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[[Category:Ambiguities]] | [[Category:Ambiguities]] | ||
The ! symbol is used to represent the factorial operation. | 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<ref>[https://twitter.com/matthras/status/1415236669553274882 Tweet by Matt Mack]</ref>: | When a factorial appears inside a sentence, it's possible to misinterpret the ! as an exclamation mark<ref>[https://twitter.com/matthras/status/1415236669553274882 Tweet by Matt Mack]</ref>: | ||
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Two ! symbols together represent the ''double factorial'', multiplying just the odd or even numbers. So juxtaposition doesn't represent composition here: \(x!! \neq (x!)!\) | 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? | Does it make it clearer that a factorial is a present if you add another punctuation symbol after the ! symbol? | ||
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"There are 6 factorial." | "There are 6 factorial." | ||
</blockquote> | </blockquote> | ||
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 !. | |||
{| | |||
|+ Product diagram | |||
|- | |||
| || || Z || || | |||
|- | |||
| || p ↙ || ↓ ! || ↘ q || | |||
|- | |||
| X || ←π₁— || X × Y || —π₂→ || Y | |||
|} | |||
Latest revision as of 16:16, 10 April 2022
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 |
