In mathematics, the phrase arbitrarily large, arbitrarily small, arbitrarily long is used in statements such as:
- "ƒ(x) is non-negative for arbitrarily large x."
which is shorthand for:
- "For every real number n, ƒ(x) is non-negative for some values of x greater than n."
This should not be confused with the phrase "sufficiently large". For instance, it is true that prime numbers can be arbitrarily large (since there are an infinite number of them), but it is not true that all sufficiently large numbers are prime. "Arbitrarily large" does not mean "infinitely large" — for instance, while prime numbers can be arbitrarily large, there is no such thing as an infinitely large prime, since all prime numbers (as well as all other integers) are finite.
In some cases, phrases such as "P(x) is true for arbitrarily large x" is used primarily for emphasis, as in "P(x) is true for all x, no matter how large x is." In such cases, the phrase "arbitrarily large" does not have the meaning indicated above, but is in fact logically synonymous with "all."
To say that there are "arbitrarily long arithmetic progressions of prime numbers" does not mean that there is any infinitely long arithmetic progression of prime numbers (there is not), nor that there is any particular arithmetic progression of prime numbers that is in some sense "arbitrarily long", but rather that no matter how large a number n is, there is some arithmetic progression of prime numbers of length at least n.
The statement "ƒ(x) is non-negative for arbitrarily large x." could be rewritten as:
- "For every real number n, there exists real number x greater than n such that ƒ(x) is non-negative.
Using "sufficiently large" instead yields:
- "There exists real number n such that for every real number x greater than n, ƒ(x) is non-negative.
Famous quotes containing the words arbitrarily and/or large:
“History is not a book, arbitrarily divided into chapters, or a drama chopped into separate acts: it has flowed forward. Rome is a continuity, called “eternal.” What has accumulated in this place acts on everyone, day and night, like an extra climate.”
—Elizabeth Bowen (1899–1973)
“Miss C_____ is ... remarkably neat in her person and is uncommonly diligent in every part of useful economy.... She hath indeed under her father’s tuition acquired ... a large share of real learning of almost all the living and dead languages. Nor was the leisure which she found for such acquirements produced by neglecting anything necessary or useful for the family, but by the most assiduous industry.”
—Sarah Fielding (1710–1768)