In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than the absolute value of the divisor. A fundamental property is that the quotient and the remainder exist and are unique, under some conditions. Because of this uniqueness, Euclidean division is often considered without referring to any method of computation, and without explicitly computing the quotient and the remainder. The methods of computation are called integer division algorithms, the best known of which being long division.
Euclidean division, and algorithms to compute it, are fundamental for many questions concerning integers, such as the Euclidean algorithm for finding the greatest common divisor of two integers,[1] and modular arithmetic, for which only remainders are considered.[2] The operation consisting of computing only the remainder is called the modulo operation,[3] and is used often in both mathematics and computer science.
Euclidean division is based on the following result, which is sometimes called Euclid's division lemma.
Given two integers a and b, with b ≠ 0, there exist unique integers q and r such that
and
where |b| denotes the absolute value of b.[4]
In the above theorem, each of the four integers has a name of its own: a is called the dividend, b is called the divisor, q is called the quotient and r is called the remainder.
The computation of the quotient and the remainder from the dividend and the divisor is called division, or in case of ambiguity, Euclidean division. The theorem is frequently referred to as the division algorithm (although it is a theorem and not an algorithm), because its proof as given below lends itself to a simple division algorithm for computing q and r (see the section Proof for more).
Division is not defined in the case where b = 0; see division by zero.
For the remainder and the modulo operation, there are conventions other than 0 ≤ r < |b|, see § Other intervals for the remainder.
Although originally restricted to integers, Euclidean division and the division theorem can be generalized to univariate polynomials over a field and to Euclidean domains.
In the case of univariate polynomials, the main difference is that the inequalities 0 ≤ ≤ --> r < | b | {\displaystyle 0\leq r<|b|} are replaced with
where deg {\displaystyle \deg } denotes the polynomial degree.
In the generalization to Euclidean domains, the inequality becomes
where f {\displaystyle f} denote a specific function from the domain to the natural numbers called a "Euclidean function".
The uniqueness of the quotient and the remainder remains true for polynomials, but it is false in general.
Although "Euclidean division" is named after Euclid, it seems that he did not know the existence and uniqueness theorem, and that the only computation method that he knew was the division by repeated subtraction.[citation needed]
Before the discovery of Hindu–Arabic numeral system, which was introduced in Europe during the 13th century by Fibonacci, division was extremely difficult, and only the best mathematicians were able to do it. Presently, most division algorithms, including long division, are based on this notation or its variants, such as binary numerals. A notable exception is Newton–Raphson division, which is independent from any numeral system.
The term "Euclidean division" was introduced during the 20th century as a shorthand for "division of Euclidean rings". It has been rapidly adopted by mathematicians for distinguishing this division from the other kinds of division of numbers.[citation needed]
Suppose that a pie has 9 slices and they are to be divided evenly among 4 people. Using Euclidean division, 9 divided by 4 is 2 with remainder 1. In other words, each person receives 2 slices of pie, and there is 1 slice left over.
This can be confirmed using multiplication, the inverse of division: if each of the 4 people received 2 slices, then 4 × 2 = 8 slices were given out in total. Adding the 1 slice remaining, the result is 9 slices. In summary: 9 = 4 × 2 + 1.
In general, if the number of slices is denoted a {\displaystyle a} and the number of people is denoted b {\displaystyle b} , then one can divide the pie evenly among the people such that each person receives q {\displaystyle q} slices (the quotient), with some number of slices r < b {\displaystyle r<b} being the leftover (the remainder). In which case, the equation a = b q + r {\displaystyle a=bq+r} holds.
If 9 slices were divided among 3 people instead of 4, then each would receive 3 and no slice would be left over, which means that the remainder would be zero, leading to the conclusion that 3 evenly divides 9, or that 3 divides 9.
Euclidean division can also be extended to negative dividend (or negative divisor) using the same formula; for example −9 = 4 × (−3) + 3, which means that −9 divided by 4 is −3 with remainder 3.
