Task: Question Answering
Input (English): Context: Abraham de Moivre (26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, one of those that link complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He was a friend of Isaac Newton, Edmond Halley, and James Stirling. Even though he faced religious persecution he remained a "steadfast Christian" throughout his life. Among his fellow Huguenot exiles in England, he was a colleague of the editor and translator Pierre des Maizeaux., England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west. The Irish Sea lies northwest of England and the Celtic Sea lies to the southwest. England is separated from continental Europe by the North Sea to the east and the English Channel to the south. The country covers five-eighths of the island of Great Britain (which lies in the North Atlantic) in its centre and south; and includes over 100 smaller islands such as the Isles of Scilly, and the Isle of Wight., Huguenots (or ) is a designation for an ethnoreligious group of French Protestants that follow the Reformed tradition., A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories, intensional definitions (which try to give the essence of a term) and extensional definitions (which proceed by listing the objects that a term describes). Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions., Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion., In mathematics , Stirling 's approximation ( or Stirling 's formula ) is an approximation for factorials . It is a very powerful approximation , leading to accurate results even for small values of n. It is named after James Stirling , though it was first stated by Abraham de Moivre . The formula as typically used in applications is \ ln ( n ! ) = n \ ln ( n ) - n + O ( \ ln ( n ) ) ( in big O notation ) . The next term in the O ( ln ( n ) ) is 1/2ln ( 2n ) ; a more precise variant of the formula is therefore n ! \ sim \ sqrt ( 2 \ pi n ) \ left ( \ frac ( n ) ( e ) \ right ) ^ n. It is also possible to give a version of Stirling 's formula with bounds valid for all positive integers n , rather than asymptotics : one has \ sqrt ( 2 \ pi ) \ n ^ ( n + \ frac12 ) e ^ ( - n ) \ le n ! \ le e \ n ^ ( n + \ frac12 ) e ^ ( - n ) for all positive integers n. Thus the ratio \ frac ( n ! ) ( n ^ ( n + \ frac12 ) e ^ ( - n ) ) is always between  2 = 2.5066 ... and e = 2.71828 ... . As an asymptotic formula , Stirling 's approximation has the property that the ratio \ frac ( n ! ) ( \ sqrt ( 2 \ pi n ) \ left ( \ frac ( n ) ( e ) \ right ) ^ n ) approaches 1 as n grows to infinity ., Trigonometry (from Greek "trignon", "triangle" and "metron", "measure") is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies., Pierre des Maizeaux, also spelled Desmaizeaux (c. 1666 or 1673June 1745), was a French Huguenot writer exiled in London, best known as the translator and biographer of Pierre Bayle., In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity), named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) and integer it holds that, A mathematician is someone who uses an extensive knowledge of mathematics in his/her work, typically to solve mathematical problems., Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework., Sir Isaac Newton (25 December 1642  20 March 1726/27) was an English mathematician, astronomer, and physicist (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time and a key figure in the scientific revolution. His book "Philosophiæ Naturalis Principia Mathematica" ("Mathematical Principles of Natural Philosophy"), first published in 1687, laid the foundations of classical mechanics. Newton also made seminal contributions to optics, and he shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus., A complex number is a number that can be expressed in the form , where and are real numbers and is the imaginary unit, satisfying the equation . In this expression, is the ' and is the ' of the complex number. If formula_1, then formula_2, Quantity is a property that can exist as a magnitude or multitude. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms of a unit of measurement. Quantity is among the basic classes of things along with quality, substance, change, and relation. Some quantities are such by their inner nature (as number), while others are functioning as states (properties, dimensions, attributes) of things such as heavy and light, long and short, broad and narrow, small and great, or much and little. A small quantity is sometimes referred to as a quantulum., Mathematics (from Greek  "máthma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics., Subject: stirling's approximation, Relation: instance_of, Options: (A) 25 (B) branch (C) complex number (D) definition (E) designation (F) editor (G) england (H) fashion (I) field (J) five (K) formula (L) four (M) identity (N) integer (O) island (P) key (Q) magnitude (R) mathematics (S) may (T) measurement (U) number (V) phrase (W) position (X) property (Y) quality (Z) quantity ([) range (\) real number (]) relation (^) relationship (_) space (`) statement (a) structure (b) study (c) theorem (d) theory (e) three (f) time (g) translator (h) triangle (i) two (j) unit of measurement (k) word (l) work
Output (English): theorem