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In ], the phrase '''arbitrarily large''', '''arbitrarily small''', '''arbitrarily long''' is used in statements such statements as:
: "ƒ(''x'') is non-negative for arbitrarily large ''x''."
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: "For every real number ''n'', ƒ(''x'') is non-negative for some values of ''x'' greater than ''n''."
This"Arbitrarily large" should not be confused with the phrase "]"; e.g. For instance, it is true that prime numbers can be arbitrarily large (since there are an ] of them),butand it is not true that allsome sufficiently large numbers are not prime. "Arbitrarily large" does not mean "]" — for instance, while prime numbers can be arbitrarily large, there is no such thing as an infinitely large prime,sinceexists because all prime numbers (as well as all other integers) are finite.<ref> Accessed 21 February 2012.</ref>
In some cases,Such phrases such as "P(''x'') is true for arbitrarily large ''x''" issometimes are so used primarily for emphasis, as in "P(''x'') is true for all ''x'', no matter how large ''x'' is."; In such cases,then the phrase "arbitrarily large" does not havelacks the meaning indicated above,butand is in fact logically synonymous with "all."
ToFrom saythe thatexistence there areof "arbitrarily long ]" doescannot notbe meaninferred that there is any infinitely long arithmetic progression of prime numbers exists (therenone is not)do, nor that there is any particular arithmetic progression of prime numbers that is in some sense is "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 ''exists.<ref> Accessed 21 February 2012</ref>
The statement "ƒ(''x'') is non-negative for arbitrarily large ''x''." could be rewritten as:
: "'''For every''' real number ''n'', '''thereexists exists'''a real number ''x'' greater thanexceeding ''n'' such that ƒ(''x'') is non-negative.
Using "sufficiently large" instead yields:
: "'''ThereSuch exists'''a real number ''n'' such exists that '''for every''' real number ''x'' greater thanexceeding ''n'', ƒ(''x'') is non-negative.
==References==
Revision as of 01:53, 21 December 2013
In mathematics, the phrase arbitrarily large, arbitrarily small, arbitrarily long is used in such statements 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."
"Arbitrarily large" should not be confused with the phrase "sufficiently large"; e.g., prime numbers can be arbitrarily large (since there are an infinite number of them) and some sufficiently large numbers are not prime. "Arbitrarily large" does not mean "infinitely large" — for instance, while prime numbers can be arbitrarily large, no infinitely large prime exists because all prime numbers (as well as all other integers) are finite.
Such phrases as "P(x) is true for arbitrarily large x" sometimes are so used primarily for emphasis as in "P(x) is true for all x, no matter how large x is"; then the phrase "arbitrarily large" lacks the meaning indicated above and is logically synonymous with "all."
From the existence of "arbitrarily long arithmetic progressions of prime numbers" cannot be inferred that any infinitely long arithmetic progression of prime numbers exists (none do, nor that any particular arithmetic progression of prime numbers in some sense is "arbitrarily long"; rather that no matter how large a number n is, some arithmetic progression of prime numbers of length at least n exists.
The statement "ƒ(x) is non-negative for arbitrarily large x." could be rewritten as:
"For every real number n exists a real number x exceeding n that ƒ(x) is non-negative.
Using "sufficiently large" instead yields:
"Such a real number n such exists that for every real number x exceeding n ƒ(x) is non-negative.