Almost all

Almost all

In mathematics, the phrase almost all has a number of specialised uses.

"Almost all" is sometimes used synonymously with "all but finitely many" (formally, a cofinite set) or "all but a countable set" (formally, a cocountable set); see almost. An example of this usage is the "Frivolous Theorem of Arithmetic", which states that "almost all natural numbers are very, very, very large". [MathWorld|urlname=FrivolousTheoremofArithmetic|title=Frivolous Theorem of Arithmetic]

When speaking about the reals, sometimes it means "all reals but a set of Lebesgue measure zero" (formally, almost everywhere). In this sense we can say "almost all reals are not a member of the Cantor set".

In number theory, if "P"("n") is a property of positive integers, and if "p"("N") denotes the number of positive integers "n" less than "N" for which "P"("n") holds, and if

:"p"("N")/"N" → 1 as "N" → ∞

(see limit), then we say that "P"("n") holds for almost all positive integers "n" (formally, asymptotically almost surely) and write:(forall^infty n) P(n).

For example, the prime number theorem states that the number of prime numbers less than or equal to "N" is asymptotically equal to "N"/ln "N". Therefore the proportion of prime integers is roughly 1/ln "N", which tends to 0. Thus, "almost all" positive integers are composite (not prime), however there are still an infinite number of primes.

Occasionally, "almost all" is used in the sense of "almost everywhere" in measure theory, or in the closely related sense of "almost surely" in probability theory.

ee also

*Sufficiently large

References


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  • Almost — Al most ([add]l m[=o]st), adv. [AS. ealm[ae]st, [ae]lm[ae]st, quite the most, almost all; eal (OE. al) all + m?st most.] Nearly; well nigh; all but; for the greatest part. [1913 Webster] Almost thou persuadest me to be a Christian. Acts xxvi. 28 …   The Collaborative International Dictionary of English

  • Almost never — Almost Al most ([add]l m[=o]st), adv. [AS. ealm[ae]st, [ae]lm[ae]st, quite the most, almost all; eal (OE. al) all + m?st most.] Nearly; well nigh; all but; for the greatest part. [1913 Webster] Almost thou persuadest me to be a Christian. Acts… …   The Collaborative International Dictionary of English

  • Almost nothing — Almost Al most ([add]l m[=o]st), adv. [AS. ealm[ae]st, [ae]lm[ae]st, quite the most, almost all; eal (OE. al) all + m?st most.] Nearly; well nigh; all but; for the greatest part. [1913 Webster] Almost thou persuadest me to be a Christian. Acts… …   The Collaborative International Dictionary of English

  • all right vs alright —   All right has multiple meanings. It can mean ok, acceptable, unhurt.   The single word spelling alright has never been accepted as standard.   However in a search on Google you ll get around 68,700,000 hits for alright and 163,000,000 for all… …   English dictionary of common mistakes and confusing words

  • all right vs alright —   All right has multiple meanings. It can mean ok, acceptable, unhurt.   The single word spelling alright has never been accepted as standard.   However in a search on Google you ll get around 68,700,000 hits for alright and 163,000,000 for all… …   English dictionary of common mistakes and confusing words

  • almost / most —    Almost means nearly all : Almost all my friends have graduated from college by now.    Most is superlative of more, meaning the greatest or to the highest degree : Chuck is the most computer savvy guy I know, or Chuck cooked a most delicious… …   Confused words

  • almost / most —    Almost means nearly all : Almost all my friends have graduated from college by now.    Most is superlative of more, meaning the greatest or to the highest degree : Chuck is the most computer savvy guy I know, or Chuck cooked a most delicious… …   Confused words

  • all — (ôl) adj. 1. Being or representing the entire or total number, amount, or quantity: »All the windows are open. Deal all the cards. See Synonyms at WHOLE(Cf. ↑whole). 2. Constituting, being, or representing the total extent or the whole: »all… …   Word Histories

  • All the Year Round — was a Victorian periodical, being a British weekly literary magazine founded and owned by Charles Dickens, published between 1859 and 1895 throughout the United Kingdom. Edited by Charles Dickens, it was the direct successor to his previous… …   Wikipedia

  • all — [ôl] adj. [ME al, all < OE eal < IE * al no s < base * al , * ol , beyond, exceeding > L ultra] 1. the whole extent or quantity of [all New England, all the gold] 2. the entire number of [all the men went] 3. every one of [all men… …   English World dictionary

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