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| {{redirect|Algebraist|the novel by Iain M. Banks|The Algebraist}}
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| {{for|beginner's introduction to algebra|Wikibooks: Algebra}}
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| [[File:Quadratic formula.svg|thumb|The [[quadratic formula]] expresses the solution of the degree two equation <math>ax^2 + bx +c=0</math> in terms of its coefficients <math>a, b, c</math>, where <math>a</math> is not zero.]]
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| '''Algebra''' (from [[Arabic language|Arabic]] ''al-jebr'' meaning "reunion of broken parts"<ref>{{cite web|title=algebra|work=[[Online Etymology Dictionary]]|url=http://www.etymonline.com/index.php?term=algebra&allowed_in_frame=0}}</ref>) is one of the broad parts of [[mathematics]], together with [[number theory]], [[geometry]] and [[mathematical analysis|analysis]]. In its most general form algebra is the study of symbols and the rules for manipulating symbols<ref>I. N. Herstein, ''Topics in Algebra'', "An algebraic system can be described as a set of objects together with some operations for combining them." p. 1, Ginn and Company, 1964</ref> and is a unifying thread of almost all of mathematics.<ref>I. N. Herstein, ''Topics in Algebra'', "...it also serves as the unifying thread which interlaces almost all of mathematics." p. 1, Ginn and Company, 1964</ref> As such, it includes everything from elementary equation solving to the study of abstractions such as [[group (mathematics)|groups]], [[ring (mathematics)|rings]], and [[field (mathematics)|fields]]. The more basic parts of algebra are called [[elementary algebra]], the more abstract parts are called [[abstract algebra]] or modern algebra. Elementary algebra is essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Much early work in algebra, as the Arabic origin of its name suggests, was done in the [[Near East]], by such mathematicians as [[Omar Khayyam]] (1048-1131).<ref>[[Omar Khayyám]]</ref><ref>{{cite web|url=http://www.britannica.com/EBchecked/topic/428267/Omar-Khayyam|title=Omar Khayyam|work=Encyclopedia Britannica|accessdate=5 October 2014}}</ref> | | If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]] |
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| Elementary algebra differs from [[arithmetic]] in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values.<ref name=citeboyer /> For example, in <math>x + 2 = 5</math> the letter <math>x</math> is unknown, but the law of inverses can be used to discover its value: <math>x=3</math>. In [[Mass–energy equivalence|<math>E=mc^2</math>]], the letters <math>E</math> and <math>m</math> are variables, and the letter <math>c</math> is a [[Constant (mathematics)|constant]]. Algebra gives methods for solving equations and expressing formulas that are much easier (for those who know how to use them) than the older method of writing everything out in words.
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| The word ''algebra'' is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an "algebra", and the word is used, for example, in the phrases [[linear algebra]] and [[algebraic topology]] (see [[#How to distinguish between different meanings of "algebra"|below]]).
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| A mathematician who does research in algebra is called an '''algebraist'''.
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| == How to distinguish between different meanings of "algebra" == | | '''source''' |
| For historical reasons, the word "algebra" has several related meanings in mathematics, as a single word or with qualifiers. Such a situation, where a single word has many meanings in the same area of mathematics, may be confusing. However the distinction is easier if one recalls that the name of a scientific area is usually singular and without an article and the name of a specific structure requires an article or the plural. Thus we have:
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| * As a single word without article, "algebra" names a broad part of mathematics (see below).
| | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
| * As a single word with article or in plural, "algebra" denotes a specific mathematical structure. See [[algebra (ring theory)]] and [[algebra over a field]]. More generally, in [[universal algebra]], it can refer to any structure.
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| * With a qualifier, there is the same distinction:
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| ** Without article, it means a part of algebra, such as [[linear algebra]], [[elementary algebra]] (the symbol-manipulation rules taught in elementary courses of mathematics as part of [[primary education|primary]] and [[secondary education]]), or [[abstract algebra]] (the study of the algebraic structures for themselves).
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| ** With an article, it means an instance of some abstract structure, like a [[Lie algebra]] or an [[associative algebra]].
