Rīmagiefung
'''Rīmagiefung''' (on Nīƿenglisce hātte ''algebra'') is dǣl [[rīmcræft]]es be þǣre cneordlǣcinge þāra laga of ƿeorcunge and gesibbnessum, and þǣm timbrum and oncnāƿnessum þe arīsaþ from heom, ēac besetednessum, manigsetednessum, efenƿeorðfindungum, and rīmagiefunglicum endebyrdnessum. Andefen eorþmetes, arāflunge, stōƿlāre, tellingalāre, and rīma flītcræftes is rīmagiefung ān þāra hēafodlicena dǣla smǣtes rīmcræftes. 

Se dǣl rīmagiefunge gehāten grundstaðoliende rīmagiefung is oft dǣl þǣre lāre in æfterran lǣrunge and inlǣdeþ þā oncnāƿnesse missenrīma þe tācniaþ rīm. Cƿidas þā sind gestaðolode on þissum missenrīmum sind gestēorede notiende þā laga ƿeorcunga þā sind gehapa rīmum, sƿilce ēacnung. Mann mæg dōn þis for missenlicum intingum, ēac efenƿeorþfindunge. Rīmagiefung is sƿīðe brādre þonne grundstaðoliende rīmagiefung and is sēo cnēorlǣcing þǣre gelimpeþ sēo nytt missenlica laga ƿeorcinge and sēo gesettung ƿeorcinga tō ōðrum þingum būtan rīmum. Ēacnung and manigfealding cann mann settan on gemǣnre endebyrdnesse laga and heora fullan, rihtan tōmearcunga inlǣdaþ rīmagiefunglice endebyrdnessa sƿilce þrēatas, hringas and līchamas, þe sind cnēorlǣhte on þǣm dǣle rīmcræftes þe hātte brād rīmagiefung.

== Stǣr ==
[[Biliþ:Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala.jpg|thumb|Tramet fram [[:en:Muḥammad ibn Mūsā al-Khwārizmī|Al-Khƿārizmī]]'s ''Sēo Sceortlice Bōc be Rīmƿeorcinge þurh Fulfillinge and Efensettunge'']]

Þā þā [[Plato]] cōm, [[Crēcisc rīmcræft]] hæfde gefēred micele andƿendunge. Þā Crēcas sciepedon eorþmetlice rīmagiefunge on þǣre þe besetednessa ƿǣron tācnod fram sīdum eorþmetlicra þinga, oftost līna, þā hæfdon stafan geþēote ƿiþ hīe.<ref>Fram "Europe in the Middle Ages" on þǣm 258 tramente: "On þǣm rīmcræftlicum sōþfōresetednessum in Euclides ''Hēafodgesceaft'' fram VII oþ IX, rīm sind tācnod fram līndǣlum on þǣm þe stafan sind gesette, and þā eorþmetlican bēhþa in al-Khƿarizmies bēc ''Rīmagiefung'' notode stæfbǣra gefēgednessa; ac ealle efenƿyrcendas in þǣm efenƿeorþfindungum þā sind gebrocena in þǣre bēc ''Rīmagiefung'' sind amearcod rīm, ǣghƿæðer þe hīe sīen tācnode fram rīmstafum oþþe sīen geƿritene on ƿordum. Sēo oncnāƿness brādnesse is abēacnod in al-Khƿarizmies amearcunge, ac hē næfde nāne ƿīsan tō tācnienne rīmagiefunglīce þā geƿunelican fōresetednessa þe sind sƿā gearƿa in eorþmete."</ref> Diophantus (þe lifode on þǣm 3. hundgēare æfter Crīste), þe hƿīlum is gehāten "se fæder rīmagiefunge", ƿæs Alexandrianisc rīmcræftmann and se ƿrītere endebyrdnesse bōca þā hātaþ ''Rīmcræft''. Þās geƿritu standaþ be þǣre arāflinge rīmagiefunglicra efenƿeorþfindunga. 

Rīmagiefunge staðol biþ gefunden on gamolum Indiscum rīmcræfte, þe [[Muhammad ibn Mūsā al-Khwārizmī|Muhammad ibn Mūsā al-Khƿārizmī]] (lifode fram þǣm 780. gēare oþ þæt 850. gēar) hefige onfēng on mōde. Hē leornode Indiscne rīmcræft and inlēd hīe in þā Allahdōmiscan dǣlas þǣre ƿorulde þurh his ƿīdcūð rīmcræftlic geƿrit, ''Bōc be Ēacnunge and Animunge æfter þǣre Ƿīsan þāra Indiscena''.<ref>http://www.brusselsjournal.com/node/4107/print</ref><ref>''A History of Mathematics: An Introduction (on ōðerre ūtsendnesse) (sōftbōc) fram Victor J katz Addison Wesley; on ōðerre ūtsendnesse (on þǣm 6 dæge Hrēðmōnaðes þæs 1998 gēares)</ref> Hē lator aƿrāt ''[[Sēo Sceortlice Bōc be Rīmweorcinge þurh Fulfillinge and Efensettung|Sēo Sceortlice Bōc be Rīmƿeorcinge þurh Fulfillinge and Efensettung]]'', þe gesette rīmagiefunge tō rīmlāriscum cræfe se is ānstandende fram [[eorþmet]]e and [[grundrīmcræft]]e.<ref>{{citation|title=Al Khwarizmi: The Beginnings of Algebra|author=Roshdi Rashed|publisher=[[Saqi Books]]|date=November 2009|isbn=0863564305}}</ref>

