Editing Recursion (section)

==Formal definitions==
[[File:Serpiente alquimica.jpg|thumb|[[Ouroboros]], an ancient symbol depicting a serpent or dragon eating its own tail]]
In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties:
* {{anchor|base case}}A simple ''base case'' (or cases) — a terminating scenario that does not use recursion to produce an answer
* {{anchor|recursive step}}A ''recursive step'' — a set of rules that reduces all successive cases toward the base case.

For example, the following is a recursive definition of a person's ''ancestor''. One's ancestor is either:
*One's parent (''base case''), ''or''
*One's parent's ancestor (''recursive step'').

The [[Fibonacci sequence]] is another classic example of recursion:

:{{math|1=Fib(0) = 0}} as base case 1,

:{{math|1=Fib(1) = 1}} as base case 2,

:For all [[integer]]s {{math|''n'' > 1}}, {{math|1=Fib(''n'') = Fib(''n'' − 1) + Fib(''n'' − 2)}}.

Many mathematical axioms are based upon recursive rules. For example, the formal definition of the [[natural number]]s by the [[Peano axioms]] can be described as: "Zero is a natural number, and each natural number has a successor, which is also a natural number."<ref>{{Cite web|url=https://www.britannica.com/science/Peano-axioms|title=Peano axioms {{!}} mathematics|website=Encyclopedia Britannica|language=en|access-date=2019-10-24}}</ref> By this base case and recursive rule, one can generate the set of all natural numbers.

Other recursively defined mathematical objects include [[factorial]]s, [[function (mathematics)|function]]s (e.g., [[recurrence relation]]s), [[set (mathematics)|sets]] (e.g., [[Cantor ternary set]]), and [[fractal]]s.

There are various more tongue-in-cheek definitions of recursion; see [[#Recursive humor|recursive humor]].