!
The ! symbol is principally used to represent the factorial operation.
When a factorial appears inside a sentence, it's possible to misinterpret the ! as an exclamation mark[1]:
"How many ways of ordering six objects are there?"
"There are 6!"
Two ! symbols together represent the double factorial, multiplying just the odd or even numbers. So juxtaposition doesn't represent composition here: \(x!! \neq (x!)!\)
A ! symbol on the left represents the number of derangements, or subfactorial. The order of precedence is not clear:
Does \(!n!\ = (!n)!\) or \(!(n!)\)?
Does \(a!b = (a!)b \) or \(a(!b)\)?
Does it make it clearer that a factorial is a present if you add another punctuation symbol after the ! symbol?
"There are 6!."
However, if you want to express surprise with an exclamation mark, it could look like a double factorial:
"There are 6!!"
Maybe ! should only be used for "factorial" in contexts that are unambiguously and clearly delimited mathematical notation, and the word "factorial" should be used in prose:
"There are 6 factorial."
The ! symbol is also widely used in category theory to indicate "the unique morphism making a diagram commute". So, for instance, the unique morphism into a terminal object, the unique morphism out of an initial object, the unique morphism into a product making the diagram commute, etc., are all denoted !.
| Z | ||||
| p ↙ | ↓ ! | ↘ q | ||
| X | ←π₁— | X × Y | —π₂→ | Y |
(The morphism ! denoting the unique arrow from an object Z equipped with morphisms p: Z → X, q: Z → Y to the universal such object X × Y making the diagram commute.)
