What is the value of ∞/2 in extended real numbers?

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Discussion Overview

The discussion revolves around the value of ∞/2 within the context of extended real numbers. Participants explore the implications of treating infinity as a number and the algebraic properties associated with it, questioning whether the value of infinity changes when halved.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • One participant asserts that ∞/2 equals ∞, seeking comments on this claim.
  • Another participant points out that in the real number system, ∞ is not defined, but in the context of limits, ∞/2 can be interpreted as ∞.
  • A question is raised about whether there is a logical or mathematical issue with the value of ∞ in this context, and if the value of ∞ changes.
  • One participant reiterates that ∞/2 equals ∞ and questions the reasoning behind expecting a different value.
  • A participant suggests that the confusion may stem from applying real number intuition to the concept of infinity, which behaves differently in mathematical contexts.
  • It is proposed that understanding the definition of ∞ clarifies why ∞ = ∞/2 is a reasonable assertion.

Areas of Agreement / Disagreement

Participants express differing views on the treatment of infinity, with some agreeing that ∞/2 equals ∞ while others question this conclusion. The discussion remains unresolved regarding the implications of treating infinity as a number.

Contextual Notes

Participants assume the discussion is framed within the extended real numbers or projective real numbers, which may influence their interpretations of infinity.

shunyadragon
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∞/2= ∞

Please comment. I do not thnk this is right.
 
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What number system are you working in? Strictly speaking, in the real number system, there is no "[itex]\infty[/itex]" but you can think of that as "shorthand" for [itex]\lim_{x\to\infty} x/2[/itex]. In that case, yes, that limit is [itex]\infty[/itex].

And, in any number system in which [itex]\infty[/itex] is defined as a number (extended reals, for example) it is true that [itex]\infty/ 2= \infty[/itex].
 
Is there any logical or math problem with the value of ∞ in this question? Does the value of ∞ change?
 
shunyadragon said:
∞/2= ∞

Please comment. I do not thnk this is right.

Why not? What would you expect a never ending number which is halved to be?
 
shunyadragon said:
Does the value of ∞ change?
No. ∞ = ∞. Why do you think ∞/2 should have a different value from ∞?

(my assumption for this post: we are working in the extended real numbers or projective real numbers)

Most likely answer: you are trying to apply your intuition about algebra of real numbers. But ∞ isn't a member of the real numbers, and so it can behave differently -- and this particular algebraic behavior is one example of that different behavior.

If you actually think about what ∞ is defined to be or what sorts of things it can quantify ∞=∞/2 should actually be rather obviously true.
 

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