Definition:Factorial/Historical Note
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Definition
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.
It can sometimes be seen rendered as $\lfloor n$.
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
- 1929: Florian Cajori: A History of Mathematical Notations: Volume $\text { 2 }$
- 1932: Clement V. Durell: Advanced Algebra: Volume $\text { I }$ ... (previous) ... (next): Chapter $\text I$ Permutations and Combinations: Factorials
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $24$
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $24$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): factorial
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): factorial