Arbitrarily large

In mathematics, the phrase arbitrarily large, arbitrarily small, arbitrarily long is used in statements such as:

: "&fnof;("x") is non-negative for arbitrarily large "x"."

which is shorthand for:

: "For every real number "n", &fnof;("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. It is also worth noting that "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 "&fnof;("x") is true for arbitrarily large "x" is used primarily for emphasis, as in "&fnof;("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".

ee also

*Sufficiently large
*Mathematical jargon

Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

Large (disambiguation) — Large is an English surname.Large may also refer to:In botany:* Large flowered bellwort, plant in the family Liliaceae * Large fruited Elm, small deciduous tree or large shrub endemic to the Far East excluding JapanIn literature:* Geraldo at… … Wikipedia
Large deviations theory — In Probability Theory, the Large Deviations Theory concerns the asymptotic behaviour of remote tails of sequences of probability distributions. Some basic ideas of the theory can be tracked back to Laplace and Cramér, although a clear unified… … Wikipedia
Large set (Ramsey theory) — For other uses of the term, see Large set. In Ramsey theory, a set S of natural numbers is considered to be a large set if and only if Van der Waerden s theorem can be generalized to assert the existence of arithmetic progressions with common… … Wikipedia
Large numbers — This article is about large numbers in the sense of numbers that are significantly larger than those ordinarily used in everyday life, for instance in simple counting or in monetary transactions. The term typically refers to large positive… … Wikipedia
Sufficiently large — In mathematics, the phrase sufficiently large is used in contexts such as::P is true for sufficiently large xwhich is actually shorthand for::there exists an ainmathbb{R} such that P is true for all xge a.This does not necessarily mean that any… … Wikipedia
law of large numbers — Date: 1911 a theorem in mathematical statistics: the probability that the absolute value of the difference between the mean of a population sample and the mean of the population from which it is drawn is greater than an arbitrarily small amount… … New Collegiate Dictionary
Mathematical jargon — The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in… … Wikipedia
List of mathematical jargon — The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in… … Wikipedia
Prime number — Prime redirects here. For other uses, see Prime (disambiguation). A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is… … Wikipedia
Robust statistics — provides an alternative approach to classical statistical methods. The motivation is to produce estimators that are not unduly affected by small departures from model assumptions. Contents 1 Introduction 2 Examples of robust and non robust… … Wikipedia

Academic Dictionaries and Encyclopedias

Arbitrarily large

Look at other dictionaries:

Share the article and excerpts

Academic Dictionaries and Encyclopedias

Wikipedia

Arbitrarily large

Look at other dictionaries:

Share the article and excerpts

Direct link