# 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**