Eli5 why we cant have negative factorials?

Question

I looked it up on google but my extent of math goes to algebra 2, but it immediately went into wacky stuff like “the gamma function”, simply put, what is the gamma function and why is it stopping stuff like -9! From being -362880

In: 2

Answer 1

The gamma function is what you get when you try to extend factorials to fractions, real numbers (positive and negative) and complex numbers….

But even the gamma function won’t help you with (-9)!.

After all, the basic rule of factorials is that n! = n times (n-1)!.

We can reverse this: (n-1)! = n! / n.

Given 1! = 1, that tells us that 0! = 1! / 1 = 1 also. But then things go downhill.

(-1)! should be 0! / 0 = 1 / 0, but we can’t divide by 0. (-1)! is not defined (even using gamma functions), and nor is (-2)!, (-3)! etc for any negative integer.