Þā staðolas rīmagiefunge cunnon ƿesan gefunden on þǣm gamolum [[Babylonisc rīmlār|Babyloniscum]] ,<ref>Struik, Dirk J. (1987). ''A Concise History of Mathematics''. New York: Dover Publications.</ref> þā forðodon forðode rīmmetende endebyrdnesse þurh þā þe hīe cūðdon dōn rīmƿeorcunga in rīmendebyrdnessiscre ƿīsan. Þā Babyloniscan forðodon endebyrdnessa tō ƿeorcienne arāflunga cnottena þe sindgeƿunelīce arāfled tōdæg be þǣre nytte [[līnlicra efenweorþfindunga|līnlicra efenƿeorþfindunga]], [[fēowerscētra efenweorþfindunga|fēoƿerscētra efenƿeorþfindunga]], and [[unrīm efenweorþfindung|unrīmra līnlicra efenƿeorþfindunga]]. Tō ungelīcnesse, þæt mǣste dǣl [[Egyptisc rīmlār|Egyptiscena]] þisre tīde, and [[Crēcisc rīmlār|Crēciscena]] and [[Cīnisc rīmlār|Cīniscena]] rīmlārmanna in þǣm [[1 þūsendgēare fōre Crīste]], geƿunelīce arāflede sƿelca efenƿeorþfindunga þurh eorþmetlica endebyrdnesse, sƿelce þās amearcod in þǣm ''[[Rhind Rīmlāriscan Papyrus]]'', [[Euclides Gesceaft|Euclides ''Gesceaft'']], and ''[[Þā Nigone Hēafodwearda on þǣm Rīmcræfte|Þā Nigone Hēafodƿearda on þǣm Rīmcræfte]]''.  Þæt eorþmetlice ƿeorc þāra Crēciscena, ēoƿod in þǣre bēc  ''Gesceaft'', macode þæt grundƿeorc for þǣre geƿuneliclǣcunge endebyrdnessa begeondan þā arāflunge sumra cnottena on maniga geƿunelica endebyrdnessa secgunge and arāflunge efenƿeorþfindunga, þēah þe þis ne scolde ƿesan gecūþ oþ þā [[rīmcræft in midealdum Alladōme|midealdan Alladōmiscan rīmcræftmenn]].

Þā Crēciscan rīmcræftmenn [[Hero Alexandrie]] and [[Diophantus]]<ref>[http://library.thinkquest.org/25672/diiophan.htm Diophantus, Father of Algebra]</ref> and Indisce rīmcræftmenn, sƿelce [[Brahmagupta]], hēoldon þā þēaƿas [[Ægypt]]es and [[Babilōn]]ie, þēah þe Diophantus ''[[Arithmetica]]'' and Brahmaguptan ''[[Brahmasphutasiddhanta]]'' standaþ on hīeran efenette.<ref>[http://www.algebra.com/algebra/about/history/ History of Algebra]</ref> Tō bȳsne, sēo forme fulle rīmcræftisce arāfling (befōne nāƿihtrīm and underran arāflunga) [[fēowerscētra efenweorþfindunga|fēoƿerscētra efenƿeorþfindunga]] ƿæs amearcod fram Brahmaguptan in his bēc ''Brahmasphutasiddhanta''.  Þǣræfter, Arabisce and Muslime rīmcræftmenn forðodon rīmagiefungisca endebyrdnessa on sƿīðe hīeran strengðe manigfealdnesse. Þēah Diophantus and þā Babylōniscan brucon oftost ānlica endebyrdnessa tō arāflenne efenƿeorþlǣcinga, Al-Khƿarizmi ƿæs se forma þe arāflede efenƿeorþlǣcinga þā hƿīle þe hē brēac brāda endebyrdnessa. Hē arāflede þā līnlican unfæstlican efenƿeorþlǣcinga, fēoƿerscēta efenƿeorþlǣcinga, unfæstlica efenƿeorþlǣcinga þǣre ōðerre endebyrdnesse and efenƿeorþlǣcinga mid manigfealdum unfæstrīmum.  

