Big-O Notation

The symbol O(x), pronounced "big-O of x," is one of the Landau symbols and is used to symbolically express the asymptotic behavior of a given function.

In particular, if n is an integer variable which tends to infinity and x is a continuous variable tending to some limit, if phi(n) and phi(x) are positive functions, and if f(n) and f(x) are arbitrary functions, then it is said that f in O(phi) provided that |f|<Aphi for some constant A and all values n and x.

Note that big-O notation is the inverse of big-omega notation, i.e., that

 f(n) in O(phi(n)) <==> phi(n) in Omega(f(n)).

Additionally, big-O notation is related to little-O notation in that f in o(phi) is stronger than and implies f in O(phi).

See also

Asymptotic, Asymptotic Notation, Big-Omega Notation, Big-Theta Notation, Landau Symbols, Little-O Notation, Little-Omega Notation

This entry contributed by Christopher Stover

Explore with Wolfram|Alpha


Hardy, G. H. and Wright, E. M. "Some Notations." §1.6 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 7-8, 1979.

Cite this as:

Stover, Christopher. "Big-O Notation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications