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Algebra () is one of the broad areas of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betw ...
s; it is a unifying thread of almost all of mathematics.
Elementary algebra Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values). This use of variables entai ...
deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics.
Abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
is the name given, mostly in
education Education is a purposeful activity directed at achieving certain aims, such as transmitting knowledge or fostering skills and character traits. These aims may include the development of understanding, rationality, kindness, and honesty. V ...
, to the study of
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
s such as groups, rings, and fields (the term is no more in common use outside educational context).
Linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...
, which deals with
linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...
s and
linear mapping In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s, is used for modern presentations of
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
, and has many practical applications (in
weather forecasting Weather forecasting is the application of science and technology to predict the conditions of the atmosphere for a given location and time. People have attempted to predict the weather informally for millennia and formally since the 19th cent ...
, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominen ...
, and some not, such as
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
. The word ''algebra'' is not only used for naming an area of mathematics and some subareas; it is also used for naming some sorts of algebraic structures, such as an algebra over a field, commonly called an ''algebra''. Sometimes, the same phrase is used for a subarea and its main algebraic structures; for example,
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...
and a
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...
. A mathematician specialized in algebra is called an algebraist.

# Etymology

The word ''algebra'' comes from the ar, الجبر, lit=reunion of broken parts,
bonesetting Traditional bone-setting is a type of a folk medicine in which practitioners engaged in joint manipulation. Before the advent of chiropractors, osteopaths and physical therapists, bone-setters were the main providers of this type of treatment. ...
, translit=al-jabr from the title of the early 9th century book '' cIlm al-jabr wa l-muqābala'' "The Science of Restoring and Balancing" by the Persian mathematician and astronomer
al-Khwarizmi Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astrono ...
. In his work, the term ''al-jabr'' referred to the operation of moving a term from one side of an equation to the other, المقابلة ''al-muqābala'' "balancing" referred to adding equal terms to both sides. Shortened to just ''algeber'' or ''algebra'' in Latin, the word eventually entered the English language during the 15th century, from either Spanish, Italian, or
Medieval Latin Medieval Latin was the form of Literary Latin used in Roman Catholic Western Europe during the Middle Ages. In this region it served as the primary written language, though local languages were also written to varying degrees. Latin functioned ...
. It originally referred to the surgical procedure of setting broken or dislocated bones. The mathematical meaning was first recorded (in English) in the 16th century.

# Different meanings of "algebra"

The word "algebra" has several related meanings in mathematics, as a single word or with qualifiers. * As a single word without an article, "algebra" names a broad part of mathematics. * As a single word with an article or in the plural, "an algebra" or "algebras" denotes a specific mathematical structure, whose precise definition depends on the context. Usually, the structure has an addition, multiplication, and scalar multiplication (see Algebra over a field). When some authors use the term "algebra", they make a subset of the following additional assumptions:
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
,
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, unital, and/or finite-dimensional. In
universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study ...
, the word "algebra" refers to a generalization of the above concept, which allows for n-ary operations. * With a qualifier, there is the same distinction: ** Without an article, it means a part of algebra, such as
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...
,
elementary algebra Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values). This use of variables entai ...
(the symbol-manipulation rules taught in elementary courses of mathematics as part of
primary Primary or primaries may refer to: Arts, entertainment, and media Music Groups and labels * Primary (band), from Australia * Primary (musician), hip hop musician and record producer from South Korea * Primary Music, Israeli record label Work ...
and
secondary education Secondary education or post-primary education covers two phases on the International Standard Classification of Education scale. Level 2 or lower secondary education (less commonly junior secondary education) is considered the second and final pha ...
), or
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
(the study of the algebraic structures for themselves). ** With an article, it means an instance of some algebraic structure, like a Lie algebra, an associative algebra, or a vertex operator algebra. ** Sometimes both meanings exist for the same qualifier, as in the sentence: ''
Commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominen ...
is the study of
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
s, which are commutative algebras over the integers''.

