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Almost All

Given a property P, if P(x)∼x as x->infty (so, using asymptotic notation, the number of numbers less than x not satisfying the property P is o(x), where o(x) is one of the so-called Landau symbols), then P is said to hold true for almost all numbers. For example, almost all positive integers are composite numbers (which is not in conflict with the second of Euclid's theorems that there are an infinite number of primes).

See also

Almost Surely, Asymptotic Notation, For All, Landau Symbols, Normal Order

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References

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 50, 1999.Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, p. 8, 1979.

Referenced on Wolfram|Alpha

Almost All

Cite this as:

Weisstein, Eric W. "Almost All." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AlmostAll.html

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