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| ** Frequently both meanings exist for the same qualifier, as in the sentence: ''[[Commutative algebra]] is the study of [[commutative ring]]s, which are [[algebra (ring theory)|commutative algebra]]s over the integers''.
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| == Algebra as a branch of mathematics == | | ==Demos== |
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| Algebra began with computations similar to those of [[arithmetic]], with letters standing for numbers.<ref name=citeboyer /> This allowed proofs of properties that are true no matter which numbers are involved. For example, in the [[quadratic equation]]
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| <math>a, b, c</math> can be any numbers whatsoever (except that <math>a</math> cannot be <math>0</math>), and the [[quadratic formula]] can be used to quickly and easily find the value of the unknown quantity <math>x</math>.
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| As it developed, algebra was extended to other non-numerical objects, such as [[vector (mathematics)|vectors]], [[matrix (mathematics)|matrices]], and [[polynomial]]s. Then the structural properties of these non-numerical objects were abstracted to define [[algebraic structure]]s such as [[group (mathematics)|groups]], [[ring (mathematics)|rings]], and [[field (mathematics)|fields]].
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| Before the 16th century, mathematics was divided into only two subfields, [[arithmetic]] and [[geometry]]. Even though some methods, which had been developed much earlier, may be considered nowadays as algebra, the emergence of algebra and, soon thereafter, of [[infinitesimal calculus]] as subfields of mathematics only dates from 16th or 17th century. From the second half of 19th century on, many new fields of mathematics appeared, most of which made use of both arithmetic and geometry, and almost all of which used algebra.
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| Today, algebra has grown until it includes many branches of mathematics, as can be seen in the [[Mathematics Subject Classification]]<ref>{{cite web|url=http://www.ams.org/mathscinet/msc/msc2010.html|title=2010 Mathematics Subject Classification|publisher=|accessdate=5 October 2014}}</ref>
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| where none of the first level areas (two digit entries) is called ''algebra''. Today algebra includes section 08-General algebraic systems, 12-[[Field theory (mathematics)|Field theory]] and [[polynomial]]s, 13-[[Commutative algebra]], 15-[[Linear algebra|Linear]] and [[multilinear algebra]]; [[matrix theory]], 16-[[associative algebra|Associative rings and algebras]], 17-[[Nonassociative ring]]s and [[Non-associative algebra|algebra]]s, 18-[[Category theory]]; [[homological algebra]], 19-[[K-theory]] and 20-[[Group theory]]. Algebra is also used extensively in 11-[[Number theory]] and 14-[[Algebraic geometry]].
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| == Etymology ==
| | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
| The word ''algebra'' comes from the [[Arabic language]] ({{lang|ar|الجبر}} ''{{transl|ar|al-jabr}}'' "restoration") from the title of the book ''[[The Compendious Book on Calculation by Completion and Balancing|Ilm al-jabr wa'l-muḳābala]]'' by [[Muḥammad ibn Mūsā al-Khwārizmī|al-Khwarizmi]]. The word entered the English language during [[Middle English|Late Middle English]] from either Spanish, Italian, or [[Medieval Latin]]. Algebra originally referred to a surgical procedure, and still is used in that sense in Spanish, while the mathematical meaning was a later development.<ref name=oed>{{cite web|title=algebra|url=http://www.oxforddictionaries.com/us/definition/english/algebra|work=Oxford English Dictionary|publisher=Oxford University Press}}</ref>
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| == History ==
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| {{Main|History of algebra|Timeline of algebra}}
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| The start of algebra as an area of mathematics may be dated to the end of 16th century, with [[François Viète]]'s work. Until the 19th century, algebra consisted essentially of the [[theory of equations]]. In the following, "Prehistory of algebra" is about the results of the theory of equations that precede the emergence of algebra as an area of mathematics.
