Almost all
Encyclopedia
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
.
A simple example is that almost all prime numbers are odd. (Two is a prime number.)
When speaking about the reals
, sometimes it means "all reals but a set of Lebesgue measure
zero" (formally, almost everywhere
). In this sense almost all reals are not a member of the Cantor set
even though the Cantor set is uncountable
.
In number theory
, if P(n) is a property of positive integer
s, and if p(N) denotes the number of positive integers n less than N for which P(n) holds, and if
(see limit
), then we say that "P(n) holds for almost all positive integers n" (formally, asymptotically almost surely) and write
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
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the phrase "almost all" has a number of specialised uses.
"Almost all" is sometimes used synonymously with "all but finitely many" (formally, a cofinite
Cofinite
In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X...
set) or "all but a countable set
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
" (formally, a cocountable
Cocountable
In mathematics, a cocountable subset of a set X is a subset Y whose complement in X is a countable set. In other words, Y contains all but countably many elements of X. For example, the irrational numbers are a cocountable subset of the reals...
set); see almost
Almost
In set theory, when dealing with sets of infinite size, the term almost or nearly is used to mean all the elements except for finitely many....
.
A simple example is that almost all prime numbers are odd. (Two is a prime number.)
When speaking about the reals
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
, sometimes it means "all reals but a set of Lebesgue measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
zero" (formally, almost everywhere
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
). In this sense almost all reals are not a member of the Cantor set
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
even though the Cantor set is uncountable
Uncountable set
In mathematics, an uncountable set is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.-Characterizations:There...
.
In number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, if P(n) is a property of positive integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s, 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
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
), then we say that "P(n) holds for almost all positive integers n" (formally, asymptotically almost surely) and write
For example, the prime number theorem
Prime number theorem
In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....
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
Composite number
A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
(not prime), however there are still an infinite number of primes.
Occasionally, "almost all" is used in the sense of "almost everywhere
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
" in measure theory, or in the closely related sense of "almost surely
Almost surely
In probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory...
" in probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
.