!

2022-04-10T16:16:10Z

Metaweta:

[[Category:Ambiguities]]

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>:

<blockquote>
"How many ways of ordering six objects are there?"

"There are 6!"
</blockquote>

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?

<blockquote>
"There are 6!."
</blockquote>

However, if you want to express surprise with an exclamation mark, it could look like a double factorial:

<blockquote>
"There are 6!!"
</blockquote>

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:

<blockquote>
"There are 6 factorial."
</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
|}

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2022-04-10T16:14:42Z

Metaweta:

Why start at x, y, z - User contributions [en-gb]

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