In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient (integer division). In algebra, the remainder is the polynomial "left over" after dividing one polynomial by another. The modulo operation is the operation that produces such a remainder when given a dividend and divisor.
Formally it is also true that a remainder is what is left after subtracting one number from another, although this is more precisely called the difference. This usage can be found in some elementary textbooks; colloquially it is replaced by the expression "the rest" as in "Give me two dollars back and keep the rest." However, the term "remainder" is still used in this sense when a function is approximated by a series expansion and the error expression ("the rest") is referred to as the remainder term.
If a and d are integers, with d non-zero, it can be proven that there exist unique integers q and r, such that a = qd + r and 0 ≤ r < |d|. The number q is called the quotient, while r is called the remainder.
The remainder, as defined above, is called the least positive remainder or simply the remainder. The integer a is either a multiple of d or lies in the interval between consecutive multiples of d, namely, q⋅d and (q + 1)d (for positive q).
At times it is convenient to carry out the division so that a is as close as possible to an integral multiple of d, that is, we can write
In this case, s is called the least absolute remainder. As with the quotient and remainder, k and s are uniquely determined except in the case where d = 2n and s = ± n. For this exception we have,
A unique remainder can be obtained in this case by some convention such as always taking the positive value of s.
In the division of 43 by 5 we have:
so 3 is the least positive remainder. We also have,
and −2 is the least absolute remainder.
These definitions are also valid if d is negative, for example, in the division of 43 by −5,
and 3 is the least positive remainder, while,
and −2 is the least absolute remainder.
In the division of 42 by 5 we have:
and since 2 < 5/2, 2 is both the least positive remainder and the least absolute remainder.
In these examples, the (negative) least absolute remainder is obtained from the least positive remainder by subtracting 5, which is d. This holds in general. When dividing by d, either both remainders are positive and therefore equal, or they have opposite signs. If the positive remainder is r1, and the negative one is r2, then
When a and d are floating-point numbers, with d non-zero, a can be divided by d without remainder, with the quotient being another floating-point number. If the quotient is constrained to being an integer, however, the concept of remainder is still necessary. It can be proved that there exists a unique integer quotient q and a unique floating-point remainder r such that a = qd + r with 0 ≤ r < |d|.
Extending the definition of remainder for floating-point numbers as described above is not of theoretical importance in mathematics; however, many programming languages implement this definition, see modulo operation.
While there are no difficulties inherent in the definitions, there are implementation issues that arise when negative numbers are involved in calculating remainders. Different programming languages have adopted different conventions:
Euclidean division of polynomials is very similar to Euclidean division of integers and leads to polynomial remainders. Its existence is based on the following theorem: Given two univariate polynomials a(x) and b(x) (with b(x) not the zero polynomial) defined over a field (in particular, the reals or complex numbers), there exist two polynomials q(x) (the quotient) and r(x) (the remainder) which satisfy:
where "deg(...)" denotes the degree of the polynomial (the degree of the constant polynomial whose value is always 0 is defined to be negative, so that this degree condition will always be valid when this is the remainder.) Moreover, q(x) and r(x) are uniquely determined by these relations.
This differs from the Euclidean division of integers in that, for the integers, the degree condition is replaced by the bounds on the remainder r (non-negative and less than the divisor, which insures that r is unique.) The similarity of Euclidean division for integers and also for polynomials leads one to ask for the most general algebraic setting in which Euclidean division is valid. The rings for which such a theorem exists are called Euclidean domains, but in this generality uniqueness of the quotient and remainder are not guaranteed.
