Size of the division of a segment in infinite intervals

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Discussion Overview

The discussion revolves around the concept of dividing a finite interval [a, b] into an infinite number of subintervals of finite size. Participants explore the implications of such divisions, questioning the nature of the sizes of these subintervals as the number of divisions approaches infinity. The conversation touches on mathematical definitions, limits, and the potential for confusion surrounding the terms used in this context.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants argue that if an interval [a, b] is divided into subintervals of finite size, it is impossible to have an infinite number of such subintervals, as the maximum number of subintervals is determined by the width of the smallest subinterval.
  • Others propose that the sizes of the subintervals could tend to zero, suggesting that it is possible to have an infinite number of subintervals with lengths that decrease, potentially forming a geometric sequence.
  • A participant questions the assumption that all subintervals must be of equal size, suggesting that overlapping intervals could still maintain finite lengths.
  • Some participants express confusion over the use of the term "finite," indicating that it is often misinterpreted as synonymous with "non-zero," which complicates the discussion.
  • One participant introduces a transformation concept where each small segment of the line is associated with a natural number, raising questions about the mathematical validity of associating a number "near infinite" with segments.
  • Concerns are raised about the convergence of the series formed by the lengths of the subintervals, with references to the definite integral as a tool for addressing such issues.
  • There is a disagreement regarding the implications of having subintervals of lengths defined by specific sequences, such as 2^{-n}, with some asserting that this leads to a limit of zero while others contest this interpretation.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the nature of the sizes of the subintervals or the mathematical implications of dividing a finite interval into an infinite number of parts. Multiple competing views remain, particularly regarding the definitions and assumptions about "finite" and "infinite."

Contextual Notes

There are limitations in the discussion related to the definitions of finite and infinite, as well as the assumptions made about the sizes of the subintervals. The mathematical steps and implications of the proposed transformations and sequences remain unresolved.

Iraides Belandria
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Dear people of this forum,
Assume we have an interval of a line [ a, b].
Now, let us divide this interval in many ( infinite ) subintervals of finite size. ? What will be the size of these many intervals, equal or unequal, when the number of subintervals tends to infinite?. ¿there is any theorem to confirm possible answer?. Thank you.
 
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Iraides Belandria said:
Assume we have an interval of a line [ a, b].
Now, let us divide this interval in many ( infinite ) subintervals of finite size.
If you divide it into a subintervals of finite size (let the smallest one be 'w' in width), then you can't have an infinite number of subintervals since you'll have at most (b-a)/w subintervals, which is finite.
 
Iraides Belandria said:
What will be the size of these many intervals, equal or unequal, when the number of subintervals tends to infinite?
Intuitively, it could be just about anything.

Technically, this isn't a well-defined question... the sizes of the intervals form a finite sequence of numbers... and as we let the number of subintervals tend to infinity, then we have a sequence of finite sequences of numbers... and they are all different lengths. I how would you define a limit of such a thing?


TD said:
(let the smallest one be 'w' in width)
You can't assume there will be a smallest! Their lengths might tend to zero. (Say... maybe the lengths form a geometric sequence)
 
You can't say anything about the sizes, other than none of them is larger than b-a.
 
Hurkyl said:
You can't assume there will be a smallest! Their lengths might tend to zero. (Say... maybe the lengths form a geometric sequence)
Not even when the size of all subintervals are finite? When you let the number of subintervals tend to infinity (and hence the size to 0), I agree (and understand), but as long as you have a finite number of subintervals? Apparently I'm missing something
 
That the OP said there were an infinite number of subintervals, perhaps? They then wanted to let the number tend to infinity, which is odd. And in cany case, who is saying that the subintervals are equally sized?
 
Well that confused me, is it possible to have a infinite number of subintervals of a finite interval [a,b] while the sizes of all those subintervals are still finite? I'd say that you have a finite number of subintervals with possibly finite sizes and that the sizes will become infinitesimally small (tending to zero) when the number tends to infinity.
 
0 is a finite length, I don't understand why there is this emphasis of finite length at all. They are sub intervals of [a,b] which automatically makes them finite.
 
You're right, thanks.
 
  • #10
Well that confused me, is it possible to have a infinite number of subintervals of a finite interval [a,b] while the sizes of all those subintervals are still finite?
Yes.

You can even require that the sizes be positive, and that the intervals overlap only at their endpoints.

I even already gave a hint as to how this may be accomplished. :smile:
 
  • #11
The source of the confusion is that too many physicists (and possibly other non-mathematicians as well) incorrectly use the term 'finite' as almost synonymous with 'non-zero' (a "finite" quantity as something that is neither infinitely large nor infinitesimally small).
 
  • #12
Gokul43201 said:
The source of the confusion is that too many physicists (and possibly other non-mathematicians as well) incorrectly use the term 'finite' as almost synonymous with 'non-zero' (a "finite" quantity as something that is neither infinitely large nor infinitesimally small).
Guiltly, although I'm not a physicist :blushing:
Usually not though, but apparently in this context I was thinking differently about 'finite' :confused:
 
  • #13
Analizing your discusion, there is an additional restriction that I did not mention in my original thread.

The requirement is that to each nonzero small segment of the line we have to associate by a one to one transformation a number 1,2,3,4,5...n where n is a huge number near infinite.
In other words, let us divide a line of total lengh L equal to 1 meter in a huge number of non zero small segments L1,L2,L3,L4,L5...Ln and such way that
L1-------->1
L2-------->2
L3-------->3
L4-------->4
...
...


Ln-------->n

¿ If n tends to a number close to infinite in such a way that Ln is a nonzero small line segment, then

L1=L2=L3=L4=...=Ln ?
 
  • #14
Iraides Belandria said:
The requirement is that to each nonzero small segment of the line we have to associate by a one to one transformation a number 1,2,3,4,5...n where n is a huge number near infinite.

that doesn't make mathematical sense. there is no concept of a number being 'near infinity'.
 
  • #15
Iraides, what's stopping me from making 1 segment of length 0.9, and then chopping the remaining interval of 0.1 into as many segments as I need?
 
  • #16
If all the sub-intervals are finite, then the partial sums of interval lengths, say starting from zero, will form a non-convergent series. That's why folks invented the definite integral.
Regards,
Reilly Atkinson

(Just to be clear, if the subintervals are all finite, then the limit L(n) ->. non-zero as n-> infinity, where L(n) is the length of the nth subinterval)

Regards,
Reilly Atkinson
 
  • #17
Even if the n'th subinterval has length 2^n?
 
  • #18
reilly said:
(Just to be clear, if the subintervals are all finite, then the limit L(n) ->. non-zero as n-> infinity, where L(n) is the length of the nth subinterval)

Regards,
Reilly Atkinson
No, that's not clear at all- it's just wrong. As matt grime pointed out, if the length of the nth interval is 2-n the "the subintervals are all finite" but the limit is 0. In the case where we have in infinite number of intervals such that the length of the nth interval is 2-n then the length of the entire interval is 1.
 
  • #19
I'm right if I say "finite AND non-zero" which I did not, so I'm not right.
Regards,
Reilly
 
  • #20
No, you are not right even with your correction. 2^{-n} gives an obvious subdivision and no subinterval has length 0 (or infinity).
 
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