# Algebra as a branch of mathematics

Algebra began with computations similar to those of
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19t ...
, with letters standing for numbers. This allowed proofs of properties that are true no matter which numbers are involved. For example, in the
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not quadr ...
:$ax^2+bx+c=0,$ $a, b, c$ can be any numbers whatsoever (except that $a$ cannot be $0$), and the
quadratic formula In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, gr ...
can be used to quickly and easily find the values of the unknown quantity $x$ which satisfy the equation. That is to say, to find all the solutions of the equation. Historically, and in current teaching, the study of algebra starts with the solving of equations, such as the quadratic equation above. Then more general questions, such as "does an equation have a solution?", "how many solutions does an equation have?", "what can be said about the nature of the solutions?" are considered. These questions led extending algebra to non-numerical objects, such as
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
s, vectors, matrices, and
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exam ...
s. The structural properties of these non-numerical objects were then formalized into
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
s such as groups, rings, and fields. Before the 16th century, mathematics was divided into only two subfields,
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19t ...
and
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
. 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 Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arith ...
as subfields of mathematics only dates from the 16th or 17th century. From the second half of the 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. Today, algebra has grown considerably and includes many branches of mathematics, as can be seen in the
Mathematics Subject Classification The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme collaboratively produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH. ...
where none of the first level areas (two digit entries) are called ''algebra''. Today algebra includes section 08-General algebraic systems, 12- Field theory and
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exam ...
s, 13-
Commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominen ...
, 15-
Linear Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
and multilinear algebra;
matrix theory In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \be ...
, 16- Associative rings and algebras, 17- Nonassociative rings and
algebras In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
, 18-
Category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
;
homological algebra Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topolo ...
, 19- K-theory and 20-
Group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen a ...
. Algebra is also used extensively in 11-
Number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
and 14-
Algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
.

# History

The use of the word "algebra" for denoting a part of mathematics dates probably from the 16th century. The word is derived from the Arabic word ''al-jabr'' that appears in the title of the treatise '' Al-Kitab al-muhtasar fi hisab al-gabr wa-l-muqabala'' (''The Compendious Book on Calculation by Completion and Balancing''), written circa 820 by Al-Kwarizmi. ''Al-jabr'' referred to a method for transforming
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
s by subtracting
like terms In mathematics, like terms are summands in a sum that differ only by a numerical factor. Like terms can be regrouped by adding their coefficients. Typically, in a polynomial expression, like terms are those that contain the same variables to t ...
from both sides, or passing one term from one side to the other, after changing its sign. Therefore, ''algebra'' referred originally to the manipulation of equations, and, by extension, to the theory of equations. This is still what historians of mathematics generally mean by ''algebra''. In mathematics, the meaning of ''algebra'' has evolved after the introduction by François Viète of symbols ( variables) for denoting unknown or incompletely specified numbers, and the resulting use of the
mathematical notation Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathem ...
for equations and
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betw ...
s. So, algebra became essentially the study of the action of operations on expressions involving variables. This includes but is not limited to the theory of equations. At the beginning of the 20th century, algebra evolved further by considering operations that act not only on numbers but also on elements of so-called
mathematical structure In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additi ...
s such as groups, fields and
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
s. This new algebra was called '' modern algebra'' by van der Waerden in his eponymous treatise, whose name has been changed to ''Algebra'' in later editions.

