Talk:0.999.../Arguments - Misplaced Pages

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This page is for mathematical arguments concerning 0.999.... Previous discussions have been archived from the main talk page, which is now reserved for editorial discussions:

It does not matter what you think and it is you who appears to have difficulty grasping even the most fundamental concepts in Mathematics. Of course there is a philosophical element; mathematics started with philosophy but you appear to be ignorant of this fact. Tell me, does a finite line have infinitely many points? And now that you mentioned definitions, please tell me what in your opinion a point is. Furthermore, your ignorance shows and if I were you I would refrain from posting comments that you have not given sufficient thought.71.248.147.163 01:28, 24 February 2006 (UTC)

That's your second. Melchoir 01:43, 24 February 2006 (UTC)

I have a lot to say, but this is really getting us nowhere, so I will cease this argument. Sorry. -- Meni Rosenfeld (talk) 06:15, 24 February 2006 (UTC)

Just to give the definitions we were asked for: A line is, in the context of elementary geometry, a one-dimensional affine space. Since here we only consider objects over the field of real numbers, all lines have infinitely many points (even uncountably many). (If we instead considered some finite field, lines would have only finitely many points, but that is irrelevant for the subject at hand.)

I could not find a definition for a "finite line", unless a line segment is meant. Since the interior of a line segment can be bijectively mapped to the entire line, a line section also has infinitely many points.

Finally, according to Euclid, a point is that "which has no parts" (or something to that effect). But even Euclid did not use that definition but rather a set of axioms points shall satisfy. In analogy to my "line" definition above, I could also define a point as a zero-dimensional affine space.

I fail to see a connection between these definitions and whether 0.9999...=1 or not. Huon 11:16, 24 February 2006 (UTC)

You fail to see what a point has to do with this fact? My, oh my. It has everything to do with it. You are trying to have 0.999... which you claim is a number on the real line equal to 1. Euclid did not define a point: "Simeon estin ou meros outhen" means in Greek: "A point has no part." This so called definition is such a load of rubbish. It is meaningless and was written down by Euclid because he could not define the concept of a point without introducing self-referential terminology. In fact a point and a line are defined in terms of each other. It is completely ridiculous to talk about rigour in modern mathematics because modern mathematics is based on this garbage. It is based on concepts and notions that are not clearly defined. This is one of the reasons real analysis has failed. If you talk about a real number line, you are talking about points and if you are talking about points, one of these points represents 0.999... and another point represents 1. Now I have in the archives shown and proved conclusively that having 0.999... < 1 does not violate any mathematics (not even the archimedean property). I asked you why you compare real numbers differently and you were not able to answer, nor anyone else as for that matter. Again, I ask you this question: Why do you compare the limit of 0.999... with 1 and why do you compare the partial sum of pi with any other number? Just start by answering this last question. We can try to redefine point in a logical, non-referential (or cyclic) way later. This is extremely difficult to do. 71.248.136.114 18:00, 27 February 2006 (UTC)

Huon, I think it is better in this context to state that a point needn't be defined at all. Every mathematical theory needs several elementary notions which are not defined and receive their meaning from axioms - otherwise the definitions would inevitably be circular. -- Meni Rosenfeld (talk) 07:57, 26 February 2006 (UTC)

Meni: My answer is both yes and no. In a purely geometrical context, you are right; lines also can be characterized axiomatically. But I assumed that we wanted to consider the special case of the "real line" and points on that line. In that case, the concept of points is more limited. Yours, Huon 13:16, 26 February 2006 (UTC)

Anon: I agree Euclid's definition of a point is useless. But neither Euclid nor any mathematician since then (that I know of) made use of that definition. I also fail to remember any set of self-referential definitions of point and line (could you provide a reference?). Now in our context of 0.9999... and 1, we do not need the complete geometric baggage. For example, we do not have more than one line, and while every totally ordered set (like the set of real numbers) could be thought of as a line, that seems to be unnecessarily complicated. But I happily agree to use geometric language if you prefer.

