relevance of it as an objection. I
think the post by "MikeVM" pretty much corresponds with my feelings having read
the thread.

>You might want to note that first that _is_ the definition,

As you are so sure, and for future reference, by both myself, and others,
perhaps you would be so kind as to let us know which accredited body has the
authority to state what any particular mathematical definition _is_, and also a
(preferably on-line) source to check these definitions, so that we don't waste
people's time in future.

>whether you like it or not, and second that whether you
>like it or not you _should_ like it,

A definition is what it is. I can't see where "liking it" comes into the
picture. I don't really care what a definition is, but I'd like to know when
I've got the correct one.

> because once you've
>convinced someone that that _is_ the definition it seems
>quite likely that you will also have succeeded in your
>quest to convince him that 0.999... = 1,

I don't see that at all.

They will only be convinced when they accept that the mathematical definition of
infinity does not imply somewhere where you can examine a property. i.e. it
isn't "somewhere where a number becomes overwhelming, or so small as to be not
worth bothering about".

Your definition does not help. I like the idea expressed elsewhere that
"enlightenment" is necessary.

>On the other hand, if you can't convince someone that (*) _is_
>the definition of 0.999..., there's really no point in discussing
>it. Because whatever actual fact he's insisting is false is
>simply _not_ the same fact as what "we" are insisting is true.

As stated elsewhere, surely it's better that someone believes a fact is true
than that they believe it's false. Whether or not they are fully aware of the
difference in some subtleties (which I have yet to see clearly explained) in the
nature of the "fact".

PRL