## Early history

The roots of algebra can be traced to the ancient
Babylonians Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c. ...
, who developed an advanced arithmetical system with which they were able to do calculations in an
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
ic fashion. The Babylonians developed formulas to calculate solutions for problems typically solved today by using
linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...
s,
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not quadr ...
s, and indeterminate linear equations. By contrast, most Egyptians of this era, as well as
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. ** Proto-Greek language, the assumed last common ancestor ...
and
Chinese mathematics Mathematics in China emerged independently by the 11th century BCE. The Chinese independently developed a real number system that includes significantly large and negative numbers, more than one numeral system ( base 2 and base 10), algebra, geomet ...
in the 1st millennium BC, usually solved such equations by geometric methods, such as those described in the ''
Rhind Mathematical Papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics. It is named after Alexander Henry Rhind, a Scottish antiquarian, who purchase ...
'', 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 developed in medieval Islam. By the time of
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institutio ...
, Greek mathematics had undergone a drastic change. The Greeks created a
geometric algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the g ...
where terms were represented by sides of geometric objects, usually lines, that had letters associated with them.See , ''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." Diophantus (3rd century AD) was an Alexandrian Greek mathematician and the author of a series of books called ''
Arithmetica ''Arithmetica'' ( grc-gre, Ἀριθμητικά) is an Ancient Greek text on mathematics written by the mathematician Diophantus () in the 3rd century AD. It is a collection of 130 algebraic problems giving numerical solutions of determinate ...
''. These texts deal with solving
algebraic equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
s, and have led, in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
, to the modern notion of
Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to ...
. Earlier traditions discussed above had a direct influence on the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī (–850). He later wrote ''
The Compendious Book on Calculation by Completion and Balancing ''The Compendious Book on Calculation by Completion and Balancing'' ( ar, كتاب المختصر في حساب الجبر والمقابلة, ; la, Liber Algebræ et Almucabola), also known as ''Al-Jabr'' (), is an Arabic mathematical treati ...
'', which established algebra as a mathematical discipline that is independent of
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
and
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19t ...
. The
Hellenistic In Classical antiquity, the Hellenistic period covers the time in Mediterranean history after Classical Greece, between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire, as signified by the Battle of Actium in ...
mathematicians
Hero of Alexandria Hero of Alexandria (; grc-gre, Ἥρων ὁ Ἀλεξανδρεύς, ''Heron ho Alexandreus'', also known as Heron of Alexandria ; 60 AD) was a Greek mathematician and engineer who was active in his native city of Alexandria, Roman Egypt. ...
and Diophantus as well as Indian mathematicians such as
Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical tre ...
, continued the traditions of Egypt and Babylon, though Diophantus' ''Arithmetica'' and Brahmagupta's '' Brāhmasphuṭasiddhānta'' are on a higher level. For example, the first complete arithmetic solution written in words instead of symbols, including zero and negative solutions, to quadratic equations was described by Brahmagupta in his book ''Brahmasphutasiddhanta,'' published in 628 AD.Bradley, Michael. ''The Birth of Mathematics: Ancient Times to 1300'', p. 86 (Infobase Publishing 2006). Later, Persian and
Arab The Arabs (singular: Arab; singular ar, عَرَبِيٌّ, DIN 31635: , , plural ar, عَرَب, DIN 31635: , Arabic pronunciation: ), also known as the Arab people, are an ethnic group mainly inhabiting the Arab world in Western Asia, ...
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's contribution was fundamental. He solved linear and quadratic equations without algebraic symbolism,
