# Why the name 'FACTORIAL'?

Factorial is defined as

$n! = n(n-1)(n-2)\cdots 1$

But why mathematicians named this thing as FACTORIAL?

Has it got something to do with factors?

• Interesting... Cajori discusses the various notations adopted over the years, but never a mention of why it was called the "factorial". Maybe this article: dx.doi.org/10.1007/BF00389433 has the answers. – J. M. is a poor mathematician Aug 27 '10 at 10:14
• I was once told that an alternative name for the factorial ! was "shriek" which makes EVEN LESS SENSE! – Seamus Aug 27 '10 at 12:00
• Seamus: It would probably interest you to know that it had been suggested at some point that n! be read as "n-admiration" (per Cajori). – J. M. is a poor mathematician Aug 27 '10 at 12:14
• this ought to be cw. – Tom Stephens Aug 27 '10 at 15:01
• I've heard of "shriek" as a name for the exclamation point symbol "!" in general, not just for its meaning of "factorial"; an emphatic word for an emphatic symbol. This seems to come from the computer science world, where it's now been superseded by "bang". See catb.org/jargon/html/B/bang.html, where the hard-to-pronounce "excl" is also cited. – Nate Eldredge Aug 27 '10 at 16:39

Well, all positive integers smaller or equal to n are factors of n!

• But that's true of other numbers (both smaller and larger) that aren't the factorial. – Sophie Alpert Apr 28 '11 at 20:54

Below is the etymology, from Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics (F). Perhaps a native French speaker can lend further insight.

FACTORIAL. The earlier term faculty was introduced around 1798 by Christian Kramp (1760-1826).

Factorial was coined (in French as factorielle) by Louis François Antoine Arbogast (1759-1803).

Kramp withdrew his term in favor of Arbogast's term. In the Preface, pp. xi-xii, of his "Éléments d'arithmétique universelle," Hansen, Cologne (1808), Kramp remarks:

...je leur avais donné le nom de facultés. Arbogast lui avait substitué la nomination plus nette et plus française de factorielles; j'ai reconnu l'avantage de cette nouvelle dénomination; et en adoptant son idée, je me suis félicité de pouvoir rendre hommage à la mémoire de mon ami. [...I've named them facultes. Arbogast has proposed the denomination factorial, clearer and more French. I've recognised the advantage of this new term, and adopting its philosophy I congratulate myself of paying homage to the memory of my friend.]

• Interesting! "faculty" is still used in German (as "Fakultät"). – Jens Nov 9 '10 at 7:23
• In Norway we call it "fakultet". – Eivind May 10 '11 at 15:50
• @Jens: I think that it only used by people who do not know the correct English word for it and use it because it sounds English and is close to the German word. – Martin Ueding May 11 '15 at 20:35
• Apparently, in Swedish too! – mccc Aug 20 '16 at 8:05