Is Zero x Infinity Really a Real Number?

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Discussion Overview

The discussion centers around the mathematical expression of zero multiplied by infinity (0 × ∞) and whether it can be considered a real number. Participants explore the implications of this expression from various mathematical perspectives, including limits, definitions of multiplication, and the nature of infinity.

Discussion Character

  • Debate/contested
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • Some participants argue that zero times anything must be zero, regardless of the nature of the other quantity, including infinity.
  • Others suggest that the expression 0 × ∞ is undefined in the real numbers and can yield different results depending on the context, particularly in limits.
  • A participant points out that multiplication is defined for specific sets of numbers and that discussing multiplication with undefined numbers lacks meaning.
  • Some participants propose that defining 0 × ∞ as zero is possible but question its utility and the implications for arithmetic involving infinity.
  • There are discussions about the philosophical implications of infinity and whether it can be treated as a quantity.
  • One participant mentions that the expression 0 × ∞ is an indeterminate form that can lead to various outcomes in calculus.
  • Another participant emphasizes the foundational nature of counting and how zero iterations of any quantity, including infinity, still results in zero.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the nature of 0 × ∞. There are competing views on whether it should be defined as zero, whether it is an indeterminate form, and how to approach the concept of infinity in mathematical operations.

Contextual Notes

The discussion highlights limitations in definitions and assumptions regarding multiplication with infinity and the implications for arithmetic operations. The varying interpretations of infinity and its role in mathematics contribute to the complexity of the topic.

  • #61
Bipolar Demon said:
Isnt multiplication just glorified addition? so according to my primitive views,=) if so how you can keep adding zero infinitely to get zero IMO.

Sometimes primitive ideas are the most informative.

I could see defining ## x * \infty = x + x+ x + \cdots ## i.e an infinite sequence. Perhaps this has already been said in this thread in a different way.

-Dave K
 
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  • #62
dkotschessaa said:
Sometimes primitive ideas are the most informative.

I could see defining ## x * \infty = x + x+ x + \cdots ## i.e an infinite sequence. Perhaps this has already been said in this thread in a different way.

-Dave K

yes but that series it would (?) seem to diverge to infinity or minus infinity, and converge to zero if x=0. IIRC
 
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  • #63
this man HATES much of set theory and calculus, and real numbers...very interesting debate:





 
  • #64
Bipolar Demon said:
yes but that series it would (?) seem to diverge to infinity or minus infinity, and converge to zero if x=0. IIRC

Well, that's would I would expect it to do!
 
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  • #65
Bipolar Demon said:
this man HATES much of set theory and calculus, and real numbers...very interesting debate:



Yeah, there are some.

I've considered being a finitist if only to be contrary.
 
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