Why do factorials not exist for negative numbers?

One of the main practical uses of the factorial is to give you the number of ways to permute objects. You can't permute #-2# objects because you can't have less than #0# objects!

Factorials are defined for whole numbers as follows:

#0! = 1#

#(n+1)! = (n+1) n!#

This allows us to define what we mean by "Factorial" for any non-negative integer.

How can this definition be extended to cover other numbers?

Gamma function

Is there a continuous function that allows us to "join the dots" and define "Factorial" for any non-negative Real number?

Yes.

#Gamma(t) = int_0^oo x^(t-1) e^(-x) dx#

Integration by parts show that #Gamma(t + 1) = t Gamma(t)#

For positive integers #n# we find #Gamma(n) = (n-1)!#

We can extend the definition of #Gamma(t)# to negative numbers using #Gamma(t) = (Gamma(t+1)) / t#, except in the case #t = 0#.

Unfortunately this means that #Gamma(t)# is not defined when #t# is zero or a negative integer. The #Gamma# function has a simple pole at #0# and negative integers.

Other options

Are there any other extensions of "Factorial" that do have values for negative integers?

Yes.

The Roman Factorial is defined as follows:

#stackrel () (|__n~|!) = { (n!, if n >= 0), ((-1)^(-n-1)/((-n-1)!), if n < 0) :}#

This is named after a mathematician S. Roman, not the Romans and is used to provide a convenient notation for the coefficients of the harmonic logarithm.