Is \frac {1}{\infty} equal to 0 or infinitely close to 0?

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Discussion Overview

The discussion centers around the expression \(\frac{1}{\infty}\) and whether it is equal to 0 or merely infinitely close to 0. Participants explore the implications of this expression from various mathematical perspectives, including standard analysis and non-standard analysis.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that \(\frac{1}{\infty} = 0\) is correct, while others find the alternative of being infinitely close to 0 less meaningful unless non-standard analysis is involved.
  • It is noted that \(\frac{1}{x} \to 0\) as \(x\) approaches infinity, but this does not imply that \(\frac{1}{\infty} = 0\), especially since infinity is not a number.
  • One participant suggests that if there is 1 unit of something in an infinitely large area, it raises questions about the existence of that unit when considering \(\frac{1}{\infty}\).
  • Another participant argues that saying \(\frac{1}{\infty} = 0\) implies non-existence, using the analogy of human population over time.
  • It is mentioned that in some mathematical systems, \(\frac{1}{\infty}\) does not have a standard meaning, while in others, it can be defined as an infinitesimal positive nonzero number or equal to 0.
  • One participant refers to the definition of infinity as an extension of the real number line, where \(\frac{1}{\infty} = 0\) is a specific definition applicable in certain contexts.
  • A later reply suggests that the concept of being infinitely close to zero is relevant in calculus, particularly in relation to the delta in integrals.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether \(\frac{1}{\infty}\) should be considered equal to 0 or infinitely close to 0. Multiple competing views remain, with some advocating for each perspective.

Contextual Notes

Limitations include the dependence on definitions of infinity and the context in which \(\frac{1}{\infty}\) is applied. The discussion reflects varying interpretations and the implications of using infinity in mathematical expressions.

Warr
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Is [tex]\frac {1}{\infty} = 0[/tex] , or is it just infinitely close to 0?
 
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= is correct, the other alternative is not too meaningful, unless you get involved with non-standard analysis.
 
mathman said:
= is correct, the other alternative is not too meaningful, unless you get involved with non-standard analysis.

Unless you're using non-standard analysis I don't think the statement [itex]\frac{1}{\infty}=0[/itex] is even meaningful.
 
Well, you could say 1/x -> 0 as x tends to infinity (though that obviously doesn'; mean 1/inifnity = 0 especially when infinity isn't even a number).
 
Lets say there is 1 unit of something in an infinitely large area...then would you say = ? Because then that says that the unit doesn't even exist...
 
No in that situation all we would be saying is that it would be meaningless to talk about the ratio of the area to the unit area.
 
No it isn't...Because that unit DOES exist. But by saying 1/inf = 0...we say it is non-existant. In the same way, human population with respect to time would be 0 if the above statement were true. This is not so...
 
1/∞ doesn't have a "standard" meaning; in some systems where infinite numbers are defined, division doesn't exist. In some others, 1/∞ is some infinitessimal positive nonzero number. In others, 1/∞=0.

If you're thinking of ∞ as that "big number that sits at the positive end of the real numbers", then you probably mean to use the extended real numbers, where 1/∞ is defined to be equal to zero.
 
In my books when infinity is defined as an extension to the Real number line, operations on infinity are also defined, included with these definitions is:

[tex]\frac 1 \infty = 0[/tex]

This is a very specific definition for a very specific application ie the real numbers. If you attempt to apply this definition out of context your results may vary.
 
  • #10
I always thought it meant infinitely close to zero, and that’s why the delta at the end of an integral doesn’t yield zero results, because the delta doesn’t actually = zero, just something infinitely small.
 

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