Intereting Posts

$\lim_{x \to 0} \dfrac{f(x)-g(x)}{g^{-1}(x)-f^{-1}(x)} = 1$ for any $f,g \in C^1$ that are tangent to $\text{id}$ at $0$ with some simple condition
How find this sum closed form?
What happens if an uncountable collection of intervals is used in the definition of the Lebesgue outer measure?
If $A \ne \varnothing$, then $\varnothing^A=\varnothing$
Is it possible to write any bounded continuous function as a uniform limit of smooth functions
(Computationally) Simple sigmoid
Intuition – Fundamental Homomorphism Theorem – Fraleigh p. 139, 136
Show that $\lim\limits_{y\downarrow 0} y\mathbb{E}=0$.
All Intersection points of two spheres having arbitary centres?
Relation between Cholesky and SVD
$z\exp(z)$ surjectivity with the Little Picard Theorem
If Gal(K,Q) is abelian then |Gal(K,Q)|=n
Prove that a matrix is invertible
The generating function for the Fibonacci numbers
Is $ \sin: \mathbb{N} \to \mathbb{R}$ injective?

Does anyone know where the factorial “!” symbol came from?

I can’t decide if it is my favorite or least favorite notation in mathematics…

- Why does 0! = 1?
- $\sum_{i=1}^n i\cdot i! = (n+1)!-1$ By Induction
- How best to explain the $\sqrt{2\pi n}$ term in Stirling's?
- Exponent of $p$ in the prime factorization of $n!$
- How can I calculate the limit of exponential divided by factorial?
- Do factorials really grow faster than exponential functions?

- Proof by induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$
- Is mathematical history written by the victors?
- Exponent of $p$ in the prime factorization of $n!$
- Documentary of mathematics.
- Primality of $n! +1$
- Provenance of Hilbert quote on table, chair, beer mug
- Ultrafilters - when did it start?
- Is zero odd or even?
- What are some examples of notation that really improved mathematics?
- “L'Hôpital's rule” vs. “L'Hospital's rule”?

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.

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*

- Is there a bijection between the reals and naturals?
- When is a module over $R$ and $S$ an $R \otimes S$-module?
- Show that If $R$ is Euclidean domain then $R$ is a field
- How to know if a point is analytics or not?
- Invertibility of the Product of Two Matrices
- Is there anything like “cubic formula”?
- How to test if a graph is fully connected and finding isolated graphs from an adjacency matrix
- Proof of the First Isomorphism Theorem for Groups
- Find the minimum value of $P=\sum _{cyc}\frac{\left(x+1\right)^2\left(y+1\right)^2}{z^2+1}$
- How to show that the Laurent series of the Riemann Zeta function has $\gamma$ as its constant term?
- How to find inverse of function?
- Is there a finite abelian group $G$ such that $\textrm{Aut}(G)$ is abelian but $G$ is not cyclic?
- Image of open donut under $\phi=z+\frac{1}{z}$
- A finite ring is a field if its units $\cup\ \{0\}$ comprise a field of characteristic $\ne 2$
- Integral $\int^1_0\frac{\ln{x} \ \mathrm{Li}_2(x)}{1-x}dx$