Now you claim that 0.9999... and 1 are represented by different points on the real line. These distinct points have a non-zero distance, say x. (The one advantage of your geometric interpretation is that we can easily form sums, differences, and products with natural numbers.) You also claim that the Archimedean property holds. Thus, there is a natural number n such that n*x>1. Especially, x*10^n>x*n>1, so x is greater than 1/10^n. Then 0.9999...=1-x<1-1/(10^n)=0.9999...9 (with n nines). Please note that every operation I performed can be done geometrically (with the exception of finding n, but you agreed that the Archimedean property holds, so n exists). Somehow I doubt you are willing to accept my line of reasoning. Please be precise in stating where you disagree and give full details.

Finally you ask: "Why do you compare the limit of 0.999... with 1 and why do you compare the partial sum of pi with any other number?" I don't quite understand the question, probably because of its terminology. To me, 0.9999... does not have a limit, it is a limit (say, of the sequence (0.9, 0.99, 0.9999, ...)). Of course I compare that limit with 1, since I cannot easily compare sequences (the set of sequences is not totally ordered, at least not canonically). Now pi can also be considered as a limit (although to give a sequence is more difficult). Of course then I also compare the limit of that sequence to other real numbers if I want to get statements abut pi. If I only need approximate statements (say, whether 0.999...>0.5 or whether pi>3), then it may suffice to compare only partial sums (as I can probably find monotonous series), but pi and 0.9999... are not treated in different ways. As an aside, there is not the partial sum of pi. Any series has infinitely many partial sums. Yours, Huon 21:39, 27 February 2006 (UTC)

The reason I state that there are infinitely many numbers between 0.999... and 1 is because you can in reality only compare 'partial sums' where most numbers (numbers are defined in terms of series) are concerned. It makes no sense in some cases to compare the limits of certain series and in other cases to compare the partial sums. You state that pi can be considered as a limit even though you can't provide a sequence - this is untrue for a limit is defined in terms of a sequence. So, pi cannot be considered a limit. Furthermore, if pi cannot be found in its 'entirety', its limit cannot be known, hence you are not comparing a limit when you compare pi to any other number. 71.248.129.191 15:53, 28 February 2006 (UTC)

If numbers are defined in terms of series, then one always has to consider them as limits of those series (or, more precisely, as the limit of the sequence of partial sums). One cannot consider them as partial sums, since some partial sums may coincide for different series. Now concernung pi, there are several series. From the article pi:

$\sum _{n=0}^{\infty }{\frac {4(-1)^{n}}{2n+1}}=\pi$ or
$\sum _{k=0}^{\infty }{\frac {1}{16^{k}}}\left=\pi$ .

The

\sum _{k=0}^{\infty }

notation means, of course, the limit of the sequence of partial sums. There are lots of further sequences converging to pi. You can choose whichever you like. The partial sums of the series given above certainly differ, their limit can be shown to be the same (although that's probably non-trivial).

I still do not understand why we now discuss pi and its properties. It may be an interesting analogy, but shouldn't we be discussing 0.9999...? My point of view is clear: 0.9999... is the limit of the sequence of partial sums (0.9, 0.99, 0.999, ...), and thus obviously equal to 1. Rasmus and I even provided proofs that every real number which is greater than all of those partial sums but not greater than 1 must be equal to 1 (see here and here). By now, you have claimed 0.9999... to be a real number (although you also claimed the opposite), you claimed there is no partial sum 1-1/(10^n) which is greater than 0.9999..., and I never heard you claim 0.9999...>1. I could cite you on the first two counts if you insist. Thus, Rasmus' and my proofs hold. Your "proof", in comparison, had a fatal flaw, trying to use induction in ways it's not designed to work. If you disagree, please repeat that proof in full detail. Yours, Huon 16:54, 28 February 2006 (UTC)