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| === Early history of algebra ===
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| [[File: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]]'']]
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| The roots of algebra can be traced to the ancient [[Babylonian mathematics|Babylonians]],<ref>{{cite book |last=Struik |first=Dirk J. |year=1987 |title=A Concise History of Mathematics |location=New York |publisher=Dover Publications |isbn=0-486-60255-9 }}</ref> who developed an advanced arithmetical system with which they were able to do calculations in an [[algorithm]]ic fashion. The Babylonians developed formulas to calculate solutions for problems typically solved today by using [[linear equation]]s, [[quadratic equation]]s, and [[indeterminate equation|indeterminate linear equations]]. By contrast, most [[Egyptian mathematics|Egyptians]] of this era, as well as [[Greek mathematics|Greek]] and [[Chinese mathematics]] in the 1st millennium BC, usually solved such equations by geometric methods, such as those described in the ''[[Rhind Mathematical Papyrus]]'', [[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, although this would not be realized until [[Mathematics in medieval Islam|mathematics developed in medieval Islam]].<ref>{{harvnb|Boyer|1991}}</ref>
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| By the time of [[Plato]], [[Greek mathematics]] had undergone a drastic change. The [[Ancient Greece|Greeks]] created a geometric algebra where terms were represented by sides of geometric objects, usually lines, that had letters associated with them.<ref name=citeboyer>{{Harv|Boyer|1991|loc="Europe in the Middle Ages" p. 258}} "In the arithmetical theorems in Euclid's ''Elements'' VII-IX, numbers had been represented by line segments to which letters had been attached, and the geometric proofs in al-Khwarizmi's ''Algebra'' made use of lettered diagrams; but all coefficients in the equations used in the ''Algebra'' are specific numbers, whether represented by numerals or written out in words. The idea of generality is implied in al-Khwarizmi's exposition, but he had no scheme for expressing algebraically the general propositions that are so readily available in geometry."</ref> [[Diophantus]] (3rd century AD) was an [[Alexandria]]n [[Greek mathematics|Greek mathematician]] and the author of a series of books called ''[[Arithmetica]]''. These texts deal with solving [[algebraic equation]]s,<ref>{{cite book |authorlink=Florian Cajori |first=Florian |last=Cajori |year=2010 |url=http://books.google.com/?id=gZ2Us3F7dSwC&pg=PA34&dq#v=onepage&q=&f=false |title=A History of Elementary Mathematics – With Hints on Methods of Teaching |page=34 |isbn=1-4460-2221-8 }}</ref> and have led, in [[number theory]] to the modern notion of [[Diophantine equation]].
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| Earlier traditions discussed above had a direct influence on [[Muhammad ibn Mūsā al-Khwārizmī]] (c. 780–850). He later wrote ''[[The Compendious Book on Calculation by Completion and Balancing]]'', which established algebra as a mathematical discipline that is independent of [[geometry]] and [[arithmetic]].<ref>{{Cite journal|title=Al Khwarizmi: The Beginnings of Algebra|author=Roshdi Rashed|publisher=[[Saqi Books]]|date=November 2009|isbn=0-86356-430-5|ref=harv|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</ref>
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| The [[Hellenistic civilization|Hellenistic]] mathematicians [[Hero of Alexandria]] and [[Diophantus]]<ref>{{cite web|url=http://library.thinkquest.org/25672/diiophan.htm|title=Diophantus, Father of Algebra|publisher=|accessdate=5 October 2014}}</ref> as well as [[Indian mathematics|Indian mathematicians]] such as [[Brahmagupta]] continued the traditions of Egypt and Babylon, though Diophantus' ''[[Arithmetica]]'' and Brahmagupta's ''[[Brahmasphutasiddhanta]]'' are on a higher level.<ref>{{cite web|url=http://www.algebra.com/algebra/about/history/|title=History of Algebra|publisher=|accessdate=5 October 2014}}</ref> For example, the first complete arithmetic solution (including zero and negative solutions) to [[quadratic equation]]s was described by Brahmagupta in his book ''Brahmasphutasiddhanta''. Later, Arabic and Muslim mathematicians developed algebraic methods to a much higher degree of sophistication. Although Diophantus and the Babylonians used mostly special ''ad hoc'' methods to solve equations, Al-Khwarizmi contribution was fundamental. He solved linear and quadratic equations without algebraic symbolism, [[negative numbers]] or [[zero]], thus he has to distinguish several types of equations.<ref name="Meri2004">{{cite book|author=Josef W. Meri|title=Medieval Islamic Civilization|url=http://books.google.com/books?id=H-k9oc9xsuAC&pg=PA31|accessdate=25 November 2012|year=2004|publisher=Psychology Press|isbn=978-0-415-96690-0|page=31}}</ref>
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| In the context where algebra is identified with the [[theory of equations]], 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>{{cite book |first=Carl B. |last=Boyer |title=A History of Mathematics |edition=Second |location= |publisher=Wiley |year=1991 |pages=178, 181 |isbn=0-471-54397-7 }}</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>{{cite book |first=Carl B. |last=Boyer |title=A History of Mathematics |edition=Second |location= |publisher=Wiley |year=1991 |page=228 |isbn=0-471-54397-7 }}</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>{{Cite book | last1=Rashed | first1=R. | last2=Armstrong | first2=Angela | year=1994 | title=The Development of Arabic Mathematics | publisher=[[Springer Science+Business Media|Springer]] | isbn=0-7923-2565-6 | oclc=29181926 | pages=11–2 | ref=harv | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</ref>