<!--
[[File:Gerolamo Cardano.jpg|thumb|200px|In 1545, the Italian mathematician [[Girolamo Cardano]] published ''[[Ars Magna (Gerolamo Cardano)|Ars magna]]'' -''The great art'', a 40-chapter masterpiece in which he gave for the first time a method for solving the general [[quartic equation]].]]
The [[Greeks|Greek]] mathematician [[Diophantus]] has traditionally been known as the "father of algebra" but in more recent times there is much debate over whether al-Khwarizmi, who founded the discipline of ''al-jabr'', deserves that title instead.<ref>Carl B. Boyer, ''A History of Mathematics, Second Edition'' (Wiley, 1991), pages 178, 181</ref> Those who support Diophantus point to the fact that the algebra found in ''Al-Jabr'' is slightly more elementary than the algebra found in ''Arithmetica'' and that ''Arithmetica'' is syncopated while ''Al-Jabr'' is fully rhetorical.<ref>Carl B. Boyer, ''A History of Mathematics, Second Edition'' (Wiley, 1991), page 228</ref> Those who support Al-Khwarizmi point to the fact that he introduced the methods of "[[Reduction (mathematics)|reduction]]" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of [[like terms]] on opposite sides of the equation) which the term ''al-jabr'' originally referred to,<ref name=Boyer-229>{{Harv|Boyer|1991|loc="The Arabic Hegemony" p. 229}} "It is not certain just what the terms ''al-jabr'' and ''muqabalah'' mean, but the usual interpretation is similar to that implied in the translation above. The word ''al-jabr'' presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word ''muqabalah'' is said to refer to "reduction" or "balancing" - that is, the cancellation of like terms on opposite sides of the equation."</ref> and that he gave an exhaustive explanation of solving quadratic equations,<ref>{{Harv|Boyer|1991|loc="The Arabic Hegemony" p. 230}} "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwarizmi's exposition that his readers must have had little difficulty in mastering the solutions."</ref> supported by geometric proofs, while treating algebra as an independent discipline in its own right.<ref>Gandz and Saloman (1936), ''The sources of al-Khwarizmi's algebra'', Osiris i, p. 263–277: "In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".</ref> His algebra was also no longer concerned "with a series of [[problem]]s to be resolved, but an [[Expository writing|exposition]] which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."<ref name=Rashed-Armstrong>{{Citation | last1=Rashed | first1=R. | last2=Armstrong | first2=Angela | year=1994 | title=The Development of Arabic Mathematics | publisher=[[Springer Science+Business Media|Springer]] | isbn=0792325656 | oclc=29181926 | pages=11–2}}</ref>

The Persian mathematician [[Omar Khayyam]] is credited with identifying the foundations of [[algebraic geometry]] and found the general geometric solution of the [[cubic equation]]. Another Persian mathematician, [[Sharaf al-Dīn al-Tūsī]], found algebraic and numerical solutions to various cases of cubic equations.<ref>{{MacTutor|id=Al-Tusi_Sharaf|title=Sharaf al-Din al-Muzaffar al-Tusi}}</ref> He also developed the concept of a [[Function (mathematics)|function]].<ref>{{Citation|last=Victor J. Katz|first=Bill Barton|title=Stages in the History of Algebra with Implications for Teaching|journal=Educational Studies in Mathematics|publisher=[[Springer Science+Business Media|Springer Netherlands]]|volume=66|issue=2|date=October 2007|doi=10.1007/s10649-006-9023-7|pages=185–201 [192]}}</ref> The Indian mathematicians [[Mahavira (mathematician)|Mahavira]] and [[Bhaskara II]], the Persian mathematician [[Al-Karaji]],<ref name="Boyer al-Karkhi ax2n">{{Harv|Boyer|1991|loc="The Arabic Hegemony" p. 239}} "Abu'l Wefa was a capable algebraist as well as a trigonometer. [...] His successor al-Karkhi evidently used this translation to become an Arabic disciple of Diophantus - but without Diophantine analysis! [...] In particular, to al-Karkhi is attributed the first numerical solution of equations of the form ax<sup>2n</sup> + bx<sup>n</sup> = c (only equations with positive roots were considered),"</ref> and the Chinese mathematician [[Zhu Shijie]], solved various cases of cubic, [[quartic equation|quartic]], [[quintic equation|quintic]] and higher-order [[polynomial]] equations using numerical methods. In the 13th century, the solution of a cubic equation by [[Fibonacci]] is representative of the beginning of a revival in European algebra. As the Islamic world was declining, the European world was ascending. And it is here that Algebra was further developed. 

[[François Viète]]’s work at the close of the 16th century marks the start of the classical discipline of algebra. In 1637 [[René Descartes]] published ''[[La Géométrie]]'', inventing [[analytic geometry]] and introducing modern algebraic notation. Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century. The idea of a [[determinant]] was developed by [[Japanese mathematics|Japanese mathematician]] [[Kowa Seki]] in the 17th century, followed independently by [[Gottfried Leibniz]] ten years later, for the purpose of solving systems of simultaneous linear equations using [[matrix (mathematics)|matrices]]. [[Gabriel Cramer]] also did some work on matrices and determinants in the 18th century. Permutations were studied by [[Joseph Lagrange]] in his 1770 paper ''Réflexions sur la résolution algébrique des équations'' devoted to solutions of algebraic equations, in which he introduced [[Lagrange resolvents]]. [[Paolo Ruffini]] was the first person to develop the theory of [[permutation group]]s, and like his predecessors, also in the context of solving algebraic equations. 

