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

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The discussion centers on the factorial function, which is typically defined for natural numbers and cannot yield negative values. However, the Gamma function serves as a natural extension of the factorial, allowing for negative inputs. This extension can produce negative outputs, as illustrated by specific negative values of n. The conversation highlights the mathematical implications of extending the factorial to include negative numbers through the Gamma function. Overall, the Gamma function provides a framework for understanding factorials beyond their conventional limits.
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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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