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 real number is any possible value along the real line.

The use of the word arbitrary indicates that a statement is universally true and applies to all cases, not just specific cases. For instance, adding an arbitrary real number to itself results in twice the original number. This is true for all real numbers. x + x =2x is true for all real numbersx However, the statement x+x=2 is not true for arbitrary values of x. It is only for a specific value of x, which is 1. x + x = 2 whenx = 1 The use of arbitrary is critical for presenting mathematical reasoning and proofs. When proving that a property holds for all elements of a set, an arbitrary element from the set is considered. By showing that the property applies to this arbitrary element, it is concluded that the property holds for every element in the set.