[[Abstract algebra]] was developed in the 19th century, initially focusing on what is now called [[Galois theory]], and on [[constructible number|constructibility]] issues.<ref>"[http://www.math.hawaii.edu/~lee/algebra/history.html The Origins of Abstract Algebra]". University of Hawaii Mathematics Department.</ref> The "[[Abstract_algebra#Modern_algebra|modern algebra]]" has deep nineteenth-century roots in the work, for example, of [[Richard Dedekind]] and [[Leopold Kronecker]] and profound interconnections with other branches of mathematics such as [[algebraic number theory]] and [[algebraic geometry]].<ref>"[http://www.msri.org/calendar/workshops/WorkshopInfo/245/show_workshop The History of Algebra in the Nineteenth and Twentieth Centuries]". Mathematical Sciences Research Institute.</ref> [[George Peacock]] was the founder of axiomatic thinking in arithmetic and algebra. [[Augustus De Morgan]] discovered [[relation algebra]] in his ''Syllabus of a Proposed System of Logic''. [[Josiah Willard Gibbs]] developed an algebra of vectors in three-dimensional space, and [[Arthur Cayley]] developed an algebra of matrices (this is a non-commutative algebra).<ref>"[http://www.cambridge.org/catalogue/catalogue.asp?ISBN=9781108005043 The Collected Mathematical Papers]". Cambridge University Press.</ref>

== Classification ==
Algebra may be divided roughly into the following categories:
* '''[[Elementary algebra]]''', in which the properties of operations on the [[real number|real number system]] are recorded using symbols as "place holders" to denote [[Constant (mathematics)|constants]] and [[Variable (mathematics)|variables]], and the rules governing [[mathematical expression]]s and [[equation]]s involving these symbols are studied. This is usually taught at school under the title ''algebra'' (or ''intermediate algebra'' and ''college algebra'' in subsequent years). University-level courses in group theory may also be called ''elementary algebra''.
* '''[[Abstract algebra]]''', sometimes also called ''modern algebra'', in which [[algebraic structure]]s such as [[group (mathematics)|groups]], [[ring (mathematics)|rings]] and [[field (mathematics)|fields]] are [[axiomatization|axiomatically]] defined and investigated. 
* '''[[Linear algebra]]''', in which the specific properties of [[vector space]]s are studied (including [[matrix (mathematics)|matrices]]);
* '''[[Universal algebra]]''', in which properties common to all algebraic structures are studied.
* '''[[Algebraic number theory]]''', in which the properties of numbers are studied through algebraic systems.  [[Number theory]] inspired much of the original abstraction in algebra.
* '''[[Algebraic geometry]]''' applies abstract algebra to the problems of geometry.
* '''[[Algebraic combinatorics]]''', in which abstract algebraic methods are used to study combinatorial questions.
In some directions of advanced study, axiomatic algebraic systems such as groups, rings, fields, and algebras over a field are investigated in the presence of a [[geometry|geometric]] structure (a [[Metric (mathematics)|metric]] or a [[topology]]) which is compatible with the algebraic structure. The list includes a number of areas of [[functional analysis]]:
<div style="-moz-column-count:2; column-count:2;">
*[[Normed linear space]]s
*[[Banach space]]s
*[[Hilbert space]]s
*[[Banach algebra]]s
*[[Normed algebra]]s
*[[Topological algebra]]s
*[[Topological group]]s
</div>

== Elementary algebra ==

{{main|Elementary algebra}}

'''Elementary algebra''' is the most basic form of algebra. It is taught to students who are presumed to have no knowledge of [[mathematics]] beyond the basic principles of [[arithmetic]]. In arithmetic, only [[number]]s and their arithmetical operations (such as +, −, ×, ÷) occur. In algebra, numbers are often denoted by symbols (such as ''a'', ''x'', or ''y''). This is useful because:
* It allows the general formulation of arithmetical laws (such as ''a'' + ''b'' = ''b'' + ''a'' for all ''a'' and ''b''), and thus is the first step to a systematic exploration of the properties of the [[real number|real number system]].
* It allows the reference to "unknown" numbers, the formulation of [[equation]]s and the study of how to solve these (for instance, "Find a number ''x'' such that 3''x'' + 1 = 10" or going a bit further "Find a number ''x'' such that ''ax''+''b''=''c''". Step which lets to the conclusion that is not the nature of the specific numbers the one that allows us to solve it but that of the operations involved).
* It allows the formulation of [[function (mathematics)|functional]] relationships (such as "If you sell ''x'' tickets, then your profit will be 3''x'' − 10 dollars, or ''f''(''x'') = 3''x'' − 10, where ''f'' is the function, and ''x'' is the number to which the function is applied.").

=== Polynomials ===
{{main|Polynomial}}

A '''polynomial''' (see the article on [[polynomial]]s for more detail) is an [[expression (mathematics)|expression]] that is constructed from one or more [[Variable (mathematics)|variables]] and constants, using only the operations of addition, subtraction, and multiplication (where repeated multiplication of the same variable is standardly denoted as exponentiation with a constant non-negative integer exponent). For example, ''x''<sup>2</sup> + 2''x'' − 3 is a polynomial in the single variable ''x''.

An important class of problems in algebra is [[factorization]] of polynomials, that is, expressing a given polynomial as a product of other polynomials. The example polynomial above can be factored as (''x'' − 1)(''x'' + 3). A related class of problems is finding algebraic expressions for the [[root of a function|roots]] of a polynomial in a single variable.

== Abstract algebra ==
{{main|Abstract algebra}} {{see also|Algebraic structure}}

'''Abstract algebra''' extends the familiar concepts found in elementary algebra and [[arithmetic]] of [[number]]s to more general concepts.