The following proof of the division theorem relies on the fact that a decreasing sequence of non-negative integers stops eventually. It is separated into two parts: one for existence and another for uniqueness of q {\displaystyle q} and r {\displaystyle r} . Other proofs use the well-ordering principle (i.e., the assertion that every non-empty set of non-negative integers has a smallest element) to make the reasoning simpler, but have the disadvantage of not providing directly an algorithm for solving the division (see § Effectiveness for more).[5]
For proving the existence of Euclidean division, one can suppose b > 0 , {\displaystyle b>0,} since, if b < 0 , {\displaystyle b<0,} the equality a = b q + r {\displaystyle a=bq+r} can be rewritten a = ( − − --> b ) ( − − --> q ) + r . {\displaystyle a=(-b)(-q)+r.} So, if the latter equality is a Euclidean division with − − --> b > 0 , {\displaystyle -b>0,} the former is also a Euclidean division.
Given b > 0 {\displaystyle b>0} and a , {\displaystyle a,} there are integers q 1 {\displaystyle q_{1}} and r 1 ≥ ≥ --> 0 {\displaystyle r_{1}\geq 0} such that a = b q 1 + r 1 ; {\displaystyle a=bq_{1}+r_{1};} for example, q 1 = 0 {\displaystyle q_{1}=0} and r 1 = a {\displaystyle r_{1}=a} if a ≥ ≥ --> 0 , {\displaystyle a\geq 0,} and otherwise q 1 = a {\displaystyle q_{1}=a} and r 1 = a − − --> a b . {\displaystyle r_{1}=a-ab.}
Let q {\displaystyle q} and r {\displaystyle r} be such a pair of numbers for which r {\displaystyle r} is nonnegative and minimal. If r < b . {\displaystyle r<b.} we have Euclidean division. Thus, we have to prove that, if r ≥ ≥ --> b , {\displaystyle r\geq b,} then r {\displaystyle r} is not minimal. Indeed, if r ≥ ≥ --> b , {\displaystyle r\geq b,} one has a = b ( q + 1 ) + ( r − − --> b ) , {\displaystyle a=b(q+1)+(r-b),} with 0 ≤ ≤ --> r − − --> b < r , {\displaystyle 0\leq r-b<r,} and r {\displaystyle r} is not minimal
This proves the existence in all cases. This provides also an algorithm for computing the quotient and the remainder, by starting from q = 0 {\displaystyle q=0} (if a ≥ ≥ --> 0 {\displaystyle a\geq 0} ) and adding 1 {\displaystyle 1} to it until a − − --> b q < b . {\displaystyle a-bq<b.} However, this algorithm is not efficient, since its number of steps is of the order of a / b {\displaystyle a/b}
The pair of integers r and q such that a = bq + r is unique, in the sense that there can be no other pair of integers that satisfy the same condition in the Euclidean division theorem. In other words, if we have another division of a by b, say a = bq' + r' with 0 ≤ r' < |b|, then we must have that
To prove this statement, we first start with the assumptions that
Subtracting the two equations yields
So b is a divisor of r′ – r. As
by the above inequalities, one gets
Since b ≠ 0, we get that r = r′ and q = q′, which proves the uniqueness part of the Euclidean division theorem.
In general, an existence proof does not provide an algorithm for computing the existing quotient and remainder, but the above proof does immediately provide an algorithm (see Division algorithm#Division by repeated subtraction), even though it is not a very efficient one as it requires as many steps as the size of the quotient. This is related to the fact that it uses only additions, subtractions and comparisons of integers, without involving multiplication, nor any particular representation of the integers such as decimal notation.
In terms of decimal notation, long division provides a much more efficient algorithm for solving Euclidean divisions. Its generalization to binary and hexadecimal notation provides further flexibility and possibility for computer implementation. However, for large inputs, algorithms that reduce division to multiplication, such as Newton–Raphson, are usually preferred, because they only need a time which is proportional to the time of the multiplication needed to verify the result—independently of the multiplication algorithm which is used (for more, see Division algorithm#Fast division methods).
The Euclidean division admits a number of variants, some of which are listed below.
In Euclidean division with d as divisor, the remainder is supposed to belong to the interval [0, d) of length |d|. Any other interval of the same length may be used. More precisely, given integers m {\displaystyle m} , a {\displaystyle a} , d {\displaystyle d} with m > 0 {\displaystyle m>0} , there exist unique integers q {\displaystyle q} and r {\displaystyle r} with d ≤ ≤ --> r < m + d {\displaystyle d\leq r<m+d} such that a = m q + r {\displaystyle a=mq+r} .
