# Why do factorials not exist for negative numbers?

Dec 26, 2015

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 $- 2$ objects because you can't have less than $0$ objects!

Dec 26, 2015

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.

$\Gamma \left(t\right) = {\int}_{0}^{\infty} {x}^{t - 1} {e}^{- x} \mathrm{dx}$

Integration by parts show that $\Gamma \left(t + 1\right) = t \Gamma \left(t\right)$

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

We can extend the definition of $\Gamma \left(t\right)$ to negative numbers using $\Gamma \left(t\right) = \frac{\Gamma \left(t + 1\right)}{t}$, except in the case $t = 0$.

Unfortunately this means that $\Gamma \left(t\right)$ 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.