'''[[Set (mathematics)|Sets]]''': Rather than just considering the different types of [[number]]s, abstract algebra deals with the more general concept of ''sets'': a collection of all objects (called [[Element (mathematics)|elements]]) selected by property, specific for the set. All collections of the familiar types of numbers are sets. Other examples of sets include the set of all two-by-two [[Matrix (mathematics)|matrices]], the set of all second-degree [[polynomials]] (''ax''<sup>2</sup> + ''bx'' + ''c''), the set of all two dimensional [[Vector (geometric)|vectors]] in the plane, and the various [[finite groups]] such as the [[cyclic group]]s which are the group of integers [[modular arithmetic|modulo]] ''n''. [[Set theory]] is a branch of [[logic]] and not technically a branch of algebra.

'''[[Binary operation]]s''': The notion of [[addition]] (+) is abstracted to give a ''binary operation'', ∗ say. The notion of binary operation is meaningless without the set on which the operation is defined. For two elements ''a'' and ''b'' in a set ''S'', ''a'' ∗ ''b'' is another element in the set; this condition is called [[Closure (mathematics)|closure]]. [[Addition]] (+), [[subtraction]] (-), [[multiplication]] (×), and [[Division (mathematics)|division]] (÷) can be binary operations when defined on different sets, as is addition and multiplication of matrices, vectors, and polynomials. 

'''[[Identity element]]s''': The numbers zero and one are abstracted to give the notion of an ''identity element'' for an operation. Zero is the identity element for addition and one is the identity element for multiplication. For a general binary operator ∗ the identity element ''e'' must satisfy ''a'' ∗ ''e'' = ''a'' and ''e'' ∗ ''a'' = ''a''. This holds for addition as ''a'' + 0 = ''a'' and 0 + ''a'' = ''a'' and multiplication ''a'' × 1 = ''a'' and 1 × ''a'' = ''a''. Not all set and operator combinations have an identity element; for example, the positive natural numbers (1, 2, 3, ...) have no identity element for addition.

'''[[Inverse elements]]''': The negative numbers give rise to the concept of ''inverse elements''. For addition, the inverse of ''a'' is −''a'', and for multiplication the inverse is 1/''a''. A general inverse element ''a''<sup>−1</sup> must satisfy the property that ''a'' ∗ ''a''<sup>−1</sup> = ''e'' and ''a''<sup>−1</sup> ∗ ''a'' = ''e''.

'''[[Associativity]]''': Addition of integers has a property called associativity. That is, the grouping of the numbers to be added does not affect the sum. For example: {{nowrap|1=(2 + 3) + 4 = 2 + (3 + 4).}} In general, this becomes (''a'' ∗ ''b'') ∗ ''c'' = ''a'' ∗ (''b'' ∗ ''c''). This property is shared by most binary operations, but not subtraction or division or [[octonion multiplication]].

'''[[Commutative operation|Commutativity]]''': Addition of integers also has a property called commutativity. That is, the order of the numbers to be added does not affect the sum. For example: 2+3=3+2. In general, this becomes ''a'' ∗ ''b'' = ''b'' ∗ ''a''. Only some binary operations have this property. It holds for the integers with addition and multiplication, but it does not hold for [[matrix multiplication]] or [[Quaternion|quaternion multiplication]] .

=== Groups—structures of a set with a single binary operation ===
{{main|Group (mathematics)}} {{see also|Group theory|Examples of groups}}

Combining the above concepts gives one of the most important structures in mathematics: a '''[[group (mathematics)|group]]'''. A group is a combination of a set ''S'' and a single [[binary operation]] ∗, defined in any way you choose, but with the following properties:
* An identity element ''e'' exists, such that for every member ''a'' of ''S'', ''e'' ∗ ''a'' and ''a'' ∗ ''e'' are both identical to ''a''.
* Every element has an inverse: for every member ''a'' of ''S'', there exists a member ''a''<sup>−1</sup> such that ''a'' ∗ ''a''<sup>−1</sup> and ''a''<sup>−1</sup> ∗ ''a'' are both identical to the identity element.
* The operation is associative: if ''a'', ''b'' and ''c'' are members of ''S'', then (''a'' ∗ ''b'') ∗ ''c'' is identical to ''a'' ∗ (''b'' ∗ ''c'').

If a group is also [[commutativity|commutative]]—that is, for any two members ''a'' and ''b'' of ''S'', ''a'' ∗ ''b'' is identical to ''b'' ∗ ''a''—then the group is said to be [[Abelian group|Abelian]].

For example, the set of integers under the operation of addition is a group. In this group, the identity element is 0 and the inverse of any element ''a'' is its negation, −''a''. The associativity requirement is met, because for any integers ''a'', ''b'' and ''c'', (''a'' + ''b'') + ''c'' = ''a'' + (''b'' + ''c'')

The nonzero [[rational number]]s form a group under multiplication. Here, the identity element is 1, since 1 × ''a'' = ''a'' × 1 = ''a'' for any rational number ''a''. The inverse of ''a'' is 1/''a'', since ''a'' × 1/''a'' = 1.

