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| In ], the phrase '''arbitrarily large''' is used in statements such as: | In ], 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: | which is shorthand for: | ||
| ⚫ | :"For every real number |
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| ⚫ | This should not be confused with the phrase "]". 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 "]" |
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| ⚫ | : "For every real number ''n'', ƒ(''x'') is non-negative for some values of ''x'' greater than ''n''." | ||
| ⚫ | In some cases, phrases such as "'' |
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| ⚫ | This should not be confused with the phrase "]". 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 "]" — 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 "ƒ(''x'') is true for arbitrarily large ''x''" is used primarily for emphasis, as in "ƒ(''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." | ||
| One can of course define terms such as "arbitrarily small," "arbitrarily long," and others in a similar manner. | |||
| ==See also== | ==See also== | ||
Revision as of 22:59, 19 July 2008
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. 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 "ƒ(x) is true for arbitrarily large x" is used primarily for emphasis, as in "ƒ(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."