|This article needs additional citations for verification. (January 2010) (Learn how and when to remove this template message)|
In mathematics, the phrase "almost all" has a number of specialised uses which extend its intuitive meaning.
A simple example is that almost all prime numbers are odd, which is based on the fact that all but one prime number are odd. (The exception is the number 2, which is prime but not odd.)
Perversely, if we allow "almost all" to mean "all but a countable set", then it follows that almost all prime numbers are even, since the set of all prime numbers is itself countable.
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.
- p(N)/N → 1 as 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.