The notation name/arity gives the name and arity of the function. "arity" is the number of arguments (i.e., parameters), so for example explode/0 means you'd just write explode without any arguments, and map/1 means you'd write something like map(f).
The fact that 0-arity functions are invoked by name, without any parentheses, makes the notation especially handy. The fact that a function name can have multiple definitions at any one time (each definition having a distinct arity) makes it easy to distinguish between them.
This notation is not used in jq programs, but it is used in the output of the (new) built-in filter, builtins/0.
By contrast, in some other programming languages, it (or some close variant, e.g. module:name/arity in Erlang) is also part of the language.
Why?
There are various difficulties which typically arise when attempting to graft a notation that's suitable for languages in which method-dispatch is based on types onto ones in which dispatch is based solely on arity.
The first, as already noted, has to do with 0-arity functions. This is especially problematic for jq as 0-arity functions are invoked in jq without parentheses.
The second is that, in general, jq functions do not require their arguments to be any one jq type. Having to write something like nth(string+number) rather than just nth/1 would be tedious at best.
This is why the manual strenuously avoids using "name(type)"-style notation. Thus we see, for example, startswith(str), rather than startswith(string). That is, the parameter names in the documentation are clearly just names, though of course they often give strong type hints.
If you're wondering why the 'name/arity' convention isn't documented in the manual, it's probably largely because the documentation was mostly written before jq supported multi-arity functions.
In summary -- any notational scheme can be made to work, but name/arity is (1) concise; (2) precise in the jq context; (3) easy-to-learn; and (4) widely in use for arity-oriented languages, at least on this planet.