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| 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>{{Cite journal|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]|last2=Barton|first2=Bill|ref=harv|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</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.
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| === History of algebra ===
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| [[File:Gerolamo Cardano (colour).jpg|thumb|200px|Italian mathematician [[Girolamo Cardano]] published the solutions to the [[cubic equation|cubic]] and [[quartic equation]]s in his 1545 book ''[[Ars Magna (Gerolamo Cardano)|Ars magna]]''.]]
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| [[François Viète]]'s work on [[new algebra]] at the close of the 16th century was an important step towards modern 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-Louis Lagrange]] in his 1770 paper ''Réflexions sur la résolution algébrique des équations'' devoted to solutions of algebraic equations, in which he introduced [[Resolvent (Galois theory)|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.
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| [[Abstract algebra]] was developed in the 19th century, deriving from the interest in solving equations, 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> [[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 noncommutative algebra).<ref>"[http://www.cambridge.org/catalogue/catalogue.asp?ISBN=9781108005043 The Collected Mathematical Papers]".Cambridge University Press.</ref>
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| == Areas of mathematics with the word algebra in their name ==
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| Some areas of mathematics that fall under the classification abstract algebra have the word algebra in their name; [[linear algebra]] is one example. Others do not: [[group theory]], [[ring theory]], and [[field theory]] are examples. In this section, we list some areas of mathematics with the word "algebra" in the name.
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| * [[Elementary algebra]], the part of algebra that is usually taught in elementary courses of mathematics.
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| * [[Abstract 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 [[linear equation]]s, [[vector space]]s and [[matrix (mathematics)|matrices]] are studied.
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| * [[Commutative algebra]], the study of [[commutative ring]]s.
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| * [[Computer algebra]], the implementation of algebraic methods as [[algorithm]]s and [[computer program]]s.
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| * [[Homological algebra]], the study of algebraic structures that are fundamental to study [[topological space]]s.
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| * [[Universal algebra]], in which properties common to all algebraic structures are studied.
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| * [[Algebraic number theory]], in which the properties of numbers are studied from an algebraic point of view.
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| * [[Algebraic geometry]], a branch of geometry, in its primitive form specifying curves and surfaces as solutions of polynomial equations.
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| * [[Algebraic combinatorics]], in which algebraic methods are used to study combinatorial questions.
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| Many mathematical structures are called '''algebras''':
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| * [[Algebra over a field]] or more generally [[Algebra (ring theory)|algebra over a ring]].<br>Many classes of algebras over a field or over a ring have a specific name:
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| ** [[Associative algebra]]
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| ** [[Non-associative algebra]]
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| ** [[Lie algebra]]
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| ** [[Hopf algebra]]
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| ** [[C*-algebra]]
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| ** [[Symmetric algebra]]
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| ** [[Exterior algebra]]
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| ** [[Tensor algebra]]
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| * In [[measure theory]],
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| ** [[Sigma-algebra]]
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| ** [[Algebra over a set]]
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| * In [[category theory]]
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| ** [[F-algebra]] and [[F-coalgebra]]
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| ** [[T-algebra]]
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| * In [[logic]],
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| ** [[Relational algebra]]: a set of [[finitary relation]]s that is [[closure (mathematics)|closed]] under certain operators.
