# Definition:Factorial/Definition 1

## Definition

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

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

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

## Examples

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

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

## Also see

- 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.

In fact, Henry Ernest Dudeney was using $\left\lvert {\kern-1pt \underline n} \right.$ as recently as the $1920$s.

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

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 19$: Combinatorial Analysis - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.18$: Sequences Defined Inductively: Exercise $2$ - 1980: David M. Burton:
*Elementary Number Theory*(revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction - 1992: Larry C. Andrews:
*Special Functions of Mathematics for Engineers*(2nd ed.) ... (previous) ... (next): $\S 1.2.4$: Factorials and binomial coefficients - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials: $(6)$