In particular, if d = − − --> ⌊ m 2 ⌋ {\displaystyle d=-\left\lfloor {\frac {m}{2}}\right\rfloor } then − − --> ⌊ m 2 ⌋ ≤ ≤ --> r < m − − --> ⌊ m 2 ⌋ {\displaystyle -\left\lfloor {\frac {m}{2}}\right\rfloor \leq r<m-\left\lfloor {\frac {m}{2}}\right\rfloor } . This division is called the centered division, and its remainder r {\displaystyle r} is called the centered remainder or the least absolute remainder.
This is used for approximating real numbers: Euclidean division defines truncation, and centered division defines rounding.
Given integers a {\displaystyle a} , m {\displaystyle m} and R , {\displaystyle R,} with m > 0 {\displaystyle m>0} and gcd ( R , m ) = 1 , {\displaystyle \gcd(R,m)=1,} let R − − --> 1 {\displaystyle R^{-1}} be the modular multiplicative inverse of R {\displaystyle R} (i.e., 0 < R − − --> 1 < m {\displaystyle 0<R^{-1}<m} with R − − --> 1 R − − --> 1 {\displaystyle R^{-1}R-1} being a multiple of m {\displaystyle m} ), then there exist unique integers q {\displaystyle q} and r {\displaystyle r} with 0 ≤ ≤ --> r < m {\displaystyle 0\leq r<m} such that a = m q + R − − --> 1 ⋅ ⋅ --> r {\displaystyle a=mq+R^{-1}\cdot r} . This result generalizes Hensel's odd division (1900).[6]
The value r {\displaystyle r} is the N-residue defined in Montgomery reduction.
Euclidean domains (also known as Euclidean rings)[7] are defined as integral domains which support the following generalization of Euclidean division:
Uniqueness of q and r is not required.[1] It occurs only in exceptional cases, typically for univariate polynomials, and for integers, if the further condition r ≥ 0 is added.
Examples of Euclidean domains include fields, polynomial rings in one variable over a field, and the Gaussian integers. The Euclidean division of polynomials has been the object of specific developments.
Frederik V is de naam voor: Frederik V van Neurenberg (1333-1398), burggraaf van Neurenberg (1357-1398) Frederik V van Baden-Durlach (1594-1659), markgraaf van Baden-Durlach (1622-1659) Frederik V van de Palts (1596-1632), keurvorst van de Palts (1610-1620) en koning van Bohemen (1619-1620) Frederik V van Denemarken (1723-1766), koning van Denemarken en Noorwegen (1746-1766) Frederik V van Hessen-Homburg (1748-1820), landgraaf van Hessen-Homburg (1751-1820) Bekijk alle artikelen waarvan de titel…
28th event hosted by the International Handball Federation 2023 World Men's Handball ChampionshipVärldmästerskapet i handboll för herrar 2023 (in Swedish)Mistrzostwa Świata w Piłce Ręcznej Mężczyzn 2023 (in Polish)Tournament detailsHost countries Poland SwedenDates11–29 JanuaryTeams32 (from 5 confederations)Venue(s)9 (in 9 host cities)Final positionsChampions Denmark (3rd title)Runner-up FranceThird place SpainFourth place SwedenTournament statis…
تماراك الإحداثيات 46°38′40″N 93°07′38″W / 46.644444444444°N 93.127222222222°W / 46.644444444444; -93.127222222222 [1] تاريخ التأسيس 26 يوليو 1921 تقسيم إداري البلد الولايات المتحدة[2] التقسيم الأعلى مقاطعة أيتكين خصائص جغرافية المساحة 9.070563 كيلومتر مربع9.261314 كيلومتر مرب
التصميم العظيم The Grand Design غلاف الكتاب اللغة الإنكليزية معلومات الكتاب المؤلف ستيفن هوكنغ وليوناردو ملودينو البلد الولايات المتحدة اللغة الانجليزية الناشر بانتام بوكس تاريخ النشر 7 سبتمبر 2010 مكان النشر نيويورك النوع الأدبي أدب علمي مبسَّط الموضوع علم التقديم نوع ا…
この記事には参考文献や外部リンクの一覧が含まれていますが、脚注による参照が不十分であるため、情報源が依然不明確です。適切な位置に脚注を追加して、記事の信頼性向上にご協力ください。(2020年5月) この記事で示されている出典について、該当する記述が具体的にその文献の何ページあるいはどの章節にあるのか、特定が求められています。ご存知の方は加
