Algebra 1398 226145754 2008-07-17T00:56:03Z Blnguyen 435516 Reverted edits by [[Special:Contributions/60.242.196.194|60.242.196.194]] ([[User talk:60.242.196.194|talk]]) to last version by Synthebot {{otheruses4|the branch of mathematics}} '''Algebra''' is a branch of [[mathematics]] concerning the study of [[structure (algebraic)|structure]], [[relation (mathematics)|relation]], and [[quantity]]. Together with [[geometry]], [[mathematical analysis|analysis]], [[combinatorics]], and [[number theory]], algebra is one of the main branches of [[mathematics]]. [[Elementary algebra]] is often part of the curriculum in [[secondary education]] and provides an introduction to the basic ideas of algebra, including effects of [[addition|add]]ing and [[multiplication|multiply]]ing [[number]]s, the concept of [[variables]], definition of [[polynomial]]s, along with [[factorization]] and determining their [[root (mathematics)|root]]s. Algebra is much broader than elementary algebra and can be generalized. In addition to working directly with numbers, algebra covers working with [[symbols]], variables, and [[set]] [[element (mathematics)|elements]]. Addition and multiplication are viewed as general [[operator|operations]], and their precise definitions lead to structures such as [[group (mathematics)|groups]], [[ring (mathematics)|rings]] and [[field (mathematics)|fields]]. == 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]]s and [[variable]]s, and the rules governing [[mathematical expression]]s and [[equation]]s involving these symbols are studied (note that this usually includes the subject matter of courses called ''intermediate algebra'' and ''college algebra''), also called second year and third year algebra; * '''[[Abstract algebra]]''', sometimes also called ''modern algebra'', in which [[algebraic structure]]s such as [[group (mathematics)|groups]], [[ring (mathematics)|rings]] and [[field (mathematics)|field]]s are [[axiomatization|axiomatically]] defined and investigated; this includes, among other fields, * '''[[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]]''' in its algebraic aspect. * '''[[Combinatorics|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"). * It allows the formulation of [[function (mathematics)|function]]al 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''' is an [[expression (mathematics)|expression]] that is constructed from one or more [[variable]]s 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 whole number exponent). For example, <math>x^2 + 2x -3\,</math> 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 <math>(x-1)(x+3)\,\!.</math> A related class of problems is finding algebraic expressions for the [[root (mathematics)|root]]s 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]]s''': 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 (spatial)|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'' gives 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'' &times; 1 = ''a'' and 1 &times; ''a'' = ''a''. However, if we take the positive natural numbers and addition, there is no identity element. '''[[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: (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 products|quaternion multiplication]] . === Groups &ndash; 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 &times; ''a'' = ''a'' &times; 1 = ''a'' for any rational number ''a''. The inverse of ''a'' is 1/''a'', since ''a'' &times; 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 1/4, 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]] <math>\mathbb{N}</math> | colspan=2|[[Integers]] <math>\mathbb{Z}</math> | colspan=4|[[Rational numbers]] <math>\mathbb{Q}</math> (also [[Real numbers|real]] <math>\mathbb{R}</math> and [[Complex numbers|complex]] <math>\mathbb{C}</math> numbers) | colspan=2|Integers mod 3: {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 | NA | 1 | NA | 0 | 1 |- | Inverse | NA | NA | -a | NA | -a | NA | <math>\begin{matrix} \frac{1}{a} \end{matrix}</math> | NA | 0,2,1, respectively | NA, 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]] (<math>\mathbb{Z}_2</math>) |} [[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&mdash;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 (''a'' + ''b'') &times; c = ''a''&times;''c''+ ''b''&times;''c'' and ''c'' &times; (''a'' + ''b'') = ''c''&times;''a'' + ''c''&times;''b'', and &times; 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 &minus;''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. == Objects called algebras == The word '''''algebra''''' is also used for various [[algebraic structures]]: * [[Algebra over a field]] or more generally [[Algebra (ring theory)|Algebra over a ring]] * [[Algebra over a set]] * [[Boolean algebra (structure)|Boolean algebra]] * [[F-algebra]] and [[F-coalgebra]] in [[category theory]] * [[Sigma-algebra]] * [[Monad (category theory)|T-Algebras of monads]]. == History == {{Main|History of algebra}} {{See also|Timeline of algebra}} [[Image:Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala.jpg|thumb|A page from [[:en:Muhammad ibn Musa al-Khwarizmi|Al-Khwārizmī]]'s ''[[The Compendious Book on Calculation by Completion and Balancing|al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala]]'']] The origins of algebra can be traced to the ancient [[Babylonian mathematics|Babylonians]],<ref>Struik, Dirk J. (1987). ''A Concise History of Mathematics''. New York: Dover Publications.</ref> who developed an advanced [[arithmetic|arithmetical system]] with which they were able to do calculations in an algebraic fashion. With the use of this system they were able to apply formulas and calculate solutions for unknown values for a class of problems typically solved today by using [[linear equation]]s, [[quadratic equation]]s, and [[indeterminate equation|indeterminate linear equation]]s. By contrast, most [[Egyptian mathematics|Egyptians]] of this era, and most [[Indian mathematics|Indian]], [[Greek mathematics|Greek]] and [[Chinese mathematics|Chinese]] mathematicians in the [[1st millennium BC|first millennium BC]], usually solved such equations by [[geometry|geometric]] methods, such as those described in the ''[[Rhind Mathematical Papyrus]]'', ''[[Sulba Sutras]]'', [[Euclid's Elements|Euclid's ''Elements'']], and ''[[The Nine Chapters on the Mathematical Art]]''. The geometric work of the Greeks, typified in the ''Elements'', provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations. Later, the [[Indian mathematics|Indian mathematicians]] developed algebraic methods to a high degree of sophistication. Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, [[Brahmagupta]] was the first to solve equations using general methods. He solved the linear indeterminate equations, quadratic equations, second order indeterminate equations and equations with multiple variable. The word "algebra" is named after the [[Arabic]] word "''al-jabr''" from the title of the book ''{{Unicode | [[The Compendious Book on Calculation by Completion and Balancing|al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala]]}}'', meaning ''The book of Summary Concerning Calculating by Transposition and Reduction'', a book written by the [[Islamic mathematics|Persian mathematician]] {{Unicode|[[Muhammad ibn Musa al-Khwarizmi|Muhammad ibn Mūsā al-khwārizmī]]}} in 820. The word ''Al-Jabr'' means ''"reunion"''. The Hellenistic mathematician [[Diophantus]] has traditionally been known as "the father of algebra" but debate now exists as to whether or not Al-Khwarizmi should take that title.