Number infinitely close to one but not one

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Discussion Overview

The discussion revolves around the concept of a number that is infinitely close to one but not equal to one. Participants explore various mathematical frameworks, including the hyperreal numbers and the implications of notation within the real number system.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that within the real number system, no number can be infinitely close to one without being equal to one.
  • Others propose the use of hyperreal numbers, suggesting that in this framework, a number can be represented as infinitely close to one but still less than one, such as 1 - epsilon, where epsilon is an infinitesimal.
  • There is a question about the notation used to represent such numbers in the hyperreal number system, with some participants expressing uncertainty about existing conventions.
  • One participant suggests that if one were to remain within the real numbers, a representation like 1 + x (where x is a non-zero real number close to zero) could be used, though this does not yield a unique number.
  • Another participant mentions the idea of naming such a number as Inf ℝ, referencing the smallest non-zero real number, which does not exist in the traditional sense.
  • Some participants discuss the significance of the notation 0.999... and its equivalence to 1, emphasizing that this representation does not yield a number less than one.
  • There are attempts to define a number that is always less than any epsilon-prime, suggesting a more abstract approach to the concept.

Areas of Agreement / Disagreement

Participants generally disagree on the existence of a number infinitely close to one that is not one, with some advocating for the hyperreal framework while others maintain that such a number cannot exist in the real number system.

Contextual Notes

The discussion highlights the limitations of the real number system in addressing the concept of infinitesimals and the challenges in notation and definition within different mathematical frameworks.

firefly431
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How would one notate a number (any algebraic entity) that is infinitely close to one but not one? I know that 0.999...=1, and thus cannot be *not* one, so what would be a suitable name for this number?
 
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There is no real number that is infinitely close to 1 yet not identical to 1. Thus, one must use a different structure, such as the hyperreal numbers. The interesting thing about switching to hyperreals is that using the same decimal numbering convention, where

.999... = 9/10 + 9/100 + 9/1000 ...,

Then 0.999... does name a number infinitely close to 1 but less than 1. I know I read a paper on this, and I will track down the reference soon.

If you do not go this route, the name would probably have to be 1 - epsilon, where epsilon is an infinitesimal.
 
Is there any special notation for writing this number? I don't know much about hyperreal numbers.
 
firefly431 said:
Is there any special notation for writing this number? I don't know much about hyperreal numbers.
This is the reference I was talking about: http://arxiv.org/abs/0811.0164
I don't know if there is any notational convention for specifying that you are using the hyperreal number system.
 
BenjaminTR said:
I don't know if there is any notational convention for specifying that you are using the hyperreal number system.

The same as for any other type of number, in this case you could write, for example, 'x is an element of *R'.

The hard part would be to learn all the theorems and rules that differ slightly between the hyperreals and the real numbers.

Since hyperreals are almost never used and one would like to express something similar it would probably be better to continue using the reals and write something like 1+x, where x is a real number arbitrarily close to 0 (but NOT 0). This is of course not a unique number.
 
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firefly431 said:
How would one notate a number (any algebraic entity) that is infinitely close to one but not one? I know that 0.999...=1, and thus cannot be *not* one, so what would be a suitable name for this number?
Do you understand that everyone is saying "as long as you are working in the real number system, there is no name for such a number, because there is NO such number!"
 
BenjaminTR said:
.999... = 9/10 + 9/100 + 9/1000 ...,

Then 0.999... does name a number infinitely close to 1 but less than 1.

No, 0.999... and 1 are the same number, so neither is less than the other. The ellipsis, ..., is significant, and indicates that the 9s repeat forever.
 
Mark44 said:
No, 0.999... and 1 are the same number, so neither is less than the other. The ellipsis, ..., is significant, and indicates that the 9s repeat forever.

He is talking about the hyperreal number system. In that number system, what he is saying is absolutely correct.
 
firefly431 said:
How would one notate a number (any algebraic entity) that is infinitely close to one but not one? I know that 0.999...=1, and thus cannot be *not* one, so what would be a suitable name for this number?

Let's stick to the real number system. The smallest non zero real number does not exist. So how do you name something that does not exist? Inf ℝ would be a possibility.

How ever this is a very meaningful question to ask about your computer, what is the smallest number it can represent? Perhaps the computer guys have a name for it. A quick google turned up RealMin in Matlab. I think you can do some more searching and find others. Take your pick.
 
  • #10
Call it anything you like

What matters is that there be an intelligible definition. Maybe something like ε such that for any ε-prime where ε ≠ ε-prime

1 - ε < 1 - ε-prime

which is to say that the magic number is always less than ε-prime. That would make it the closest number to 1.
 
  • #11
Well, the above needs some fixing. The closest number to 1 is ε where ε ≠ 1 and where for any ε-prime such that ε-prime ≠ ε and ε-prime ≠1

abs(1 - ε) < abs(1 - ε-prime)
 
  • #12
Mark44 said:
No, 0.999... and 1 are the same number, so neither is less than the other. The ellipsis, ..., is significant, and indicates that the 9s repeat forever.

micromass said:
He is talking about the hyperreal number system. In that number system, what he is saying is absolutely correct.

I stand corrected.
 

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