Re: Demonstrating that 0.999... = 1

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 09/24/04 

Date: Fri, 24 Sep 2004 07:11:37 -0500

On Fri, 24 Sep 2004 10:40:03 GMT, prl@cyclops.com (PRL) wrote:

>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.

Things don't mean anything, things are just things. Symbols mean
things. "1/9" is a symbol, denoting the number 1/9. What that
symbol means is the same as what 1/9 is.

>>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_ ?

What a symbol denotes is the same as what the symbol 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.

That's correct.

>>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.

Find a book on mathematics. Ok, try the section on infinite decimals
in Ross "Elementary Analysis: the Theory of Calculus".

Or if you want an online reference look at the place in the FAQ
that someone gave a reference to elsewhere in the thread.

Various people have pointed out that various people might mean
other things by "0.999...". That's true, but irrrlevant.
We're talking about what to say to people who don't believe
that "0.999...=1". What a mathematician means by that statement
is what I said - if the objector things it means something
else then the statement he's objecting to is not something
that I say is true.

>>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.

Have you ever found someone who does not believe the statement
"You can make the finite decimal 0.999.9 as close to 1 as you
want, just by taking enough decimals"?

I've never seen anyone who doubts that. The relevance is that
if someone doesn't doubt that then that person _does_ believe
that 0.999...=1 - the confusion is over what the symbols mean,
not about the actual facts.

>They will only be convinced when they accept that the mathematical definition of
>infinity does not imply somewhere where you can examine a property.

This is nonsense - the fact that 0.999...=1 has nothing whatever to
do with infinity.

>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

************************

David C. Ullrich

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