<ref>Carl B. Boyer, ''A History of Mathematics, Second Edition'' (Wiley, 1991), pages 178, 181</ref> Those who support Al-Khwarizmi point to the fact that much of his work on [[Reduction (mathematics)|reduction]] is still in use today and that he gave an exhaustive explanation of solving quadratic equations, while treating algebra as an independent discipline. Those who support [[Diophantus]] point to the fact that the algebra found in ''Al-Jabr'' is 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> Another Persian mathematician, [[Omar Khayyam]], developed [[algebraic geometry]] and found the general geometric solution of the [[cubic equation]]. The Indian mathematicians [[Mahavira (mathematician)|Mahavira]] and [[Bhaskara II]], and the Chinese mathematician [[Zhu Shijie]], solved various cases of cubic, [[quartic equation|quartic]], [[quintic equation|quintic]] and higher-order [[polynomial]] equations. 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 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. [[Abstract algebra]] was developed in the 19th century, initially focusing on what is now called [[Galois theory]], and on [[constructible number|constructibility]] issues. The stages of the development of symbolic algebra are roughly as follows: * Rhetorical algebra, which was developed by the Babylonians and remained dominant up to the 16th century; * Geometric constructive algebra, which was emphasised by the [[Vedic civilization|Vedic Indian]] and classical Greek mathematicians; * Syncopated algebra, as developed by [[Diophantus]], [[Brahmagupta]] and the ''[[Bakhshali Manuscript]]''; and * Symbolic algebra, which was initiated by [[Abū al-Hasan ibn Alī al-Qalasādī]]<ref name=Qalasadi/> and sees its culmination in the work of [[Gottfried Leibniz]]. [[Image:Diophantus-cover.jpg|right|thumb|200px|Cover of the 1621 edition of Diophantus's ''Arithmetica'', translated into [[Latin]] by [[Claude Gaspard Bachet de Méziriac]].]]A timeline of key algebraic developments are as follows: * Circa 1800 BC: The [[First Babylonian Dynasty|Old Babylonian]] [[Strassburg tablet]] seeks the solution of a quadratic elliptic equation. * Circa 1600 BC: The ''[[Plimpton 322]]'' tablet gives a table of [[Pythagorean triples]] in [[Babylonia]]n [[Cuneiform script]]. * Circa 800 BC: Indian mathematician [[Baudhayana]], in his ''Baudhayana [[Sulba Sutras|Sulba Sutra]]'', discovers Pythagorean triples algebraically, finds geometric solutions of linear equations and quadratic equations of the forms ax<sup>2</sup> = c and ax<sup>2</sup> + bx = c, and finds two sets of positive integral solutions to a set of simultaneous [[Diophantine equation]]s. * Circa 600 BC: Indian mathematician [[Apastamba]], in his ''Apastamba Sulba Sutra'', solves the general linear equation and uses simultaneous Diophantine equations with up to five unknowns. * Circa 300 BC: In Book II of his Elements, [[Euclid]] gives a geometric construction with Euclidean tools for the solution of the quadratic equation for positive real roots. The construction is due to the Pythagorean School of geometry. * Circa 300 BC: A geometric construction for the solution of the cubic is sought (doubling the cube problem). It is now well known that the general cubic has no such solution using [[Euclidean tools]]. * Circa 100 BC: Algebraic equations are treated in the Chinese mathematics book ''[[The Nine Chapters on the Mathematical Art|Jiuzhang suanshu]]'' (''The Nine Chapters on the Mathematical Art''), which contains solutions of linear equations solved using the [[False position method|rule of double false position]], geometric solutions of quadratic equations, and the solutions of matrices equivalent to the modern method, to solve systems of simultaneous linear equations. * Circa 100 BC: The ''[[Indian mathematics#Bakhshali Manuscript (200 BCE - 400 CE)|Bakhshali Manuscript]]'' written in [[Middle kingdoms of India|ancient India]] uses a form of algebraic notation using letters of the alphabet and other signs, and contains cubic and quartic equations, algebraic solutions of [[linear equations]] with up to five unknowns, the general algebraic formula for the quadratic equation, and solutions of indeterminate quadratic equations and simultaneous equations. * Circa 150 AD: [[Hero of Alexandria]] treats algebraic equations in three volumes of mathematics. * Circa 200: [[Diophantus]], who lived in Egypt and is often considered the "father of algebra", writes his famous ''[[Arithmetica]]'', a work featuring solutions of algebraic equations and on the theory of numbers. * 499: Indian mathematician [[Aryabhata]], in his treatise ''Aryabhatiya'', obtains whole-number solutions to linear equations by a method equivalent to the modern one, describes the general integral solution of the indeterminate linear equation and gives integral solutions of simultaneous indeterminate linear equations. * Circa 625: Chinese mathematician [[Wang Xiaotong]] finds numerical solutions of cubic equations. * 628: Indian mathematician [[Brahmagupta]], in his treatise ''Brahma Sputa Siddhanta'', invents the [[chakravala method|''chakravala'' method]] of solving indeterminate quadratic equations, including [[Pell's equation]], and gives rules for solving linear and quadratic equations. * 820: The word ''algebra'' is derived from operations described in the treatise written by the [[Persian people|Persian]] mathematician {{Unicode|[[Muhammad ibn Musa al-Khwarizmi|Muḥammad ibn Mūsā al-Ḵwārizmī]]}} titled ''Al-Kitab al-Jabr wa-l-Muqabala'' (meaning "The Compendious Book on Calculation by Completion and Balancing") on the systematic solution of [[linear equation|linear]] and [[quadratic equation]]s. Al-Khwarizmi is often considered as the "father of algebra", much of whose works on reduction was included in the book and added to many methods we have in algebra now. * Circa 850: [[Persian people|Persian]] mathematician [[al-Mahani]] conceived the idea of reducing geometrical problems such as [[Doubling the cube|duplicating the cube]] to problems in algebra. * Circa 850: Indian mathematician [[Mahavira (mathematician)|Mahavira]] solves various quadratic, cubic, quartic, quintic and higher-order equations, as well as indeterminate quadratic, cubic and higher-order equations. * Circa 990: [[Persian people|Persian]] [[Abu Bakr al-Karaji]], in his treatise ''al-Fakhri'', further develops algebra by extending Al-Khwarizmi's methodology to incorporate integral powers and integral roots of unknown quantities. He replaces geometrical operations of algebra with modern arithmetical operations, and defines the [[monomial]]s x, x<sup>2</sup>, x<sup>3</sup>, ... and 1/x, 1/x<sup>2</sup>, 1/x<sup>3</sup>, ... and gives rules for the products of any two of these. * Circa 1050: Chinese mathematician [[Jia Xian]] finds numerical solutions of polynomial equations. * 1072: [[Persian people|Persian]] mathematician [[Omar Khayyam]] develops algebraic geometry and, in the ''Treatise on Demonstration of Problems of Algebra'', gives a complete classification of cubic equations with general geometric solutions found by means of intersecting conic sections. * 1114: Indian mathematician [[Bhaskara]], in his ''Bijaganita'' (''Algebra''), recognizes that a positive number has both a positive and negative [[square root]], and solves various cubic, quartic and higher-order polynomial equations, as well as the general quadratic indeterminant equation. * 1202: Algebra is introduced to [[Europe]] largely through the work of [[Leonardo Fibonacci]] of [[Pisa]] in his work ''[[Liber Abaci]]''. * Circa 1300: Chinese mathematician [[Zhu Shijie]] deals with [[polynomial algebra]], solves quadratic equations, simultaneous equations and equations with up to four unknowns, and numerically solves some quartic, [[Quintic equation|quintic]] and higher-order polynomial equations. * Circa 1400: Indian mathematician [[Madhava of Sangamagramma]] finds [[iterative method]]s for approximate solution of non-linear equations. * Circa 1450: Arab mathematician [[Abū al-Hasan ibn Alī al-Qalasādī]] took "the first steps toward the introduction of [[Mathematical notation|algebraic symbolism]]." He represented [[Table of mathematical symbols|mathematical symbols]] using characters from the [[Arabic alphabet]].