The integers under the multiplication operation, however, do not form a group. This is because, in general, the multiplicative inverse of an integer is not an integer. For example, 4 is an integer, but its multiplicative inverse is ¼, which is not an integer.

The theory of groups is studied in [[group theory]]. A major result in this theory is the [[classification of finite simple groups]], mostly published between about 1955 and 1983, which is thought to classify all of the [[finite set|finite]] [[simple group]]s into roughly 30 basic types.

{| class="wikitable"
|-
| colspan=11|Examples
|-
!Set:
| colspan=2|[[Natural numbers]] '''N'''
| colspan=2|[[Integers]] '''Z'''
| colspan=4|[[Rational numbers]] '''Q''' (also [[Real numbers|real]] '''R''' and [[Complex numbers|complex]] '''C''' numbers)
| colspan=2|Integers [[modular arithmetic|modulo]] 3: '''Z'''<sub>3</sub> = {0, 1, 2} 
|-
!Operation
| + 
| × (w/o zero)
| +
| × (w/o zero)
| +
| −
| × (w/o zero)
| ÷ (w/o zero)
| +
| × (w/o zero)
|-
!Closed
| Yes
| Yes
| Yes
| Yes
| Yes
| Yes
| Yes
| Yes
| Yes
| Yes
|-
| Identity
| 0
| 1
| 0
| 1
| 0
| N/A
| 1
| N/A
| 0
| 1
|-
| Inverse
| N/A
| N/A
| −''a''
| N/A
| −''a''
| N/A
| 1/''a''
| N/A
| 0, 2, 1, respectively
| N/A, 1, 2, respectively
|-
| Associative
| Yes
| Yes
| Yes
| Yes
| Yes
| No
| Yes
| No
| Yes
| Yes
|-
| Commutative
| Yes
| Yes
| Yes
| Yes
| Yes
| No
| Yes
| No
| Yes
| Yes
|-
| Structure
| [[monoid]]
| [[monoid]]
| [[Abelian group]]
| [[monoid]]
| [[Abelian group]]
| [[quasigroup]]
| [[Abelian group]]
| [[quasigroup]]
| [[Abelian group]]
| [[Abelian group]] ('''Z'''<sub>2</sub>)
|}

[[Semigroup]]s, [[quasigroup]]s, and [[monoid]]s are structures similar to groups, but more general. They comprise a set and a closed binary operation, but do not necessarily satisfy the other conditions. A [[semigroup]] has an ''associative'' binary operation, but might not have an identity element. A [[monoid]] is a semigroup which does have an identity but might not have an inverse for every element. A [[quasigroup]] satisfies a requirement that any element can be turned into any other by a unique pre- or post-operation; however the binary operation might not be associative.

All groups are monoids, and all monoids are semigroups.

=== Rings and fields—structures of a set with two particular binary operations, (+) and (×)  ===
{{main|ring (mathematics)|field (mathematics)}} {{see also|Ring theory|Glossary of ring theory|Field theory (mathematics)|glossary of field theory}}

Groups just have one binary operation.  To fully explain the behaviour of the different types of numbers, structures with two operators need to be studied. The most important of these are [[Ring (mathematics)|rings]], and [[Field (mathematics)|fields]]. 

'''[[Distributivity]]''' generalised the ''distributive law'' for numbers, and specifies the order in which the operators should be applied, (called  the [[Order of operations|precedence]]). For the integers {{nowrap|1=(''a'' + ''b'') × ''c'' = ''a'' × ''c'' + ''b'' × ''c''}} and {{nowrap|1=''c'' × (''a'' + ''b'') = ''c'' × ''a'' + ''c'' × ''b'',}} and × is said to be ''distributive'' over +.

A '''[[Ring (mathematics)|ring]]''' has two binary operations (+) and (×), with × distributive over +. Under the first operator (+) it forms an ''Abelian group''. Under the second operator (×) it is associative, but it does not need to have identity, or inverse, so division is not allowed. The additive (+) identity element is written as 0 and the additive inverse of ''a'' is written as −''a''.

The integers are an example of a ring. The integers have additional properties which make it an '''[[integral domain]]'''.

A '''[[Field (mathematics)|field]]''' is a ''ring'' with the additional property that all the elements excluding 0 form an ''Abelian group'' under ×. The multiplicative (×) identity is written as 1 and the multiplicative inverse of ''a'' is written as ''a''<sup>−1</sup>.

The rational numbers, the real numbers and the complex numbers are all examples of fields.
-->


==  Ǣrendgeƿritunclu  ==
{{reflist}}

== Fruman ==
* Donald R. Hill, ''Islamic Science and Engineering'' (Edinburgh University Press, 1994).
* Ziauddin Sardar, Jerry Ravetz, and Borin Van Loon, ''Introducing Mathematics'' (Totem Books, 1999).
* George Gheverghese Joseph, ''The Crest of the Peacock: Non-European Roots of Mathematics'' ([[Penguin Books]], 2000).
* John J O'Connor and Edmund F Robertson, ''[[MacTutor History of Mathematics archive]]'' ([[University of St Andrews]], 2005).
* I.N. Herstein: ''Topics in Algebra''. ISBN 0-471-02371-X
* R.B.J.T. Allenby: ''Rings, Fields and Groups''. ISBN 0-340-54440-6
* [[L. Euler]]: ''[http://web.mat.bham.ac.uk/C.J.Sangwin/euler/ Elements of Algebra] {{Webarchive|url=https://web.archive.org/web/20110413234352/http://web.mat.bham.ac.uk/C.J.Sangwin/euler/ |date=2011-04-13 }}'', ISBN 978-1-899618-73-6
* Isaac Asimov ''Realm of Algebra'' (Houghton Mifflin), 1961

