Question

What is 0!
(This is using Factorial Notation)

Answer 1

0! := 1

Answer 2

How does that work? I am mind blanking right now...

Answer 3

I know that 9! is 1x2x3x4x5x6x7x8x9
So, would that mean that 0! is 0x0??

Answer 4

In some sense, it is an invented notion. We define 0! = 1 because it is useful for us.

That said, I think the best way to think about it is similar to a^0. The product is considered to be empty, in a sense, so the only thing left around is the multiplicative identity 1.
Or 1! = 1 * 0!
This also has to work out, so 0! = 1 could be defined here as well.

Answer 5

So, there is no real reasoning it just is? Just some rule they made up to prevent paradoxes?

Answer 6

Not quite "no real reasoning", it is done so rather than a choice like 0! = 0 because more often than not, it is more useful to think of 0! as 1 so that our formulas tend to work out (ex binomial theorem and coefficients/ combinations).

Answer 7

I like thinking of a^0 = 1 because a^x/a^x = a^(x-x) = a^0 = 1.

As for factorials, as AccessDenied is saying, it's just a definition to make factorials useful.

Answer 8

Numberphile has a lot of cool videos on stuff like this if you want to look into it:
https://www.youtube.com/watch?v=Mfk_L4Nx2ZI Zero factorial