Csdrhrt

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

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

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 $n(n - 1)(n - 2) ... 3cdot 2cdot 1$. 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 $n(n - 1)(n - 2) ... 3cdot 2cdot 1$. I had to do it since I've nominated this product a large number of times in my demonstrations.

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

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

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

Yesterday I gave a short answer to the question What does this symbol denote? and I cited a 1921 historical survey paper by Cajori. Since this paper is not freely available on the internet (in fact, my copy is a photocopy from a library bound volume of the journal), I decided to expand my answer by including excerpts from the paper, with links, almost all of which are freely available (at least in the U.S.), to all papers and books cited in the excerpts. However, because that question has been marked as a duplicate of the question here, and the question here seems to be more visible and a better depository for these excerpts, I’m posting the excerpts here instead.

Besides Cajori’s paper, I want to call attention to Pat Ballew’s 24 January 2014 blog entry Notes on the History of the Factorial, which also gives a fairly thorough historical survey.

Florian Cajori, History of symbols for $underline{n}=$ factorial, Isis 3 #3 (Summer 1921), 414-418.

(first few sentences of the paper, on p. 414) As simple a matter as the notations for the designation of « $n$-factorial » (that is, of the product $1.;2.;3 ldots n)$ involves in the history of its development questions of pedagogical and philosophical interest. Is it desirable to introduce a young student into a large amount of algebraic symbolism or should there be restrictions? May the symbolism not be so extensive as to retard rather than accelerate the progress of algebra as a science? We are led to the consideration of these questions by noting the history of factorial notations. It has been known $left({}^1right)$ for some years that the notation $n!$ representing the product $1.;2 ldots n,$ was introduced over a century ago by CHRISTIAN KRAMP [= Christian (Chrétien) Kramp (1760-1826)], professor at the University of Strasbourg, in his Éléments d'arithmétique universelle ou d'Algèbre [= Élémens d'Arithmétique Universelle; see the paragraph just before Article 188 on p. 219], Cologne, 1808 $[cdots]$

Note: Footnote $left({}^1right)$ on p. 414 mentions two references, published in 1903 and in 1904, in support of the claim about Kramp. In the lower half of p. 414, Cajori mentions Legendre's use of $Gamma(n+1)$ and Gauss's use of $Pi(n),$ both around 1811, and some other more obscure notations used in the first half of the 1800s.

(from upper half of p. 415) Relating to the origin of the notation $underline{big|n}$ for « $n$-factorial », nothing has been given in histories, except the statement that it has been in use in England. We are glad to be able to throw light upon the history of this symbol, and to give some details regarding the spread of it, and of its rival $n!,$ in the United States and other countries. The notation $underline{big|n}$ was suggested in 1827 by THOMAS JARRETT (1805-1882) who had just graduated from St Catherine’s College in Cambridge, England, with the degree of B. A. It occurs in a paper « On Algebraic Notation » that was printed in 1830 $left({}^3right).$

Note: Footnote $left({}^3right)$ on p. 415 specifies p. 67 of volume III of Transactions of the Cambridge Philosophical Society.

(from bottom of p. 415 to top of p. 416) For a quarter of a century the notation $underline{big|n}$ was neglected. In 1846, Rev. HARVEY GOODWIN used it freely in an article, « On the Geometrical Representation of Roots of Algebraic Equations », that was printed in 1849 $left({}^4right).$ [Footnote $left({}^4right)$ specifies p. 343 of volume 8 of Cambridge Philosophical Transactions.] In 1847 GOODWIN published his Elementary Course in Mathematics, a popular educational manual which reached several editions, but, to our surprise, he did not make use of any contracted notation for factorial $underline{n}$ in this text. In fact, the symbol $underline{big|n}$ made no substantial headway in England until it was adopted by ISAAC TODHUNTER about 1860, and was used in his popular texts.