<ref name=Qalasadi>{{MacTutor Biography|id=Al-Qalasadi|title= Abu'l Hasan ibn Ali al Qalasadi}}</ref> * 1535: Nicolo Fontana [[Tartaglia]] and others mathematicians in Italy independently solved the general cubic equation.<ref name="Stewart">Stewart, Ian, ''Galois Theory, Third Edition'' (Chapman & Hall/CRC Mathematics, 2004).</ref> * 1545: Girolamo [[Cardano]] publishes ''Ars magna'' -''The great art'' which gives Fontana's solution to the general quartic equation.<ref name="Stewart" /> * 1572: [[Rafael Bombelli]] recognizes the complex roots of the cubic and improves current notation. * 1591: Francois [[Viete]] develops improved symbolic notation for various powers of an unknown and uses vowels for unknowns and consonants for constants in ''In artem analyticam isagoge''. * 1631: [[Thomas Harriot]] in a posthumous publication uses exponential notation and is the first to use symbols to indicate "less than" and "greater than". * 1682: [[Gottfried Leibniz|Gottfried Wilhelm Leibniz]] develops his notion of symbolic manipulation with formal rules which he calls ''characteristica generalis''. * 1680s: Japanese mathematician [[Kowa Seki]], in his ''Method of solving the dissimulated problems'', discovers the [[determinant]], and [[Bernoulli number]]s.<ref>{{cite web|url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Seki.html|title=Takakazu Seki Kowa|work=MacTutor History of Mathematics archive|author=O'Connor, John J., and Edmund F. Robertson}}</ref> * 1750: [[Gabriel Cramer]], in his treatise ''Introduction to the analysis of algebraic curves'', states [[Cramer's rule]] and studies [[algebraic curves]], matrices and determinants. * 1824: [[Niels Henrik Abel]] proved that the general quintic equation is insoluble by radicals.<ref name="Stewart" /> * 1832: Galois theory is developed by [[Évariste Galois]] in his work on abstract algebra.<ref name="Stewart" /> == See also == {{Wikibooks}} {{wiktionary}} * [[List of basic algebra topics]] * [[List of mathematics articles]] * [[Fundamental theorem of algebra]] * [[Computer algebra system]] * [[Order of operations]] ==Notes== {{reflist}} ==References== * 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]'', ISBN 978-1-89961-873-6 == External links == *[http://www.gresham.ac.uk/event.asp?PageId=45&EventId=620 4000 Years of Algebra], lecture by Robin Wilson, at [[Gresham College]], 17th October 2007 (available for MP3 and MP4 download, as well as a text file). *{{sep entry|algebra|Algebra|Vaughan Pratt}} {{Mathematics-footer}} {{Algebra-footer}} [[Category:Algebra| ]] [[Category:Arabic words and phrases]] [[af:Algebra]] [[ar:جبر]] [[an:Alchebra]] [[ast:Álxebra]] [[az:Cəbr]] [[bn:বীজগণিত]] [[zh-min-nan:Tāi-sò͘]] [[ba:Алгебра]] [[be:Алгебра]] [[bg:Алгебра]] [[ca:Àlgebra]] [[cs:Algebra]] [[co:Algebra]] [[cy:Algebra]] [[da:Algebra]] [[de:Algebra]] [[et:Algebra]] [[el:Άλγεβρα]] [[eml:Algebra]] [[es:Álgebra]] [[eo:Algebro]] [[eu:Aljebra]] [[fa:جبر]] [[fr:Algèbre]] [[fy:Algebra]] [[gd:Ailseabra]] [[gl:Álxebra]] [[zh-classical:代數學]] [[ko:대수학]] [[hi:बीजगणित]] [[io:Algebro]] [[id:Aljabar]] [[ia:Algebra]] [[is:Algebra]] [[it:Algebra]] [[he:אלגברה]] [[jv:Aljabar]] [[kn:ಬೀಜಗಣಿತ]] [[ka:ალგებრა]] [[ht:Aljèb]] [[lo:ພຶດຊະຄະນິດ]] [[la:Algebra]] [[lv:Algebra]] [[lt:Algebra]] [[lij:Algebra]] [[jbo:turki'icmaci]] [[hu:Algebra]] [[mk:Алгебра]] [[ml:ബീജഗണിതം]] [[mr:बीजगणित]] [[ms:Algebra]] [[nl:Algebra]] [[ja:代数学]] [[no:Algebra]] [[nn:Algebra]] [[nov:Algebra]] [[pms:Àlgebra]] [[pl:Algebra]] [[pt:Álgebra]] [[ro:Algebră]] [[qu:Qillqanancha kamay]] [[ru:Алгебра]] [[sco:Algebra]] [[sq:Algjebra]] [[scn:Àlgibbra]] [[simple:Algebra]] [[sk:Algebra]] [[sl:Algebra]] [[sr:Алгебра]] [[fi:Algebra]] [[sv:Algebra]] [[tl:Aldyebra]] [[ta:இயற்கணிதம்]] [[th:พีชคณิต]] [[vi:Đại số]] [[tg:Алгебра]] [[tr:Cebir]] [[tk:Algebra]] [[uk:Алгебра]] [[fiu-vro:Algõbra]] [[vls:Algebra]] [[yi:אלגעברע]] [[yo:Áljẹ́brà]] [[zh-yue:代數學]] [[zh:代数]]