# 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.