Can n be negative? What values of n allow for negative numbers?

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Discussion Overview

The discussion centers around the question of whether the variable n can take on negative values, particularly in the context of the factorial function and its extensions. Participants explore the implications of defining n as negative and the mathematical frameworks that allow for such definitions.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants assert that the factorial function, traditionally defined for natural numbers, cannot yield negative values.
  • Others introduce the concept of the Gamma function as an extension of the factorial, suggesting that it can take on negative values under certain conditions.
  • A participant provides specific examples of negative values of n and their corresponding factorial-like expressions, indicating that negative inputs can lead to complex results.
  • A later reply expresses appreciation for the exploration of negative values in this context, indicating interest in the topic.

Areas of Agreement / Disagreement

Participants do not reach a consensus; there are competing views regarding the definition and implications of negative values for n in relation to the factorial function and its extensions.

Contextual Notes

The discussion highlights the limitations of traditional definitions of factorial and the need for careful consideration of mathematical extensions like the Gamma function. The specific conditions under which negative values can be used remain unresolved.

gikiian
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If yes, then for what values of n?
 
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We usually only define ##n!## for natural numbers ##n##. Then ##n!## can never be negative.

There is however a very natural extension of the factorial, this is called the Pi function (which is just a translation of the Gamma function). This can indeed become negative as you can see from the graph: http://en.wikipedia.org/wiki/Gamma_function
 
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If we extend the domain we can have negative numbers for example
-1~(-3.74768264672741260139148848269149969586163939513235551205229915)!
-1~(-3.45702473822080062303945414765117954323659790903378442096479450)!
 
THAT is what I wanted to see. Thank you very much Sir.
 

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