?????

Because it makes for a nice pattern.

And because you need to divide by 0! in series.

There's probably other reasons.

https://en.wikipedia.org/wiki/Empty_product

>>8751998
there's exactly one permutation of the empty set

the amount of ways you can choose a set out of a group can be calculated using factorials. For example, the amount of ways to arrange 5 different objects is 5*4*3*2*1=5! The amount of ways to arrange 1 object is 1!=1. The amount of ways to arrange zero objects is 0!=1, it's the state of having no object, the empty set.

>>8752050
>>8752051
>>8752060
makes sense, thanks anons

File: calc.png (5KB, 642x125px) Image search: [Google]

5KB, 642x125px

What is even happening when you apply the factorial function to complex numbers? What does it even look like when you apply x! to the entire complex plane?

>>8752140
It's called the $\Gamma$ function

It's useful to define factorials this way.

>>8752140
There's an integral called the gamma function, it can be analytically continued along the negatives

>>8752166
>analytically continued along the negatives
Can it be argued that analytic continuation is a sign that our math isn't properly defined for negative values for some functions?