Everything about Almost All totally explained
In
mathematics, the phrase
almost all has a number of specialised uses.
"Almost all" is sometimes used synonymously with "all but
finitely many" or "all but a
countable 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.
When speaking about the
reals, sometimes it means "all reals but a set of
Lebesgue measure zero". 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" 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.
Further Information
Get more info on 'Almost All'.
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