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| ** [[Boolean algebra]], a structure abstracting the computation with the [[truth value]]s ''false'' and ''true''. See also [[Boolean algebra (structure)]].
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| ** [[Heyting algebra]]
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| == Elementary algebra ==
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| {{main|Elementary algebra}}
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| [[File:algebraic equation notation.svg|thumb|right|Algebraic expression notation:<br/> 1 – power (exponent)<br/> 2 – coefficient<br/> 3 – term<br/> 4 – operator<br/> 5 – constant term<br/> ''x'' ''y'' ''c'' – variables/constants]]
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| '''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 represented by symbols called [[variable (mathematics)|variables]] (such as ''a'', ''n'', ''x'', ''y'' or ''z''). This is useful because:
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| * 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]].
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| * 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''". This step leads to the conclusion that it is not the nature of the specific numbers that allows us to solve it, but that of the operations involved.)
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| * It allows the formulation of [[function (mathematics)|functional]] relationships. (For instance, "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".)
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| === Polynomials ===
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| [[File:Polynomialdeg3.svg|The [[graph of a function|graph]] of a polynomial function of degree 3.|thumb|upright]]
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| {{main|Polynomial}}
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| A '''polynomial''' is an [[expression (mathematics)|expression]] that is the sum of a finite number of non-zero [[term (mathematics)|terms]], each term consisting of the product of a constant and a finite number of [[Variable (mathematics)|variables]] raised to whole number powers. For example, ''x''<sup>2</sup> + 2''x'' − 3 is a polynomial in the single variable ''x''. A '''polynomial expression''' is an expression that may be rewritten as a polynomial, by using commutativity, associativity and distributivity of addition and multiplication. For example, (''x'' − 1)(''x'' + 3) is a polynomial expression, that, properly speaking, is not a polynomial. A '''polynomial function''' is a function that is defined by a polynomial, or, equivalently, by a polynomial expression. The two preceding examples define the same polynomial function.
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| Two important and related problems in algebra are the [[factorization of polynomials]], that is, expressing a given polynomial as a product of other polynomials that can not be factored any further, and the computation of [[polynomial greatest common divisor]]s. 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.
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| === Teaching algebra ===
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| {{see also|Mathematics education}}
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| It has been suggested that elementary algebra should be taught as young as eleven years old,<ref>{{Cite web |title=Hull's Algebra |work=New York Times |date=July 16, 1904 |url=http://query.nytimes.com/mem/archive-free/pdf?res=F10714FB395E12738DDDAF0994DF405B848CF1D3 |format=[[pdf]] |accessdate=September 21, 2012}}</ref> though in recent years it is more common for public lessons to begin at the eighth grade level (≈ 13 y.o. ±) in the United States.<ref>{{Cite web |last=Quaid |first=Libby |title=Kids misplaced in algebra |work=[[Associated Press]] |date=September 22, 2008 |url=http://www.usatoday.com/news/nation/2008-09-22-357650952_x.htm |format=Report |accessdate=September 23, 2012}}</ref>
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| Since 1997, [[Virginia Tech]] and some other universities have begun using a personalized model of teaching algebra that combines instant feedback from specialized computer software with one-on-one and small group tutoring, which has reduced costs and increased student achievement.<ref>{{cite news|url=http://www.nytimes.com/2012/09/07/us/ut-arlington-adopts-new-way-to-tackle-algebra.html|title=THE TEXAS TRIBUNE; U.T.-Arlington Adopts New Way to Tackle Algebra|last=Hamilton|first=Reeve|date=7 September 2012|work=The New York Times|accessdate=10 September 2012}}</ref>
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| == Abstract algebra ==
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| {{Main|Abstract algebra|Algebraic structure}}
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| '''Abstract algebra''' extends the familiar concepts found in elementary algebra and [[arithmetic]] of [[number]]s to more general concepts. Here are listed fundamental concepts in abstract algebra.
