Encyclopedia > Almost all

  Article Content

Almost all

In mathematics, the phrase almost all has three specialised uses:

1) "Almost all" is sometimes used synonymous with "all but finitely many". For instance, one might say "almost all prime numbers are odd".

2) 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 tends to 0 as N tends to ∞ (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.

3) Occasionally, "almost all" is used in the sense of "almost everywhere" in measure theory.

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
Canadian Music Hall of Fame

... musicians for their lifetime achievements in music. The ceremony is held each year in Toronto as part of the Juno Awards. Complete list of Inductees 1978 Guy ...

This page was created in 22.1 ms