Arbitrarily large


In mathematics, the phrases arbitrarily large, arbitrarily small and arbitrarily long are used in statements to make clear of the fact that an object is large, small and long with little limitation or restraint, respectively. The use of "arbitrarily" often occurs in the context of real numbers, though its meaning can differ from that of "sufficiently" and "infinitely".

Examples

The statement
is a shorthand for:
In the common parlance, the term "arbitrary long" is often used in the context of sequence of numbers. For example, to say that there are "arbitrarily long arithmetic progressions of prime numbers" does not mean that there exists any infinitely long arithmetic progression of prime numbers, nor that there exists any particular arithmetic progression of prime numbers that is in some sense "arbitrarily long". Rather, the phrase is used to refer to the fact that no matter how large a number ' is, there exists some arithmetic progression of prime numbers of length at least '.
Similar to arbitrary large, one can also define the phrase " holds for arbitrarily small real numbers", as follows:
In other words:

Arbitrarily large vs. sufficiently large vs. infinitely large

While similar, "arbitrarily large" is not equivalent to "sufficiently large". For instance, while it is true that prime numbers can be arbitrarily large, it is not true that all sufficiently large numbers are prime.
As another example, the statement " is non-negative for arbitrarily large '." could be rewritten as:
However, using "sufficiently large", the same phrase becomes:
Furthermore, "arbitrarily large" also does not mean "infinitely large". For example, although prime numbers can be arbitrarily large, an infinitely large prime number does not exist—since all prime numbers are finite.
In some cases, phrases such as "the proposition is true for arbitrarily large '" are used primarily for emphasis, as in " is true for all ', no matter how large ' is." In these cases, the phrase "arbitrarily large" does not have the meaning indicated above. Instead, the usage in this case is in fact logically synonymous with "all".