Aluminium alloys (or aluminum alloys; see spelling differences) are alloys in which aluminium (Al) is the predominant metal. The typical alloying elements are copper, magnesium, manganese, silicon, tin and zinc. There are two principal classifications, namely casting alloys and wrought alloys, both of which are further subdivided into the categories heat-treatable and non-heat-treatable. About 85% of aluminium is used for wrought products, for example rolled plate, foils and extrusions. Cast aluminium alloys yield cost-effective products due to the low melting point, although they generally have lower tensile strengths than wrought alloys. The most important cast aluminium alloy system is Al–Si, where the high levels of silicon (4.0–13%) contribute to give good casting characteristics. Aluminium alloys are widely used in engineering structures and components where light weight or corrosion resistance is required.Alloys composed mostly of aluminium have been very important in aerospace manufacturing since the introduction of metal-skinned aircraft. Aluminium-magnesium alloys are both lighter than other aluminium alloys and much less flammable than alloys that contain a very high percentage of magnesium.Aluminium alloy surfaces will develop a white, protective layer of aluminium oxide if left unprotected by anodizing and/or correct painting procedures. In a wet environment, galvanic corrosion can occur when an aluminium alloy is placed in electrical contact with other metals with more positive corrosion potentials than aluminium, and an electrolyte is present that allows ion exchange. Referred to as dissimilar-metal corrosion, this process can occur as exfoliation or as intergranular corrosion. Aluminium alloys can be improperly heat treated. This causes internal element separation, and the metal then corrodes from the inside out.Aluminium alloy compositions are registered with The Aluminum Association. Many organizations publish more specific standards for the manufacture of aluminium alloy, including the Society of Automotive Engineers standards organization, specifically its aerospace standards subgroups, and ASTM International.Baron Amherst of Hackney
Baron Amherst of Hackney (), in the County of London, is a title in the Peerage of the United Kingdom. It was created on 26 August 1892 for the former Conservative Member of Parliament William Tyssen-Amherst, with remainder, in default of male issue, to his eldest daughter Mary and her issue male. Tyssen-Amherst had previously represented West Norfolk and South West Norfolk in the House of Commons. He was succeeded according to the special remainder by his daughter Mary. She was the wife of Colonel Lord William Cecil, third son of William Cecil, 3rd Marquess of Exeter. As of 2017 the title is held by their great-great-grandson, the fifth Baron, who succeeded his father in 2009. As a male-line descendant of the third Marquess of Exeter he is also in remainder to this peerage and its subsidiary titles the earldom of Exeter and barony of Burghley.
Rear Admiral Sir Nigel Cecil was the son of Hon. Henry Mitford Amherst Cecil, fourth son of the second Baroness and Lord William Cecil. The champion racehorse trainer Sir Henry Cecil was the son of Henry Cecil, a younger brother of the third Baron.The family seat now is Hawthorn House, near Lymington, Hampshire.Baronet
A baronet ( or ; abbreviated Bart or Bt) or the rare female equivalent, a baronetess (, , or ; abbreviation Btss), is the holder of a baronetcy, a hereditary title awarded by the British Crown. The practice of awarding baronetcies was originally introduced in England in the 14th century and was used by James I of England in 1611 as a means of raising funds.
A baronetcy is the only British hereditary honour that is not a peerage, with the exception of the Anglo-Irish Black Knight, White Knight and Green Knight (of which only the Green Knight is extant). A baronet is addressed as "Sir" (just as is a knight) or "Dame" in the case of a baronetess but ranks above all knighthoods and damehoods in the order of precedence, except for the Order of the Garter, the Order of the Thistle, and the dormant Order of St Patrick. Baronets are conventionally seen to belong to the lesser nobility even though William Thoms claims that "The precise quality of this dignity is not yet fully determined, some holding it to be the head of the nobiles minores, while others, again, rank Baronets as the lowest of the nobiles majores, because their honour, like that of the higher nobility, is both hereditary and created by patent."Comparisons with continental titles and ranks are tenuous due to the British system of primogeniture and the fact that claims to baronetcies must be proven; currently the Official Roll of the Baronetage is overseen by the Ministry of Justice. In practice this means that the UK Peerage and Baronetage consists of about 2000 families (some Peers are also Baronets), which is roughly 0.01% of UK families. In some continental countries the nobility consisted of about 5% of the population, and in most countries titles are no longer recognised or regulated by the state.British Waterways
British Waterways, often shortened to BW, was a statutory corporation wholly owned by the government of the United Kingdom. It served as the navigation authority for the majority of canals and a number of rivers and docks in England, Scotland and Wales.On 2 July 2012 all of British Waterways' assets and responsibilities in England and Wales were transferred to the newly founded charity the Canal & River Trust. In Scotland, British Waterways continues to operate as a standalone public corporation under the trading name Scottish Canals.
The British Waterways Board was initially established as a result of the Transport Act 1962 and took control of the inland waterways assets of the British Transport Commission in 1963. British Waterways was sponsored by the Department for Environment, Food and Rural Affairs (DEFRA) in England and Wales, and by the Scottish Government in Scotland.British Waterways managed and maintained 2,200 miles (3,541 km) of canals, rivers and docks within the United Kingdom including the buildings, structures and landscapes alongside these waterways. Half of the United Kingdom population lives within five miles of a canal or river once managed by British Waterways. In addition to the watercourses, British Waterways also cared for and owned 2,555 listed structures including seventy scheduled ancient monuments. A further 800 areas have special designation and a further hundred are Sites of Special Scientific Interest (SSSIs).