==  Ūƿearde hlencan  ==

* [http://www.gresham.ac.uk/event.asp?PageId=45&EventId=620 4000 Years of Algebra] {{Webarchive|url=https://web.archive.org/web/20071004172100/http://www.gresham.ac.uk/event.asp?PageId=45&EventId=620 |date=2007-10-04 }}, tǣcing fram Robin Ƿilson, æt [[Gresham College]] on þǣm 17 dæge Ƿinterfylleðes þæs 2007 gēares (gēare for MP3 and MP4 onbehlæsting, and geƿrit).

[[Flocc:Rīmcræft]]
[[Flocc:Rīmagiefung]]
[[flocc:Nīwlicu word]]
Rīmagiefung (on Nīƿenglisce hātte algebra) is dǣl rīmcræftes be þǣre cneordlǣcinge þāra laga of ƿeorcunge and gesibbnessum, and þǣm timbrum and oncnāƿnessum þe arīsaþ from heom, ēac besetednessum, manigsetednessum, efenƿeorðfindungum, and rīmagiefunglicum endebyrdnessum. Andefen eorþmetes, arāflunge, stōƿlāre, tellingalāre, and rīma flītcræftes is rīmagiefung ān þāra hēafodlicena dǣla smǣtes rīmcræftes. 

Se dǣl rīmagiefunge gehāten grundstaðoliende rīmagiefung is oft dǣl þǣre lāre in æfterran lǣrunge and inlǣdeþ þā oncnāƿnesse missenrīma þe tācniaþ rīm. Cƿidas þā sind gestaðolode on þissum missenrīmum sind gestēorede notiende þā laga ƿeorcunga þā sind gehapa rīmum, sƿilce ēacnung. Mann mæg dōn þis for missenlicum intingum, ēac efenƿeorþfindunge. Rīmagiefung is sƿīðe brādre þonne grundstaðoliende rīmagiefung and is sēo cnēorlǣcing þǣre gelimpeþ sēo nytt missenlica laga ƿeorcinge and sēo gesettung ƿeorcinga tō ōðrum þingum būtan rīmum. Ēacnung and manigfealding cann mann settan on gemǣnre endebyrdnesse laga and heora fullan, rihtan tōmearcunga inlǣdaþ rīmagiefunglice endebyrdnessa sƿilce þrēatas, hringas and līchamas, þe sind cnēorlǣhte on þǣm dǣle rīmcræftes þe hātte brād rīmagiefung.

 Stǣr 
thumb|Tramet fram Al-Khƿārizmī's Sēo Sceortlice Bōc be Rīmƿeorcinge þurh Fulfillinge and Efensettunge

Þā þā Plato cōm, Crēcisc rīmcræft hæfde gefēred micele andƿendunge. Þā Crēcas sciepedon eorþmetlice rīmagiefunge on þǣre þe besetednessa ƿǣron tācnod fram sīdum eorþmetlicra þinga, oftost līna, þā hæfdon stafan geþēote ƿiþ hīe.Fram "Europe in the Middle Ages" on þǣm 258 tramente: "On þǣm rīmcræftlicum sōþfōresetednessum in Euclides Hēafodgesceaft fram VII oþ IX, rīm sind tācnod fram līndǣlum on þǣm þe stafan sind gesette, and þā eorþmetlican bēhþa in al-Khƿarizmies bēc Rīmagiefung notode stæfbǣra gefēgednessa; ac ealle efenƿyrcendas in þǣm efenƿeorþfindungum þā sind gebrocena in þǣre bēc Rīmagiefung sind amearcod rīm, ǣghƿæðer þe hīe sīen tācnode fram rīmstafum oþþe sīen geƿritene on ƿordum. Sēo oncnāƿness brādnesse is abēacnod in al-Khƿarizmies amearcunge, ac hē næfde nāne ƿīsan tō tācnienne rīmagiefunglīce þā geƿunelican fōresetednessa þe sind sƿā gearƿa in eorþmete." Diophantus (þe lifode on þǣm 3. hundgēare æfter Crīste), þe hƿīlum is gehāten "se fæder rīmagiefunge", ƿæs Alexandrianisc rīmcræftmann and se ƿrītere endebyrdnesse bōca þā hātaþ Rīmcræft. Þās geƿritu standaþ be þǣre arāflinge rīmagiefunglicra efenƿeorþfindunga. 

