Dividing Infinity by Infinity: What is the Result?

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Discussion Overview

The discussion revolves around the concept of dividing infinity by infinity, exploring whether this operation can yield a defined result or if it remains undefined. Participants engage with the theoretical implications and mathematical interpretations of infinity in this context.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that dividing infinity by infinity could yield 1.
  • Others argue that the result is undefined, emphasizing that infinity is not a number and cannot be treated like ordinary numbers.
  • A participant illustrates that different limits can approach infinity in various ways, leading to different results for quotients, thus questioning the possibility of a unique value for infinity divided by infinity.
  • Another participant clarifies that in standard arithmetic, there is no definition for infinity over infinity, although some extended number systems might allow for such operations.

Areas of Agreement / Disagreement

Participants express disagreement on whether dividing infinity by infinity can yield a defined result, with some asserting it is undefined while others suggest it could be defined as 1 in certain contexts.

Contextual Notes

The discussion highlights limitations in the definitions of infinity and the conditions under which different interpretations may apply. There is an acknowledgment of various mathematical frameworks that might treat infinity differently.

adjacent
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What would happen if,I divide ∞ by ∞.What would be the answer. Would it be 1?
 
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If would be undefined.
 
phinds said:
If would be undefined.
I don't understand,What do you mean?
 
Infinity is not a number and you can't calculate with it like you can with usual numbers. Consider the values of quotients [itex]\frac{x+1}{2x+1}[/itex] and [itex]\frac{2x+1}{3x+1}[/itex] when ##x## is let to grow without bound. In both expressions the numerator and denumerator approach infinity when ##x\rightarrow\infty##, but the value of first expression approaches ##1/2## and the value of second expression approaches ##2/3##. You can't define a unique value for the quotient ##\infty/\infty##.
 
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adjacent said:
I don't understand,What do you mean?

What is it about the word "undefined" that you do not understand? It means "would have no definition". Is that not clear?
 
In terms of "ordinary arithmetic", "infinity" is NOT a number and so there simply is NO definition for "infinity over infinity". There are ways of extending the number systems to give various types of "infinity" some of which then allow you to divide infinity by infinity. Most of those, but possibly not all, define division so that infinity by infinity is 1.
 

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