Through its charitable arm The Waterways Trust, British Waterways maintained a museum of its history at the National Waterways Museum's three sites at Gloucester Docks, Stoke Bruerne and Ellesmere Port. Since the transfer of the assets and responsibilities of British Waterways to the Canal & River Trust the Waterways Trust in England and Wales has merged with the Canal & River Trust. It continues, however, as an independent charity in Scotland.Charitable trust
A charitable trust is an irrevocable trust established for charitable purposes and, in some jurisdictions, a more specific term than "charitable organization". A charitable trust enjoys a varying degree of tax benefits in most countries. It also generates good will. Some important terminology in charitable trusts is the term ‘corpus’ (Latin for ‘body’) which refers to the assets with which the trust is funded and the term ‘donor’ which is the person donating assets to a charity.Chinese remainder theorem
The Chinese remainder theorem is a theorem of number theory, which states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime.
The earliest known statement of the theorem is by the Chinese mathematician Sunzi in Sunzi Suanjing in the 3rd century AD.
The Chinese remainder theorem is widely used for computing with large integers, as it allows replacing a computation for which one knows a bound on the size of the result by several similar computations on small integers.
The Chinese remainder theorem (expressed in terms of congruences) is true over every principal ideal domain. It has been generalized to any commutative ring, with a formulation involving ideals.Decapoda
The Decapoda or decapods (literally "ten-footed") are an order of crustaceans within the class Malacostraca, including many familiar groups, such as crayfish, crabs, lobsters, prawns, and shrimp. Most decapods are scavengers. The order is estimated to contain nearly 15,000 species in around 2,700 genera, with around 3,300 fossil species. Nearly half of these species are crabs, with the shrimp (about 3000 species) and Anomura including hermit crabs, porcelain crabs, squat lobsters (about 2500 species) making up the bulk of the remainder. The earliest fossil decapod is the Devonian Palaeopalaemon.Divisibility rule
A divisibility rule is a shorthand way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any radix, or base, and they are all different, this article presents rules and examples only for decimal, or base 10, numbers. Martin Gardner explained and popularized these rules in his September 1962 "Mathematical Games" column in Scientific American.Division (mathematics)
Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication. The mathematical symbols used for the division operator are the obelus (÷) and the slash (/).
At an elementary level the division of two natural numbers is – among other possible interpretations – the process of calculating the number of times one number is contained within another one. This number of times is not always an integer, and this led to two different concepts.
The division with remainder or Euclidean division of two natural numbers provides a quotient, which is the number of times the second one is contained in the first one, and a remainder, which is the part of the first number that remains, when in the course of computing the quotient, no further full chunk of the size of the second number can be allocated.
For a modification of this division to yield only one single result, the natural numbers must be extended to rational numbers or real numbers. In these enlarged number systems, division is the inverse operation to multiplication, that is a = c ÷ b means a × b = c, as long as b is not zero—if b = 0, then this is a division by zero, which is not defined.Both forms of divisions appear in various algebraic structures. Those in which a Euclidean division (with remainder) is defined are called Euclidean domains and include polynomial rings in one indeterminate. Those in which a division (with a single result) by all nonzero elements is defined are called fields and division rings. In a ring the elements by which division is always possible are called the units; e.g., within the ring of integers the units are 1 and –1.Euclidean algorithm
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two numbers, the largest number that divides both of them without leaving a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c. 300 BC).
It is an example of an algorithm, a step-by-step procedure for performing a calculation according to well-defined rules,
and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations.
The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 252 − 105 = 147. Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that occurs, they are the GCD of the original two numbers. By reversing the steps, the GCD can be expressed as a sum of the two original numbers each multiplied by a positive or negative integer, e.g., 21 = 5 × 105 + (−2) × 252. The fact that the GCD can always be expressed in this way is known as Bézout's identity.
The version of the Euclidean algorithm described above (and by Euclid) can take many subtraction steps to find the GCD when one of the given numbers is much bigger than the other. A more efficient version of the algorithm shortcuts these steps, instead replacing the larger of the two numbers by its remainder when divided by the smaller of the two (with this version, the algorithm stops when reaching a zero remainder). With this improvement, the algorithm never requires more steps than five times the number of digits (base 10) of the smaller integer. This was proven by Gabriel Lamé in 1844, and marks the beginning of computational complexity theory. Additional methods for improving the algorithm's efficiency were developed in the 20th century.