Overview of the beer culture in North Korea BeerCraft beer at the Taedonggang Microbrewery No. 3 in PyongyangKorean nameChosŏn'gŭl맥주Hancha麥酒Revised RomanizationmaekjuMcCune–ReischauermaekchuIPA[mɛk̚.t͈ɕu] North Korea has at least ten major breweries and many microbreweries that supply a wide range of beer products. The top brand is the light lager Taedonggang by the state-owned Taedonggang Brewing Company. The country's problems with goods distribution and power output ha…
The topic of this article may not meet Wikipedia's notability guideline for books. Please help to demonstrate the notability of the topic by citing reliable secondary sources that are independent of the topic and provide significant coverage of it beyond a mere trivial mention. If notability cannot be shown, the article is likely to be merged, redirected, or deleted.Find sources: Fitzpatrick's War – news · newspapers · books · scholar · JSTOR (June 2014) …
ザ・ハイスクール ヒーローズTHE HIGH SCHOOL HEROES別名 ハイヒロジャンル 連続ドラマ脚本 高橋悠也吉高寿男筧昌也監督 及川拓郎竹園元筧昌也出演者 岩﨑大昇佐藤龍我那須雄登浮所飛貴藤井直樹金指一世箭内夢菜長田成哉関智一和田優希(Jr.SP/ジャニーズJr.) 佐藤新(IMPACTors)檜山光成(少年忍者/ジャニーズJr.)南彩加戸次重幸中山美穂柳葉敏郎音楽 沢田完オープニング…
Union Army officer. Hollon RichardsonPhotograph circa 1861District Attorney of Chippewa County, WisconsinIn officeJanuary 1, 1861 – August 1861Preceded byA. K. GreggSucceeded byRodman Palmer Personal detailsBorn(1835-12-25)December 25, 1835Poland, Ohio, U.S.DiedDecember 24, 1916(1916-12-24) (aged 80)Keyport, Washington, U.S.Resting placeMount Pleasant Cemetery, Seattle, WashingtonPolitical partyRepublicanSpouse Leonora Colista Robinson (m. 1862…
Historic building in Exeter, England Benedictine Priory of St NicholasSt Nicholas Priory, ExeterReligionAffiliationRoman CatholicEcclesiastical or organizational statusevents venueStatusevents and private hireLocationLocationExeter, EnglandLocation within Devon and the United KingdomGeographic coordinates50°43′18″N 3°32′06″W / 50.7218°N 3.5350°W / 50.7218; -3.5350ArchitectureTypePrioryCompleted1087 The Benedictine Priory of St Nicholas or just St Nicholas Prio…
List of events ← 1676 1675 1674 1677 in Denmark → 1678 1679 1680 Decades: 1650s 1660s 1670s 1680s 1690s See also:Other events of 1677List of years in Denmark Events from the year 1677 in Denmark. Incumbents Monarch – Christian V[1] Grand Chancellor – Frederik Ahlefeldt Events 1–2 July: Battle of Køge Bay. 31 December – Christian V establishes the County oif Samsø for mistress Sophie Amalie Moth from the manors of Brattingsborg and Bisgård. Scanian War 31 May–1 Ju…
SMK Yadika 13InformasiJenisSwastaAkreditasiAKepala SekolahNovita Yusnaini, SSKetua KomiteGabriel Mala, STJurusan atau peminatanTeknik komputer dan jaringan, Teknik Kendaraan Ringan Otomotif, Akuntansi Keuangan Lembaga, Otomatisasi Tata Kelola PerkantoranRentang kelasX-XIIKurikulumKurikulum 13 RevisiAlamatLokasiJln.Villa Indah 1 Kp. Kebon Desa Jejalen Jaya,Kecamatan Tambun Selatan, Kabupaten Bekasi, Jawa Barat, IndonesiaTel./Faks.0812 1934 9338Situs webhttp://www.smkyadika13.…
This article's lead section may be too short to adequately summarize the key points. Please consider expanding the lead to provide an accessible overview of all important aspects of the article. (April 2023) Anthracene Names IUPAC name Anthracene Identifiers CAS Number 120-12-7 Y 3D model (JSmol) Interactive image Beilstein Reference 1905429 ChEBI CHEBI:35298 Y ChEMBL ChEMBL333179 Y ChemSpider 8111 Y DrugBank DB07372 Y ECHA InfoCard 100.003.974 EC Number 217-004-5 Gmelin…
