The Most Excellent Order of the British Empire is an order of chivalry established on 4 June 1917 by George V of the United Kingdom. The Order is composed of five classes in civil and military divisions. In descending order of seniority, these are:

Only the highest two ranks automatically cause an individual to become a knight or dame, an honour allowing the recipient to use the title "Sir" (male) or "Dame" (female) before their first name (though men can be knighted separately from this and other Orders of Chivalry). Honorary knighthoods, given to individuals who are not nationals of a realm where Queen Elizabeth II is Head of State, permit use of the honour as a post-nominal but not as a title before their name. Awards in the Order of the British Empire in the Commonwealth Realms were discontinued with the establishment of national systems of honours and awards such as the Order of Australia, the Order of Canada and the New Zealand Order of Merit. Foreign recipients are classified as honorary members of the Order they receive, and do not contribute to the numbers restricted to that Order as full members do.

The British Empire comprised the dominions, colonies, protectorates, mandates and other territories ruled or administered by the United Kingdom. It originated with the overseas colonies and trading posts established by England in the late 16th and early 17th centuries. At its height, it was the largest empire in history and, for over a century, was the foremost global power. By 1922 the British Empire held sway over about 458 million people, one-fifth of the world's population at the time, and covered more than 33,700,000 km² (13,012,000 sq mi), almost a quarter of the Earth's total land area. As a result, its political, linguistic and cultural legacy is widespread. At the peak of its power, it was often said that "the sun never sets on the British Empire" because its span across the globe ensured that the sun was always shining on at least one of its numerous territories.

During the Age of Discovery in the 15th and 16th centuries, Portugal and Spain pioneered European exploration of the globe, and in the process established large overseas empires. Envious of the great wealth these empires generated, England, France and the Netherlands began to establish colonies and trade networks of their own in the Americas and Asia. A series of wars in the 17th and 18th centuries with the Netherlands and France left England (and then, following union between England and Scotland in 1707, Great Britain) the dominant colonial power in North America and India. The loss of the Thirteen Colonies in North America in 1783 after a war of independence deprived Britain of some of its oldest and most populous colonies.

In mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann–Landau notation (after Edmund Landau and Paul Bachmann), or asymptotic notation. In computer science, big O notation is used to classify algorithms by how they respond (e.g., in their processing time or working space requirements) to changes in input size.

Big O notation characterizes functions according to their growth rates: different functions with the same growth rate may be represented using the same O notation. A description of a function in terms of big O notation usually only provides an upper bound on the growth rate of the function. Associated with big O notation are several related notations, using the symbols o, Ω, ω, and Θ, to describe other kinds of bounds on asymptotic growth rates.

Big O notation is also used in many other fields to provide similar estimates.