The Euclidean algorithm has many theoretical and practical applications. It is used for reducing fractions to their simplest form and for performing division in modular arithmetic. Computations using this algorithm form part of the cryptographic protocols that are used to secure internet communications, and in methods for breaking these cryptosystems by factoring large composite numbers. The Euclidean algorithm may be used to solve Diophantine equations, such as finding numbers that satisfy multiple congruences according to the Chinese remainder theorem, to construct continued fractions, and to find accurate rational approximations to real numbers. Finally, it can be used as a basic tool for proving theorems in number theory such as Lagrange's four-square theorem and the uniqueness of prime factorizations. The original algorithm was described only for natural numbers and geometric lengths (real numbers), but the algorithm was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials of one variable. This led to modern abstract algebraic notions such as Euclidean domains.ISO 639-5
ISO 639-5:2008 "Codes for the representation of names of languages—Part 5: Alpha-3 code for language families and groups" is a highly incomplete international standard published by the International Organization for Standardization (ISO). It was developed by ISO Technical Committee 37, Subcommittee 2, and first published on May 15, 2008. It is part of the ISO 639 series of standards.Irish Republican Army
The Irish Republican Army (IRA) are paramilitary movements in Ireland in the 20th and the 21st century dedicated to Irish republicanism, the belief that all of Ireland should be an independent republic from British rule and free to form their own government. The original Irish Republican Army formed in 1917 from those Irish Volunteers who did not enlist in the British Army during World War I, members of the Irish Citizen Army and others. Irishmen formerly in the British Army returned to Ireland and fought in the Irish War of Independence. During the Irish War of Independence it was the army of the Irish Republic, declared by Dáil Éireann in 1919. Some Irish people dispute the claims of more recently created organisations that insist that they are the only legitimate descendants of the original IRA, often referred to as the "Old IRA".
The playwright and former IRA member Brendan Behan once said that the first issue on any Irish organisation's agenda was "the split". For the IRA, that has often been the case. The first split came after the Anglo-Irish Treaty in 1921, with supporters of the Treaty forming the nucleus of the National Army of the newly created Irish Free State, while the anti-treaty forces continued to use the name Irish Republican Army. After the end of the Irish Civil War (1922–23), the IRA was around in one form or another for forty years, when it split into the Official IRA and the Provisional IRA in 1969. The latter then had its own breakaways, namely the Real IRA and the Continuity IRA, each claiming to be the true successor of the Army of the Irish Republic.
The Irish Republican Army (1919–1922) (in later years, known as the "Old" IRA), recognised by the First Dáil as the legitimate army of the Irish Republic in April 1921 and fought the Irish War of Independence. On ratification by the Dáil of the Anglo-Irish Treaty, it split into pro-Treaty forces (the National Army, also known as the Government forces or the Regulars) and anti-Treaty forces (the Republicans, Irregulars or Executive forces) after the Treaty. These two went on to fight the Irish Civil War.
The Irish Republican Army (1922–1969), the anti-treaty IRA which fought and lost the civil war and which thereafter refused to recognise either the Irish Free State or Northern Ireland, deeming them both to be creations of British imperialism. It existed in one form or another for over 40 years before splitting in 1969.
The Official IRA (OIRA), the remainder of the IRA after the 1969 split with the Provisionals; was primarily Marxist in its political orientation. It is now inactive in the military sense, while its political wing, Official Sinn Féin, became the Workers' Party of Ireland.
The Provisional IRA (PIRA) broke from the OIRA in 1969 over abstentionism and how to deal with the increasing violence in Northern Ireland. Although opposed to the OIRA's Marxism, it came to develop a left-wing orientation and increasing political activity.
The Continuity IRA (CIRA) broke from the PIRA in 1986, because the latter ended its policy on abstentionism (thus recognising the authority of the Republic of Ireland).
The Real IRA (RIRA), a 1997 breakaway from the PIRA consisting of members opposed to the Northern Ireland peace process.
In April 2011, former members of the Provisional IRA announced a resumption of hostilities, and that "they had now taken on the mantle of the mainstream IRA." They further claimed "We continue to do so under the name of the Irish Republican Army. We are the IRA." and insisted that they "were entirely separate from the Real IRA, Óglaigh na hÉireann (ONH), and the Continuity IRA." They claimed responsibility for the April assassination of PSNI constable Ronan Kerr as well as responsibility for other attacks that had previously been claimed by the Real IRA and ONH.