Rīmagiefunge staðol biþ gefunden on gamolum Indiscum rīmcræfte, þe Muhammad ibn Mūsā al-Khƿārizmī (lifode fram þǣm 780. gēare oþ þæt 850. gēar) hefige onfēng on mōde. Hē leornode Indiscne rīmcræft and inlēd hīe in þā Allahdōmiscan dǣlas þǣre ƿorulde þurh his ƿīdcūð rīmcræftlic geƿrit, Bōc be Ēacnunge and Animunge æfter þǣre Ƿīsan þāra Indiscena.http://www.brusselsjournal.com/node/4107/print<ref>A History of Mathematics: An Introduction (on ōðerre ūtsendnesse) (sōftbōc) fram Victor J katz Addison Wesley; on ōðerre ūtsendnesse (on þǣm 6 dæge Hrēðmōnaðes þæs 1998 gēares)</ref> Hē lator aƿrāt Sēo Sceortlice Bōc be Rīmƿeorcinge þurh Fulfillinge and Efensettung, þe gesette rīmagiefunge tō rīmlāriscum cræfe se is ānstandende fram eorþmete and grundrīmcræfte.

Þā staðolas rīmagiefunge cunnon ƿesan gefunden on þǣm gamolum Babyloniscum ,Struik, Dirk J. (1987). A Concise History of Mathematics. New York: Dover Publications. þā forðodon forðode rīmmetende endebyrdnesse þurh þā þe hīe cūðdon dōn rīmƿeorcunga in rīmendebyrdnessiscre ƿīsan. Þā Babyloniscan forðodon endebyrdnessa tō ƿeorcienne arāflunga cnottena þe sindgeƿunelīce arāfled tōdæg be þǣre nytte līnlicra efenƿeorþfindunga, fēoƿerscētra efenƿeorþfindunga, and unrīmra līnlicra efenƿeorþfindunga. Tō ungelīcnesse, þæt mǣste dǣl Egyptiscena þisre tīde, and Crēciscena and Cīniscena rīmlārmanna in þǣm 1 þūsendgēare fōre Crīste, geƿunelīce arāflede sƿelca efenƿeorþfindunga þurh eorþmetlica endebyrdnesse, sƿelce þās amearcod in þǣm Rhind Rīmlāriscan Papyrus, Euclides Gesceaft, and Þā Nigone Hēafodƿearda on þǣm Rīmcræfte.  Þæt eorþmetlice ƿeorc þāra Crēciscena, ēoƿod in þǣre bēc  Gesceaft, macode þæt grundƿeorc for þǣre geƿuneliclǣcunge endebyrdnessa begeondan þā arāflunge sumra cnottena on maniga geƿunelica endebyrdnessa secgunge and arāflunge efenƿeorþfindunga, þēah þe þis ne scolde ƿesan gecūþ oþ þā midealdan Alladōmiscan rīmcræftmenn.

Þā Crēciscan rīmcræftmenn Hero Alexandrie and DiophantusDiophantus, Father of Algebra and Indisce rīmcræftmenn, sƿelce Brahmagupta, hēoldon þā þēaƿas Ægyptes and Babilōnie, þēah þe Diophantus Arithmetica and Brahmaguptan Brahmasphutasiddhanta standaþ on hīeran efenette.History of Algebra Tō bȳsne, sēo forme fulle rīmcræftisce arāfling (befōne nāƿihtrīm and underran arāflunga) fēoƿerscētra efenƿeorþfindunga ƿæs amearcod fram Brahmaguptan in his bēc Brahmasphutasiddhanta.  Þǣræfter, Arabisce and Muslime rīmcræftmenn forðodon rīmagiefungisca endebyrdnessa on sƿīðe hīeran strengðe manigfealdnesse. Þēah Diophantus and þā Babylōniscan brucon oftost ānlica endebyrdnessa tō arāflenne efenƿeorþlǣcinga, Al-Khƿarizmi ƿæs se forma þe arāflede efenƿeorþlǣcinga þā hƿīle þe hē brēac brāda endebyrdnessa. Hē arāflede þā līnlican unfæstlican efenƿeorþlǣcinga, fēoƿerscēta efenƿeorþlǣcinga, unfæstlica efenƿeorþlǣcinga þǣre ōðerre endebyrdnesse and efenƿeorþlǣcinga mid manigfealdum unfæstrīmum.  

  Ǣrendgeƿritunclu  

 Fruman 
 Donald R. Hill, Islamic Science and Engineering (Edinburgh University Press, 1994).
 Ziauddin Sardar, Jerry Ravetz, and Borin Van Loon, Introducing Mathematics (Totem Books, 1999).
 George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics (Penguin Books, 2000).
 John J O'Connor and Edmund F Robertson, MacTutor History of Mathematics archive (University of St Andrews, 2005).
 I.N. Herstein: Topics in Algebra. ISBN 0-471-02371-X
 R.B.J.T. Allenby: Rings, Fields and Groups. ISBN 0-340-54440-6
 L. Euler: Elements of Algebra , ISBN 978-1-899618-73-6
 Isaac Asimov Realm of Algebra'' (Houghton Mifflin), 1961

  Ūƿearde hlencan  

 4000 Years of Algebra , tǣcing fram Robin Ƿilson, æt Gresham College on þǣm 17 dæge Ƿinterfylleðes þæs 2007 gēares (gēare for MP3 and MP4 onbehlæsting, and geƿrit).

Flocc:Rīmcræft
Flocc:Rīmagiefung
flocc:Nīwlicu word