In mathematics, arbitrary refers to a choice or value that is not restricted by specific rules or conditions. It can be anything from a set or a range of possibilities. For example, an arbitrary value is any possible value along the real line.

In mathematical proofs, the term arbitrary is used to show generality. It indicates that a statement or conclusion applies to any choice, not just specific cases. For instance, saying let $x$ be an arbitrary real number means the proof works for any real number, not just a specific one.