Question

Why do factorials not exist for negative numbers?

Answer 1

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!

Answer 2

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 `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.