negative numbers In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed ma ...
or
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usua ...
, thus he had to distinguish several types of equations. In the context where algebra is identified with the theory of equations, the Greek mathematician Diophantus has traditionally been known as the "father of algebra" and in the context where it is identified with rules for manipulating and solving equations, Persian mathematician al-Khwarizmi is regarded as "the father of algebra".See , page 263–277: "In a sense, al-Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because al-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". It is open to debate whether Diophantus or al-Khwarizmi is more entitled to be known, in the general sense, as "the father of algebra". 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. Those who support Al-Khwarizmi point to the fact that he introduced the methods of " reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of
like terms In mathematics, like terms are summands in a sum that differ only by a numerical factor. Like terms can be regrouped by adding their coefficients. Typically, in a polynomial expression, like terms are those that contain the same variables to t ...
on opposite sides of the equation) which the term ''al-jabr'' originally referred to,See , ''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". and that he gave an exhaustive explanation of solving quadratic equations, supported by geometric proofs while treating algebra as an independent discipline in its own right. His algebra was also no longer concerned "with a series of problems to be resolved, but an
exposition Exposition (also the French for exhibition) may refer to: *Universal exposition or World's Fair *Expository writing ** Exposition (narrative) *Exposition (music) *Trade fair * ''Exposition'' (album), the debut album by the band Wax on Radio *Expos ...
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". Another Persian mathematician
Omar Khayyam Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm Nīsābūrī (18 May 1048 – 4 December 1131), commonly known as Omar Khayyam ( fa, عمر خیّام), was a polymath, known for his contributions to mathematics, astronomy, philosophy, an ...
is credited with identifying the foundations of
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
and found the general geometric solution of the
cubic equation In algebra, a cubic equation in one variable is an equation of the form :ax^3+bx^2+cx+d=0 in which is nonzero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
. His book ''Treatise on Demonstrations of Problems of Algebra'' (1070), which laid down the principles of algebra, is part of the body of Persian mathematics that was eventually transmitted to Europe. Yet another Persian mathematician, Sharaf al-Dīn al-Tūsī, found algebraic and numerical solutions to various cases of cubic equations. He also developed the concept of a function. The Indian mathematicians
Mahavira Mahavira (Sanskrit: महावीर) also known as Vardhaman, was the 24th '' tirthankara'' (supreme preacher) of Jainism. He was the spiritual successor of the 23rd ''tirthankara'' Parshvanatha. Mahavira was born in the early part of the 6 ...
and Bhaskara II, the Persian mathematician
Al-Karaji ( fa, ابو بکر محمد بن الحسن الکرجی; c. 953 – c. 1029) was a 10th-century Persian mathematician and engineer who flourished at Baghdad. He was born in Karaj, a city near Tehran. His three principal surviving works ...
,See , ''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 ax2n + bxn = c (only equations with positive roots were considered)," and the Chinese mathematician
Zhu Shijie Zhu Shijie (, 1249–1314), courtesy name Hanqing (), pseudonym Songting (), was a Chinese mathematician and writer. He was a Chinese mathematician during the Yuan Dynasty. Zhu was born close to today's Beijing. Two of his mathematical works h ...
, solved various cases of cubic, quartic, quintic and higher-order
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exam ...
equations using numerical methods. In the 13th century, the solution of a cubic equation by
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Wester ...
is representative of the beginning of a revival in European algebra. Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī (1412–1486) took "the first steps toward the introduction of algebraic symbolism". He also computed Σ''n''2, Σ''n''3 and used the method of successive approximation to determine square roots.