Escudo de la República Socialista de Croacia InformaciónEntidad República Socialista de CroaciaFecha de adopción 17 de enero de 1947DescripciónBlasón Un escudo ajedrezado blanco y rojo, así como un sol naciente sobre el marTenante Tallos de trigo con un yunqueOtros elementos Estrella roja[editar datos en Wikidata] El Emblema de la República Socialista de Croacia fue adoptado el 17 de enero de 1947 por el gobierno de la República Socialista de Croacia. Está basado en el emblem…
This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: The Tin Princess – news · newspapers · books · scholar · JSTOR (August 2015) (Learn how and when to remove this template message) The Tin Princess First editionAuthorPhilip PullmanCountryUnited KingdomLanguageEnglishSeriesSally Lockhart seriesGenreJuvenile Fiction/ Action & Adv…
American wine importer Kermit LynchBornDecember 1941Bakersfield, CaliforniaOccupation(s)Wine importer, authorSpouseGail Skoff Kermit Lynch on the set of Wine Library TV Kermit Lynch (born December 1941 Bakersfield, California) is an American wine importer and author based in Berkeley, California. He is the author of Adventures on the Wine Route, which won the Veuve Clicquot Wine Book of the Year award, as well as Inspiring Thirst. He also co-owns Domaine Les Pallières in Gigondas along with the…
Donnybrook StadiumEnergia ParkThe new grandstand, shortly after completion, in February 2008Donnybrook StadiumLocation within DublinLocationDonnybrook, DublinCoordinates53°19′15.00″N 6°14′00.00″W / 53.3208333°N 6.2333333°W / 53.3208333; -6.2333333OwnerIrish Rugby Football UnionCapacity6,000[1]SurfaceSynthetic grass[2]ConstructionOpened1964Renovated2008TenantsBective Rangers RFC, Old Wesley RFC, Ireland Women (2016–present)[2][3 …
Chevet and mosaics The Basilica of Notre-Dame du Port is a Romanesque basilica, formerly a collegiate church, in the Port quarter of Clermont-Ferrand, between Place Delille and the cathedral. From the 10th century to the French Revolution it was served by a community of canons, regular until the 13th century, and thereafter secular. History Chevet under snow West door According to tradition, the church was founded by the Bishop of Clermont, Avitus of Clermont, in the 6th century and was rebuilt …
Motor vehicle MephistophelesOverviewManufacturerFiatProduction1923Body and chassisClassRacing carRelatedFiat SB4PowertrainEngine21.7L Fiat A.12Transmission4-speed manualDimensionsKerb weight2 tons The Fiat Mephistopheles (known in Italian as Mefistofele) is a one-off racing car created by Ernest A.D. Eldridge in 1923 by combining a Fiat racing car chassis and Fiat aeroplane engine. The name is from the demon of the same name. The name alluded to the infernal noise emitted from the unmuffled…
Form of popular music characterized by slow tempos and relaxed moodsChill out redirects here. For other uses, see Chill out (disambiguation). Chill-out (shortened as chill; also typeset as chillout or chill out) is a loosely defined form of popular music characterized by slow tempos and relaxed moods.[1][2] The definition of chill-out music has evolved throughout the decades, and generally refers to anything that might be identified as a modern type of easy listening. The term ch…
Lokasi Pengunjung: 18.117.138.193