Why do factorials not exist for negative numbers?
2 Answers
There would be a contradiction with its function if it existed.
Explanation:
One of the main practical uses of the factorial is to give you the number of ways to permute objects. You can't permute
It depends what you mean...
Explanation:
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
For positive integers
We can extend the definition of
Unfortunately this means that
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.