Largest Number in (0,1)? PF's Take

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Discussion Overview

The discussion revolves around the properties of the open interval (0,1), specifically addressing the concepts of largest and smallest numbers within this interval. Participants explore theoretical implications and mathematical reasoning related to these properties.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that there is no largest number in the open interval (0,1), emphasizing that any candidate for a largest number can always be exceeded by another number within the interval.
  • One participant questions the existence of the smallest number in the interval, suggesting that it could be represented as an infinitesimal or a non-terminating decimal.
  • Another participant proposes that if a minimum number exists, it leads to a contradiction since half of that number would also be a positive number within the interval, implying no minimum exists.
  • There is a contention regarding the representation of numbers in the interval, with some arguing against the validity of non-terminating decimals having a last digit.

Areas of Agreement / Disagreement

Participants generally agree that the open interval (0,1) has no largest or smallest member, but there are differing views on the implications of this and the nature of numbers within the interval.

Contextual Notes

Discussions include assumptions about the nature of numbers, such as the properties of infinitesimals and the representation of non-terminating decimals, which remain unresolved.

JCienfuegos
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Since we all agree that 1 = 0.999..., what does PF think the largest number in the open interval (0,1) is?
 
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What do you think the smallest number in the interval is?
 
JCienfuegos said:
Since we all agree that 1 = 0.999..., what does PF think the largest number in the open interval (0,1) is?

Just because you can ask a question, doesn't mean there is an answer. (0,1) doesn't have a largest element. It's as simple as that.
 
I always imagined that the smallest number less than one was the infinitesimal 0.9..., but that cannot be, because that equals one. The only other answer I could come come up with was that only terminating decimals of the form 0.999...9 would be in the interval, but that's no good either.

I guess the smallest number would be of the form 0.000...01 and it too would be non-terminating.
 
JCienfuegos said:
I guess the smallest number would be of the form 0.000...01 and it too would be non-terminating.
But that number appears to terminate. Is there anything after the 1?

You are overthinking this. We agree that (0,1) does not contain 0. So suppose it has a minimum, call it M. This is a positive number. Then M/2 is a smaller positive number. This is a contradiction, so there is no minimum after all.
 
The open interval (0, 1) has NO largest or smallest member. You cannot talk about "the form 0.000...01 and it too would be non-terminating". If is non-terminating, it does not have a last digit so you cannot assert that it is 1.
 

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