The New IRA, which was formed as a merger between the Real IRA and other republican groups in 2012. (see Real IRA)Largest remainder method
The largest remainder method (also known as Hare–Niemeyer method, Hamilton method or as Vinton's method) is one way of allocating seats proportionally for representative assemblies with party list voting systems. It contrasts with various divisor methods.Modulo operation
In computing, the modulo operation finds the remainder after division of one number by another (sometimes called modulus).
Given two positive numbers, a (the dividend) and n (the divisor), a modulo n (abbreviated as a mod n) is the remainder of the Euclidean division of a by n. For example, the expression "5 mod 2" would evaluate to 1 because 5 divided by 2 leaves a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0 because the division of 9 by 3 has a quotient of 3 and leaves a remainder of 0; there is nothing to subtract from 9 after multiplying 3 times 3. (Note that doing the division with a calculator will not show the result referred to here by this operation; the quotient will be expressed as a decimal fraction.)
Although typically performed with a and n both being integers, many computing systems allow other types of numeric operands. The range of numbers for an integer modulo of n is 0 to n − 1. (a mod 1 is always 0; a mod 0 is undefined, possibly resulting in a division by zero error in programming languages.) See modular arithmetic for an older and related convention applied in number theory.
When either a or n is negative, the naive definition breaks down and programming languages differ in how these values are defined.Remainder (law)
In property law of the United Kingdom and the United States and other common law countries, a remainder is a future interest given to a person (who is referred to as the transferee or remainderman) that is capable of becoming possessory upon the natural end of a prior estate created by the same instrument. Thus, the prior estate must be one that is capable of ending naturally, for example upon the expiration of a term of years or the death of a life tenant. A future interest following a fee simple absolute cannot be a remainder because of the preceding infinite duration.
For example, a person, D, gives ("conveys") a piece of real property called Blackacre "to A for life, and then to B and her heirs". A receives a life estate in Blackacre and B holds a remainder, which can become possessory when the prior estate naturally terminates (A's death). However, B cannot claim the property until A's death.
There are two types of remainders in property law, vested and contingent. A vested remainder is held by a specific person without any conditions precedent; a contingent remainder is one for which the holder has not been identified, or for which a condition precedent must be satisfied.Rock Bottom Remainders
The Rock Bottom Remainders are an American rock charity supergroup, consisting of published writers, most of them both amateur musicians and popular English-language book, magazine, and newspaper authors. The band took its self-mocking name from the publishing term "remaindered book", a work of which the unsold remainder of the publisher's stock of copies is sold at a reduced price. Their performances collectively raised $2 million for charity from their concerts.
The band's members have included Dave Barry, Stephen King, Amy Tan, Cynthia Heimel, Sam Barry, Ridley Pearson, Scott Turow, Joel Selvin, James McBride, Mitch Albom, Roy Blount Jr., Barbara Kingsolver, Robert Fulghum, Matt Groening, Tad Bartimus, Greg Iles, Aron Ralston and honorary member Maya Angelou among others, as well as professional musicians such as multi-instrumentalist (and author) Al Kooper, drummer Josh Kelly, guitarist Roger McGuinn and saxophonist Erasmo Paulo. Founder Kathi Kamen Goldmark died on May 24, 2012.Taylor's theorem
In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial. For analytic functions the Taylor polynomials at a given point are finite-order truncations of its Taylor series, which completely determines the function in some neighborhood of the point. It can be thought of as the extension of linear approximation to higher order polynomials, and in the case of k equals 2 is often referred to as a quadratic approximation. The exact content of "Taylor's theorem" is not universally agreed upon. Indeed, there are several versions of it applicable in different situations, and some of them contain explicit estimates on the approximation error of the function by its Taylor polynomial.
Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1712. Yet an explicit expression of the error was not provided until much later on by Joseph-Louis Lagrange. An earlier version of the result was already mentioned in 1671 by James Gregory.
Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathematical analysis. Within pure mathematics it is the starting point of more advanced asymptotic analysis and is commonly used in more applied fields of numerics, as well as in mathematical physics. Taylor's theorem also generalizes to multivariate and vector valued functions on any dimensions n and m. This generalization of Taylor's theorem is the basis for the definition of so-called jets, which appear in differential geometry and partial differential equations.Texas Legends
The Texas Legends are an NBA G League team based in Frisco, Texas, and the minor league affiliate of the Dallas Mavericks. The franchise began as the Colorado 14ers in 2006, before relocating to Frisco in 2009 and becoming the Texas Legends for the 2010–11 season. The Legends play their home games at the Dr Pepper Arena.