In
mathematics, the phrase
almost all has a number of specialised uses.
"Almost all" is sometimes used synonymously with "all but
finitely many"; see
almost.
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". 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.
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.
See also
Category:Mathematical terminology