Definition:Factorial
Definition
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:
- $n! = \displaystyle \prod_{k=1}^n k = 1 \times 2 \times \cdots \times \left({n-1}\right) \times n$
where $\prod$ denotes product notation.
The first few factorials are:
$\begin{array}{r|r} n & n! \\ \hline 0 & 1 \\ 1 & 1 \\ 2 & 2 \\ 3 & 6 \\ 4 & 24 \\ 5 & 120 \\ 6 & 720 \\ 7 & 5 \ 040 \\ 8 & 40 \ 320 \\ 9 & 362 \ 880 \\ 10 & 3 \ 628 \ 800 \\ \end{array}$
...etc. This sequence is A000142 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Historical Note
The symbol used here, 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. For example, Gauss used $\Pi \left({n}\right)$ for $n!$.
Of major importance, however, is the Gamma function, which is an extension of the concept of the factorial to the complex plane.
Sources
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 19$
- Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): $\S 1.2.5$
- Murray R. Spiegel: Mathematical Handbook of Formulas and Tables (1968): $3.1, \ 3.2$