(from lower half of p. 416) In the United States $underline{big|n}$ was probably introduced through TODHUNTER'S texts. In the first volume (1874) of J. E. HENDRICKS' [= Joel Evans Hendricks (1818-1893)] Analyst, (Des Moines, Iowa) both the notation $underline{big|n}$ and $n!$ are used by different writers. The latter notation, though simpler, was used in elementary texts of this country [= United States] less frequently than the first. The notation $underline{big|n}$ was adopted by such prominent text book writers as JOSEPH FICKLIN (Complete Algebra, copyright 1874), CHARLES DAVIES (Revised BOURDON'S Elements of Algebra, 1877), EDWARD OLNEY (1881) and about the same time by GEORGE A. WENTWORTH, WEBSTER WELLS, E. A. BOWSER [= Edward Albert Bowser (1845-1910)] and others. Thus it became firmly rooted in this country $left({}^4right).$ Among the French and German authors $underline{big|n}$ has met with no favor whatever [means: it was rarely used]. The notation $n!$ found wide adoption in Germany $[cdots]$

(footnote $left({}^4right)$ on p. 416) In a few publications JARRETT’S factorial symbol is given in the modified form $underline{nbig|}.$ See, for example, THOMAS CRAIG’S Treatise on Linear Differential Equations, vol. 1, p. 463, New York 1889, and WEBSTER’S New International Dictionary of the English Language, Springfield 1919, under the word « factorial ».

(from top half of p. 417) G. CHRYSTAL'S Algebra [= Algebra. An Elementary Text-Book for the Higher Classes of Secondary Schools and for Colleges, Part II, footnote on p. 4], 1889, came out for [means: advocated for] $n!,$ though in the $text{XIX}^{text{th}}$ century it was much less frequent in England than $underline{big|n}.$ In the United States $n!$ was used by W. P. G. Bartlett [= William Pitt Greenwood Bartlett (1837-1865)] $left({}^3right)$ as early as the year 1858. It was adopted mainly by a group of men who had studied at Harvard $[cdots]$

(footnote $left({}^3right)$ on p. 417) J. D. RUNKLE’S, Mathematical Monthly, Vol. I, No. 3, p. 84-87, Cambridge, Mass., 1858.

(lower half of p. 417 to end of the paper) In the present century [= early 1900s] the notation $n!$ has gained almost complete ascendancy over its rivals. It is far more convenient to the printer. Remarkable is the fact that many writings, both advanced and elementary, do not use any contracted notation for $n$-factorial; the expanded notation $1,; 2, ; 3ldots ; n$ ["$1.;2.;3 ldots n$" probably intended] is adhered to. The facts are that a short mode of designation is not so imperative here as it is for « square root », « cube root » or « the $n^{text{th}}$ power ». We have seen that HARVEY GOODWIN of Caius College, Cambridge, made liberal use of $underline{big|n}$ in a research article, but avoided it in his Elementary Course. Instinctively he shrunk from the introduction of it in elementary instruction. We have here the issue relating to the early and profuse of symbolism in mathematics: Is it desirable? In the case of $n$-factorial some writers of elementary books of recognized standing avoid it. More than this, it has been avoided by many writers in the field of advanced mathematics, such as $[cdots]$ Of course, I am not prepared to say that these writers never used $n!$ or $underline{big|n};$ I claim only that usually they avoided those symbols. These considerations are a part of the general question of the desirability of the use of symbols in mathematics to the extent advocated by the school of G. PEANO in Italy and of A. N. WHITEHEAD and B. RUSSELL in England. The feeling against such a « scab of symbols » seems to be strong and wide-spread. If the adoption of only one symbol, like our $n!$ or $underline{big|n},$ were involved, the issue would seem trivial, but when dozens of symbols are offered, a more serious situation arises. Certain types of symbols are indispensable; others possess only questionable value. Rich meaning is conveyed instantaneously by $frac{dy}{dx},$ $int y , dx,$ but $underline{big|n}$ and $n!$ serve no other purpose than to save a bit of space. Writers who accept in toto the program of expressing all theorems and all reasoning by a severely contracted symbolism, must frame notations for matters that can more conveniently be expressed by ordinary words or in less specialized symbolism. We know that intellectual food is sometimes more easily digested, if not taken in the most condensed form. It will be asked, to what extent can specialized notations be adopted with profit? To this question we reply, only experience can tell. It is one of the functions of the history of mathematics to record such experiences. Some light, therefore, may be expected from the study of the history of mathematics, as to what constitutes the most profitable and efficient course to pursue in the future. The history of mathematics can reduce to a minimum the amount of future experimentation. Hence algebraic notations deserve more careful historic treatment than they have hitherto received.