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| '''[[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 groups of integers [[modular arithmetic|modulo]] ''n''. [[Set theory]] is a branch of [[logic]] and not technically a branch of algebra.
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| '''[[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 are addition and multiplication of matrices, vectors, and polynomials.
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| '''[[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 sets and operator combinations have an identity element; for example, the set of positive natural numbers (1, 2, 3, ...) has no identity element for addition.
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| '''[[Inverse elements]]''': The negative numbers give rise to the concept of ''inverse elements''. For addition, the inverse of ''a'' is written −''a'', and for multiplication the inverse is written ''a''<sup>−1</sup>. A general two-sided inverse element ''a''<sup>−1</sup> satisfies the property that ''a'' ∗ ''a''<sup>−1</sup> = 1 and ''a''<sup>−1</sup> ∗ ''a'' = 1 .
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| '''[[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]].
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| '''[[Commutative operation|Commutativity]]''': Addition and multiplication of real numbers are both commutative. That is, the order of the numbers does not affect the result. For example: 2 + 3 = 3 + 2. In general, this becomes ''a'' ∗ ''b'' = ''b'' ∗ ''a''. This property does not hold for all binary operations. For example, [[matrix multiplication]] and [[Quaternion|quaternion multiplication]] are both non-commutative.
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| === Groups ===
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| {{main|Group (mathematics)}} {{see also|Group theory|Examples of groups}}
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| 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:
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| * An identity element ''e'' exists, such that for every member ''a'' of ''S'', ''e'' ∗ ''a'' and ''a'' ∗ ''e'' are both identical to ''a''.
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| * 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.
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| * The operation is associative: if ''a'', ''b'' and ''c'' are members of ''S'', then (''a'' ∗ ''b'') ∗ ''c'' is identical to ''a'' ∗ (''b'' ∗ ''c'').
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| 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]].
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| 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'')
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| 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.
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| 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.
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| 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 separates the [[finite set|finite]] [[simple group]]s into roughly 30 basic types.
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| [[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 either a unique left-multiplication or right-multiplication; however the binary operation might not be associative.
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| All groups are monoids, and all monoids are semigroups.
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| {| class="wikitable"
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| |-
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| | colspan=11|Examples
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| |-
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| !Set:
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| | colspan=2|[[Natural numbers]] '''N'''
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| | colspan=2|[[Integers]] '''Z'''
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| | colspan=4|[[Rational numbers]] '''Q''' (also [[Real numbers|real]] '''R''' and [[Complex numbers|complex]] '''C''' numbers)
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| | colspan=2|Integers [[modular arithmetic|modulo]] 3: '''Z'''<sub>3</sub> = {0, 1, 2}
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| |-
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| !Operation
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| | +
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| | × (w/o zero)
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| | +
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| | × (w/o zero)
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| | +
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| | −
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| | × (w/o zero)
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| | ÷ (w/o zero)
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| | +
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| | × (w/o zero)
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| |-
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| !Closed
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| | Yes
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| | Yes
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| | Yes
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| | Yes
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| | Yes
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| | Yes
| |
| | Yes
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| | Yes
| |
| | Yes
| |
| | Yes
| |
| |-
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| | Identity
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| | 0
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| | 1
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| | 0
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| | 1
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| | 0
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| | N/A
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| | 1
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| | N/A
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| | 0
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| | 1
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| |-
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| | Inverse
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| | N/A
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| | N/A
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| | −''a''
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| | N/A
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| | −''a''
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| | N/A
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| | 1/''a''
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| | N/A
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| | 0, 2, 1, respectively
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| | N/A, 1, 2, respectively
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| |-
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| | Associative
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| | Yes
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| | Yes
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| | Yes
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| | Yes
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| | Yes
| |
| | No
| |
| | Yes
| |
| | No
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| | Yes
| |
| | Yes
| |
| |-
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| | Commutative
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| | Yes
| |
| | Yes
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| | Yes
| |
| | Yes
| |
| | Yes
| |
| | No
| |
| | Yes
| |
| | No
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| | Yes
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| | Yes
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| |-
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| | Structure
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| | [[monoid]]
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| | [[monoid]]
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| | [[abelian group]]
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| | [[monoid]]
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| | [[abelian group]]
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| | [[quasigroup]]
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| | [[abelian group]]
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| | [[quasigroup]]
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| | [[abelian group]]
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| | [[abelian group]] ('''Z'''<sub>2</sub>)
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| |}
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| === Rings and fields ===
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| {{main|Ring (mathematics)|Field (mathematics)}} {{see also|Ring theory|Glossary of ring theory|Field theory (mathematics)|Glossary of field theory}}
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| 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]].
