Definition:Factorial/Definition 2

From ProofWiki
Jump to navigation Jump to search

Definition

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


The factorial of $n$ is defined as:

\(\displaystyle n!\) \(=\) \(\displaystyle \prod_{k \mathop = 1}^n k\)
\(\displaystyle \) \(=\) \(\displaystyle 1 \times 2 \times \cdots \times \paren {n - 1} \times n\)

where $\displaystyle \prod$ denotes product notation.


Examples

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

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

This sequence is A000142 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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.


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.
-- 1929: Florian Cajori: A History of Mathematical Notations: Volume 2


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


Sources