Definition:Factorial

Let $$n \in \N$$.

Then the factorial of $$n$$ is defined as:
 * $$n! = \begin{cases}

1 & : n = 0 \\ n \left({n - 1}\right)! & : n > 0 \end{cases}$$

That is:
 * $$n! = \displaystyle \prod_{k=1}^n k = 1 \times 2 \times \cdots \times \left({n-1}\right) \times n$$.

The first few factorials are:

...etc.

Historical Note
The symbol used here, which is now universal, was introduced by Christian Kramp in his 1808 work Elements d'arithmétique universelle.

Before that, various symbols were used whose existence is now unimportant.