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| 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 required. The additive (+) identity element is written as 0 and the additive inverse of ''a'' is written as −''a''.
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| '''[[Distributivity]]''' generalises the ''distributive law'' for numbers. 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 +.
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| The integers are an example of a ring. The integers have additional properties which make it an '''[[integral domain]]'''.
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| 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>.
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| The rational numbers, the real numbers and the complex numbers are all examples of fields.
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| == See also ==
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| <!-- Please place all see also references to the following pages.-->
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| {{portal|Algebra}}
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| * [[Outline of algebra]]
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| * [[Outline of linear algebra]]
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| * [[Algebra tile]]
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| == Notes ==
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| {{reflist|30em}}
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| == References ==
| |
| *{{Citation
| |
| | first=Carl B.
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| | last=Boyer
| |
| | authorlink=Carl Benjamin Boyer
| |
| | title=A History of Mathematics
| |
| | edition=Second Edition
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| | publisher=John Wiley & Sons, Inc.
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| | year=1991
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| | isbn=0-471-54397-7
| |
| }}
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| * Donald R. Hill, ''Islamic Science and Engineering'' (Edinburgh University Press, 1994).
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| * Ziauddin Sardar, Jerry Ravetz, and Borin Van Loon, ''Introducing Mathematics'' (Totem Books, 1999).
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| * George Gheverghese Joseph, ''The Crest of the Peacock: Non-European Roots of Mathematics'' ([[Penguin Books]], 2000).
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| * John J O'Connor and Edmund F Robertson, [http://www-history.mcs.st-andrews.ac.uk/Indexes/Algebra.html ''History Topics: Algebra Index'']. In ''[[MacTutor History of Mathematics archive]]'' ([[University of St Andrews]], 2005).
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| * I.N. Herstein: ''Topics in Algebra''. ISBN 0-471-02371-X
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| * R.B.J.T. Allenby: ''Rings, Fields and Groups''. ISBN 0-340-54440-6
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| * [[L. Euler]]: ''[http://web.mat.bham.ac.uk/C.J.Sangwin/euler/ Elements of Algebra]'', ISBN 978-1-899618-73-6
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| * {{cite book|last=Asimov|first=Isaac|title=Realm of Algebra|year=1961|publisher=Houghton Mifflin|authorlink=Isaac Asimov}}
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| == External links ==
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| {{Wiktionary|algebra}}
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| {{Wikibooks|Algebra}}
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| * [http://www.khanacademy.org/math/algebra Khan Academy: Conceptual videos and worked examples]
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| * [https://www.khanacademy.org/math/algebra/introduction-to-algebra/overview_hist_alg/v/origins-of-algebra Khan Academy: Origins of Algebra, free online micro lectures]
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| * [http://algebrarules.com Algebrarules.com: An open source resource for learning the fundamentals of Algebra]
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| * [http://www.gresham.ac.uk/event.asp?PageId=45&EventId=620 4000 Years of Algebra], lecture by Robin Wilson, at [[Gresham College]], October 17, 2007 (available for MP3 and MP4 download, as well as a text file).
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| * {{sep entry|algebra|Algebra|Vaughan Pratt}}
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| {{Algebra |expanded}}
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| {{Areas of mathematics |collapsed}}
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| [[Category:Algebra| ]]
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