# Definition:Factorial

## Definition

Let $n \in \Z_{\ge 0}$ be a positive integer.

### Definition 1

The **factorial of $n$** is defined inductively as:

- $n! = \begin{cases} 1 & : n = 0 \\ n \paren {n - 1}! & : n > 0 \end{cases}$

### Definition 2

The **factorial of $n$** is defined as:

\(\ds n!\) | \(=\) | \(\ds \prod_{k \mathop = 1}^n k\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 1 \times 2 \times \cdots \times \paren {n - 1} \times n\) |

where $\displaystyle \prod$ denotes product notation.

## Examples

The factorials of the first few positive integers are as follows:

$\begin{array}{r|r} n & n! \\ \hline 0 & 1 \\ 1 & 1 \\ 2 & 2 \\ 3 & 6 \\ 4 & 24 \\ 5 & 120 \\ 6 & 720 \\ 7 & 5 \, 040 \\ 8 & 40 \, 320 \\ 9 & 362 \, 880 \\ 10 & 3 \, 628 \, 800 \\ \end{array}$

## Definition for Multiindices

Let $\alpha$ be a multiindex, indexed by a set $J$ such that for each $j \in J$, $\alpha_j \geq 0$.

Then we define:

- $\displaystyle\alpha! = \prod_{j \mathop \in J} \alpha_j!$

where the factorial on the right is a factorial of natural numbers.

Note that by definition, all by finitely many of the $\alpha_j$ are zero, so the product over $J$ is convergent.

## Also known as

While the canonical vocalisation of $n!$ is **$n$ factorial**, it can often be found referred to as **$n$ bang** or (usually by schoolchildren) **$n$ shriek**.

Some mathematicians prefer **$n$ gosh**.

Some early sources favour **factorial $n$**

## Also see

- Definition:Gamma Function: the extension to the complex plane

- Results about
**factorials**can be found here.

## Historical Note

The symbol $!$ used on $\mathsf{Pr} \infty \mathsf{fWiki}$ for the factorial, which is now universal, was introduced by Christian Kramp in his $1808$ work *Élémens d'arithmétique universelle*.

Before that, various symbols were used whose existence is now of less importance.

Notations for $n!$ in history include the following:

- $\sqbrk n$ as used by Euler
- $\mathop{\Pi} n$ as used by Gauss
- $\left\lvert {\kern-1pt \underline n} \right.$ and $\left. {\underline n \kern-1pt} \right\rvert$, once popular in England and Italy.

Augustus De Morgan declared his reservations about Kramp's notation thus:

*Amongst the worst barbarisms is that of introducing symbols which are quite new in mathematical, but perfectly understood in common, language. Writers have borrowed from the Germans the abbreviation $n!$ ... which gives their pages the appearance of expressing admiration that $2$, $3$, $4$, etc., should be found in mathematical results.*

The use of $n!$ for non-integer $n$ is uncommon, as the Gamma function tends to be used instead.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $6$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $24$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $6$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $24$