Misplaced Pages

This is an old revision of this page, as edited by Huon (talk | contribs) at 13:16, 26 February 2006. The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

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 Previous discussions have been archived from the main talk page, which is now reserved for editorial discussions:

• Archive 1 (2005-05-06 to 2005-11-16)
• Archive 2 (2005-11-16 to 2005-12-07)
• Archive 3 (2005-12-07 to 2005-12-09)
• Archive 4 (2005-12-09 to 2005-12-20)
• Archive 5 (2005-12-18 to 2006-02-11)
• Archive 6 (2006-02-11 to 2006-02-23)

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)

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)