In mathematics, the phrase almost all has a number of specialised uses. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and
"Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set"; see almost. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. In Mathematics, especially in Set theory, when dealing with sets of infinite size the term almost or nearly is used to mean all the elements except An example of this usage is the Frivolous Theorem of Arithmetic, which states that almost all natural numbers are very, very, very large. [1]
When speaking about the reals, sometimes it means "all reals but a set of Lebesgue measure zero". In Mathematics, the real numbers may be described informally in several different ways In Mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a Length, Area or Volume to In this sense we can say "almost all reals are not a member of the Cantor set". In Mathematics, the Cantor set, introduced by German Mathematician Georg Cantor in 1883 (but discovered in 1875 by Henry John Stephen Smith
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" and write
. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close"
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. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 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. A composite number is a positive Integer which has a positive Divisor other than one or itself
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. In Measure theory (a branch of Mathematical analysis) one says that a property holds almost everywhere if the set of elements for which the property does In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with In Probability theory, one says that an event happens almost surely (a Probability theory is the branch of Mathematics concerned with analysis of random phenomena
See also
References
- ^ Eric W. Weisstein, Frivolous Theorem of Arithmetic at MathWorld. In Mathematics, the phrase sufficiently large is used in contexts such as P is true for sufficiently large x which is actually shorthand Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
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