Endless or Finite? 0.000...01 Closest to Zero?

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Discussion Overview

The discussion centers around the concept of the smallest positive number closest to zero, particularly focusing on the idea of representing this number as 0.000...01, which raises questions about the nature of infinity and the properties of real numbers. Participants explore theoretical implications, mathematical notation, and philosophical considerations regarding the existence of such a number.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that 0.000...01 represents the closest positive number to zero, suggesting that it consists of infinite zeros followed by one.
  • Others argue that the notation used is problematic, asserting that a decimal number cannot have an infinite sequence of zeros in that manner, and that the concept of a "closest" positive number may be flawed.
  • A participant questions the idea of infinity not being real while simultaneously suggesting the existence of an infinite number of zeros, highlighting a potential contradiction.
  • There is a discussion about the nature of zero, with some suggesting it is an infinitesimal number, smaller than any positive real number.
  • One participant introduces the idea of using fractions (e.g., 1/10, 1/100) to illustrate the concept of small numbers, emphasizing that one can always add more zeros.
  • Another participant points out that the assumption of a smallest positive number may be misguided, suggesting that such a number does not exist within the real numbers.
  • There is mention of the rigorous construction of real numbers and the possibility of extensions to the real numbers that could allow for the existence of a smallest positive number.

Areas of Agreement / Disagreement

Participants express multiple competing views regarding the existence of a smallest positive number and the nature of infinity. The discussion remains unresolved, with differing opinions on the implications of these concepts.

Contextual Notes

Limitations include the ambiguity in notation and definitions, as well as the philosophical assumptions underlying the discussion about the nature of numbers and infinity.

kurushio95
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I think infinite is not real. Think about it. What is the closest positive number to zero that you can have? it would be 0.000...01, right? infinite zeros plus one. What do you guys think? I didn't know where this would belong, so I bunged it here.
 
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kurushio95 said:
What is the closest positive number to zero that you can have?
Why do you think there is a closest?

it would be 0.000...01, right? infinite zeros plus one.
That cannot a decimal number. The places in a decimal number are indexed by the integers; therefore, each place must be at a finite distance from the decimal point.

Also, your notation is poor. It suggests that you have a consecutive sequence of zeroes, which must necessarily be finite. If it were infinite, then it must consist of two or more disconnected sequences. I use a pipe (|) to denote these breaks -- it is, in fact, possible to have an infinite connected sequence of zeroes:
000...​
and it is possible to have an infinite connected sequence of zeroes followed by a 1:
000... | 1​
however, there is no number to the immediate left of the 1. If you want every place (aside from the left endpoint) to have a number to its immediate left, you can build a sequence like
000... | ...0001​
but these two components do not connect in the middle. Of course, this and the previous sequence cannot denote a decimal number.
 
Last edited:
kurushio95 said:
I think infinite is not real. Think about it. What is the closest positive number to zero that you can have? it would be 0.000...01, right? infinite zeros plus one. What do you guys think? I didn't know where this would belong, so I bunged it here.

Ok, so you think infinite isn't real, and yet you think that you can have 0.[infinite 0s]1?? How can you have an infinite number of 0s if infinity isn't real? A self-contradictory stance that proves itself is interesting, but of little philosophical or scientific value, IMO.
 
I forgot; I had some nonmathematical comments too.

kurushio95 said:
I think infinite is not real.
What does any of this have to do with reality?


infinite zeros plus one.
This is not what you meant. The zeroes are not infinite; it's the quantity of zeroes that is infinite. You meant to say something like "infinitely many zeroes".

In fact, zero has the opposite quality: zero is smaller than every positive real number, and is thus an infinitessimal number. (It is, of course, the only infinitessimal real number)
 
Hurkyl said:
it must consist of two or more disconnected sequences.
Disconnected is an awkward word; to use here. I think I implied what I meant, but shame on me for not explicitly defining it!

What I mean is that if you have a zero in one sequence, you cannot reach the other sequence by iteratively performing the operation "move from this zero to an adjacent zero". Maybe I should have called this "stepwise disconnected".
 
like i said, i didn't know where a question like this would go. I guess the best way i could explain it is to ask what the smallest positive number is that is not zero. I'm having a hard time phrasing my thoughts into words, in case you couldn't tell.

P.S. I'm only in high school math, so I wasn't expecting to be 100% correct in my notation. thanks for pointing it out though and showing me the correct way.
 
It may be easier to think about small numbers like this with fractions. Ie, 1/10, 1/100, 1/1000, etc. No matter how many zeroes you put in there, you can always put in one more.
 
kurushio95 said:
I guess the best way i could explain it is to ask what the smallest positive number is that is not zero. I'm having a hard time phrasing my thoughts into words, in case you couldn't tell.
The problem may be that you don't realize you have made an assumption. When you ask
what is the smallest positive number that is not zero​
you have assumed that such a thing exists.

The truth is that such a thing does not exist; one of the best ways of fighting off these misguided assumptions is to try and prove them. By failing When you to prove that there really is a smallest nonzero positive number, you will begin to cast off this error that is leading you astray.
 
Part of the confusion arises from the intuitive assumption that the real numbers somehow describe our physically real space. This is probably the original motivation behind the real numbers, but once one learns about the rigorous construction of the real numbers, one inevitably understands that they are not necessarily the same thing as our space.

After learning that there is also different extensions to the real numbers, I've started to think if such extension could exist also, where the smallest positive number would exist. It couldn't be a field extension at least, but perhaps some other extension, that would still contain the field of real numbers as a subset.

kurushio95, want to proceed with you philosophy? Then get rigor! :biggrin:

(... although it would probably be better to first understand real numbers well)
 

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