Does anyone know where the factorial "!" symbol came from?

I can't decide if it is my favorite or least favorite notation in mathematics...

  • This is due to euclid, you find it already in the "elements" – user88576 May 19 '14 at 22:39
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    I'd bet money that was added in a much later translation of Euclid, considering that the exclamation mark didn't appear until the late Middle Ages. – colormegone May 19 '14 at 22:55
up vote 13 down vote accepted

Earliest Uses of Various Mathematical Symbols will help you for the origin of math symbols.

Factorial is in the category "probability and statistics" and we can read:

The notation $n!$ was introduced by Christian Kramp (1760-1826) in 1808. In his Élémens d'arithmétique universelle (1808), Kramp wrote [in old French]:

Je me sers de la notation trés simple $n!$ pour désigner le produit de nombres décroissans depuis n jusqu'à l'unité, savoir $n(n - 1)(n - 2) ... 3\cdot 2\cdot 1$. L'emploi continuel de l'analyse combinatoire que je fais dans la plupart de mes démonstrations, a rendu cette notation indispensable.

My translation:

I used the very simple notation $n!$ to refer to the product of decreasing integers from n to 1, ie $n(n - 1)(n - 2) ... 3\cdot 2\cdot 1$. I had to do it since I've nominated this product a large number of times in my demonstrations.

  • 3
    Can someone translate? My old French is a little rusty. – user132181 May 20 '14 at 8:29
  • It's probably not the best translation but the idea is here. – Fabien May 20 '14 at 9:59
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    I employ the very simple notation $n!$ for denoting the product of the decreasing integers from $n$ to the unity, that is, $n(n-1)(n-2)\dots 3\cdot 2\cdot 1$. The continued usage of combinatorial analysis I make in the greater part of my proofs has made this notation indispensable. – egreg May 20 '14 at 11:11
  • "The continued usage of combinatorial analysis ... has made this notation indispensable." So say we all! – colormegone May 21 '14 at 18:23

As noted in Fabien's answer, the first stop for questions about notation is Cajori's A History of Mathematical Notations. Section 713 there contains an excerpt with Augustus de Morgan's observations on notation, including his opinion of the use of "!" for the factorial. He, for one, was not a fan. Here's an excerpt of the excerpt:

"Mathematical notation, like language, has grown up without much looking to, at the dictates of convenience and with the sanction of the majority. Resemblance, real or fancied, has been the first guide, and analogy has succeeded....

Among the worst of 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 $n!$ to signify $1\,.\,2\,.\,3\,.\,.\,.\,.\,(n-1)\,.\,n$, which gives their pages the appearance of expressing surprise and admiration that $2$, $3$, $4$, etc., should be found in mathematical results."

According to Ian Stewart, the symbol "!" was introduced because of printability. Before 1808

$\underline{n\big|} = n \cdot (n-1) \cdots 3 \cdot 2$

was [widely?] used to denote the factorial. Because it was hard to print [in non-computer ages], the French mathematician Christian Kramp chose "!".

Source: Professor Stewart's Hoard of Mathematical Treasures

  • 4
    I think that should be $\underline{\big|n}$, not $\underline{n\big|}$. There is an example in a paper by Hilbert, from 1894 (second page, half-way down, "$D_{\alpha\beta} = \ldots$"). And it is indeed badly typeset, but then our own computer-age efforts are hardly exemplary. – TonyK Aug 22 '14 at 10:52
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    In the above-mentioned source it's called $\underline{n\big|}$. – RomeoAndJuliet Aug 22 '14 at 10:55
  • Perhaps Stewart got it wrong. See for instance this MathWorld page. – TonyK Aug 22 '14 at 10:59
  • My Maths teacher used $\underline{n\big|} $ as a notation, because he liked it, and fairly comfortable to calculate factorial that way. But again, its just a notation :) – MonK Aug 22 '14 at 12:25
  • @TonyK Cajori agrees with Hilbert that it should be $\underline{\big|n}$. He also says this notation was introduced in 1827, and became somewhat popular only after Todhunter used it in his texts around 1860. – Per Manne Aug 22 '14 at 14:22

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