Mathematics

=== Algebra ===
{{Main|Algebra}}
[[File:Quadratic formula.svg|thumb|The [[quadratic formula]], which concisely expresses the solutions of all [[quadratic equation]]s]]  [[File:Rubik's cube.svg|thumb|The [[Rubik's Cube group]] is a concrete application of [[group theory]].<ref>{{cite book |last=Joyner |first=David |year=2008 |chapter=The (legal) Rubik's Cube group |title=Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys |pages=219–232 |edition=2nd |publisher=[[Johns Hopkins University Press]] |isbn=978-0-8018-9012-3 |lccn=2008011322 |oclc=213765703}}</ref>]]

Algebra is the art of manipulating [[equation]]s and formulas. Diophantus (3rd century) and [[Muhammad ibn Musa al-Khwarizmi|al-Khwarizmi]] (9th century) were the two main precursors of algebra.<ref>{{cite journal |last1=Christianidis |first1=Jean  |last2=Oaks |first2=Jeffrey |date=May 2013 |title=Practicing algebra in late antiquity: The problem-solving of Diophantus of Alexandria |journal=Historia Mathematica |volume=40 |issue=2 |pages=127–163 |doi=10.1016/j.hm.2012.09.001 |doi-access=free |eissn=1090-249X |issn=0315-0860 |lccn=75642280 |oclc=2240703 |s2cid=121346342}}</ref>{{sfn|Kleiner|2007|loc="History of Classical Algebra" pp. 3–5}} Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term ''algebra'' is derived from the [[Arabic]] word ''al-jabr'' meaning 'the reunion of broken parts'<ref>{{Cite web |last=Lim |first=Lisa |date=December 21, 2018 |title=Where the x we use in algebra came from, and the X in Xmas |website=[[South China Morning Post]] |url=https://www.scmp.com/magazines/post-magazine/short-reads/article/2178856/where-x-we-use-algebra-came-and-x-xmas |url-access=limited |url-status=live |archive-url=https://web.archive.org/web/20181222003908/https://www.scmp.com/magazines/post-magazine/short-reads/article/2178856/where-x-we-use-algebra-came-and-x-xmas |archive-date=December 22, 2018 |access-date=February 8, 2024}}</ref> that he used for naming one of these methods in the title of [[The Compendious Book on Calculation by Completion and Balancing|his main treatise]].

Algebra became an area in its own right only with [[François Viète]] (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers.<ref>{{cite journal |last=Oaks |first=Jeffery A. |year=2018 |title=François Viète's revolution in algebra |journal=[[Archive for History of Exact Sciences]] |volume=72 |issue=3 |pages=245–302 |doi=10.1007/s00407-018-0208-0 |eissn=1432-0657 |issn=0003-9519 |lccn=63024699 |oclc=1482042 |s2cid=125704699 |url=https://researchoutreach.org/wp-content/uploads/2019/02/Jeffrey-Oaks.pdf |url-status=live |archive-url=https://web.archive.org/web/20221108162134/https://researchoutreach.org/wp-content/uploads/2019/02/Jeffrey-Oaks.pdf |archive-date=November 8, 2022 |access-date=February 8, 2024}}</ref> Variables allow mathematicians to describe the operations that have to be done on the numbers represented using [[mathematical formulas]].

Until the 19th century, algebra consisted mainly of the study of [[linear equation]]s (presently ''[[linear algebra]]''), and polynomial equations in a single [[unknown (algebra)|unknown]], which were called ''algebraic equations'' (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as [[matrix (mathematics)|matrices]], [[modular arithmetic|modular integers]], and [[geometric transformation]]s), on which generalizations of arithmetic operations are often valid.{{sfn|Kleiner|2007|loc="History of Linear Algebra" pp. 79–101}} The concept of [[algebraic structure]] addresses this, consisting of a [[set (mathematics)|set]] whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called ''modern algebra'' or [[abstract algebra]], as established by the influence and works of [[Emmy Noether]].<ref>{{cite book |last=Corry |first=Leo |author-link=Leo Corry |year=2004 |chapter=Emmy Noether: Ideals and Structures |title=Modern Algebra and the Rise of Mathematical Structures |pages=247–252 |edition=2nd revised |publisher=Birkhäuser Basel |publication-place=Germany |isbn=3-7643-7002-5 |lccn=2004556211 |oclc=51234417 |url={{GBurl|id=WdGbeyehoCoC|p=247}} |access-date=February 8, 2024}}</ref> (The latter term appears mainly in an educational context, in opposition to [[elementary algebra]], which is concerned with the older way of manipulating formulas.)

Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include:<ref name=MSC/>
* [[group theory]];
* [[field (mathematics)|field theory]];
* [[vector space]]s, whose study is essentially the same as [[linear algebra]];
* [[ring theory]];
* [[commutative algebra]], which is the study of [[commutative ring]]s, includes the study of [[polynomial]]s, and is a foundational part of [[algebraic geometry]];
* [[homological algebra]];
* [[Lie algebra]] and [[Lie group]] theory;
* [[Boolean algebra]], which is widely used for the study of the logical structure of [[computer]]s.

The study of types of algebraic structures as [[mathematical object]]s is the purpose of [[universal algebra]] and [[category theory]].<ref>{{cite book |last=Riche |first=Jacques |editor1-last=Beziau |editor1-first=J. Y. |editor2-last=Costa-Leite |editor2-first=Alexandre |year=2007 |chapter=From Universal Algebra to Universal Logic |pages=3–39 |title=Perspectives on Universal Logic |publisher=Polimetrica International Scientific Publisher |publication-place=Milano, Italy |isbn=978-88-7699-077-9 |oclc=647049731 |url={{GBurl|id=ZoRN9T1GUVwC|p=3}} |access-date=February 8, 2024}}</ref> The latter applies to every [[mathematical structure]] (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as [[topological space]]s; this particular area of application is called [[algebraic topology]].<ref>{{cite book |last=Krömer |first=Ralph |year=2007 |title=Tool and Object: A History and Philosophy of Category Theory |pages=xxi–xxv, 1–91 |series=Science Networks - Historical Studies |volume=32 |publisher=[[Springer Science & Business Media]] |publication-place=Germany |isbn=978-3-7643-7523-2 |lccn=2007920230 |oclc=85242858 |url={{GBurl|id=41bHxtHxjUAC|pg=PR20}} |access-date=February 8, 2024}}</ref>