Is Dividing by Infinity Mathematically Valid?

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Discussion Overview

The discussion centers around the mathematical validity of dividing by infinity, exploring whether such an operation is meaningful or permissible within arithmetic and calculus. It includes considerations of limits and the implications of treating infinity as a number.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants assert that dividing by infinity is not mathematically valid and that infinity cannot be used in arithmetic expressions.
  • One participant mentions that in limit expressions, such as ##\lim_{x \to \infty} \frac{1}{x}##, the limit approaches 0, but emphasizes that we do not write 1/∞.
  • Another participant states that the literal answer to "what is 1/infinity" is "an undefined quantity," similar to n/0, and warns against treating undefined quantities as numbers.
  • A different viewpoint is presented where a participant questions if one can substitute values that behave like infinity in arithmetic expressions, leading to a speculative conclusion that 1/infinity could equal -12 based on a controversial manipulation of series.

Areas of Agreement / Disagreement

Participants generally disagree on the validity of dividing by infinity, with some asserting it is undefined while others explore hypothetical scenarios involving substitutions and series.

Contextual Notes

The discussion highlights limitations in treating infinity as a number and the potential for contradictions when undefined quantities are manipulated. There are unresolved mathematical steps and assumptions regarding the behavior of infinite series.

member 529879
What is 1 / infinity? Is it even possible to divide by infinity?
 
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Scheuerf said:
What is 1 / infinity? Is it even possible to divide by infinity?
No, and you can't use ∞ in arithmetic expressions.

Where expressions such as this often show up is in limit expressions, such as ##\lim_{x \to \infty} \frac{1}{x}##. The value of this limit is 0. We don't write 1/∞.
 
To expand just slightly on Mark's post, the literal answer to "what is 1/infinity" is "an undefined quantity" (as is n/0).

One of the problems with allowing these undefined quantities to be used as though they were numbers is that by using them you can readily prove that any number is the same as any other number.
 
If you can't use infinity in arithmetic expressions, can you substitute things equal to infinity that behave well in those expressions?
Like this? :)

1+2+3... the partial sums of the infinite series grow without bound = infinity
1+2+3...can be summed to equal -1/12
if two things equal to the same thing are themselves equal
then
1/infinity = 1/(-1/12) = -12
 

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