## Modern history

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 René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...
published ''
La Géométrie ''La Géométrie'' was published in 1637 as an appendix to ''Discours de la méthode'' (''Discourse on the Method''), written by René Descartes. In the ''Discourse'', he presents his method for obtaining clarity on any subject. ''La Géométrie ...
'', inventing
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engine ...
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 In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...
was developed by Japanese mathematician Seki Kōwa in the 17th century, followed independently by
Gottfried Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
ten years later, for the purpose of solving systems of simultaneous linear equations using matrices. Gabriel Cramer also did some work on matrices and determinants in the 18th century. Permutations were studied by
Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaLagrange resolvents. Paolo Ruffini was the first person to develop the theory of
permutation group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to i ...
s, and like his predecessors, also in the context of solving algebraic equations.
Abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
was developed in the 19th century, deriving from the interest in solving equations, initially focusing on what is now called
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
, and on constructibility issues. 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 Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...
developed an algebra of vectors in three-dimensional space, and
Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, Cayley enjoyed solving complex maths problems ...
developed an algebra of matrices (this is a noncommutative algebra).

# Areas of mathematics with the word algebra in their name

Some subareas of algebra have the word algebra in their name;
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...
is one example. Others do not:
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen a ...
,
ring theory In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...
, and field theory are examples. In this section, we list some areas of mathematics with the word "algebra" in the name. *
Elementary algebra Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values). This use of variables entai ...
, the part of algebra that is usually taught in elementary courses of mathematics. *
Abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, in which
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
s such as groups, rings and fields are axiomatically defined and investigated. *
Linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...
, in which the specific properties of
linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...
s,
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
s and matrices are studied. *
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...
, a branch of algebra abstracting the computation with the truth values ''false'' and ''true''. *
Commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominen ...
, the study of
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
s. *
Computer algebra In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions ...
, the implementation of algebraic methods as
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
s and
computer program A computer program is a sequence or set of instructions in a programming language for a computer to execute. Computer programs are one component of software, which also includes documentation and other intangible components. A computer program ...
s. *
Homological algebra Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topolo ...
, the study of algebraic structures that are fundamental to study
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
s. *
Universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study ...
, in which properties common to all algebraic structures are studied. *
Algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic o ...
, in which the properties of numbers are studied from an algebraic point of view. *
Algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, a branch of geometry, in its primitive form specifying curves and surfaces as solutions of
polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
s. * Algebraic combinatorics, in which algebraic methods are used to study combinatorial questions. * Relational algebra: a set of finitary relations that is closed under certain operators. Many mathematical structures are called algebras: * Algebra over a field or more generally
algebra over a ring In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
.
Many classes of algebras over a field or over a ring have a specific name: ** Associative algebra ** Non-associative algebra ** Lie algebra **
Composition algebra In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies :N(xy) = N(x)N(y) for all and in . A composition algebra includes an involution ...
**
Hopf algebra Hopf is a German surname. Notable people with the surname include: * Eberhard Hopf (1902–1983), Austrian mathematician * Hans Hopf (1916–1993), German tenor * Heinz Hopf (1894–1971), German mathematician * Heinz Hopf (actor) (1934–2001), Sw ...
**
C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuou ...
** Symmetric algebra **
Exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
**
Tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
* In
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
, ** Sigma-algebra ** Algebra over a set * In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
** F-algebra and F-coalgebra ** T-algebra * In logic, ** Relation algebra, a residuated Boolean algebra expanded with an involution called converse. **
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...
, a complemented distributive lattice. **
Heyting algebra In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of '' ...

# 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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
beyond the basic principles of
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19t ...
. In arithmetic, only
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
s and their arithmetical operations (such as +, −, ×, ÷) occur. In algebra, numbers are often represented by symbols called variables (such as ''a'', ''n'', ''x'', ''y'' or ''z''). 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 system. * It allows the reference to "unknown" numbers, the formulation of
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
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 allow us to solve it, but that of the operations involved.) * It allows the formulation of 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".)

## Polynomials

A polynomial is an expression that is the sum of a finite number of non-zero terms, each term consisting of the product of a constant and a finite number of variables raised to whole number powers. For example, ''x''2 + 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. Two important and related problems in algebra are the
factorization of polynomials In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same doma ...
, that is, expressing a given polynomial as a product of other polynomials that cannot be factored any further, and the computation of polynomial greatest common divisors. The example polynomial above can be factored as (''x'' − 1)(''x'' + 3). A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable.

## Education

It has been suggested that elementary algebra should be taught to students as young as eleven years old, 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. However, in some US schools, algebra instruction starts in ninth grade.

# Abstract algebra

Abstract algebra extends the familiar concepts found in elementary algebra and
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19t ...
of
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
s to more general concepts. Here are the listed fundamental concepts in abstract algebra. Sets: Rather than just considering the different types of
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
s, abstract algebra deals with the more general concept of ''sets'': collections of objects called elements. All collections of the familiar types of numbers are sets. Other examples of sets include the set of all two-by-two matrices, the set of all second-degree
polynomials In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
(''ax''2 + ''bx'' + ''c''), the set of all two dimensional vectors of a plane, and the various finite groups such as the
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary ...
s, which are the groups of integers modulo ''n''.
Set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
is a branch of logic and not technically a branch of algebra.
Binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...
s: The notion of
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...
(+) is generalized to the notion of ''binary operation'' (denoted here by ∗). 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.
Addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...
(+),
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 take ...
(−),
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...
(×), and
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting ...
(÷) can be binary operations when defined on different sets, as are addition and multiplication of matrices, vectors, and polynomials.
Identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
s: The numbers zero and one are generalized 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'', and is necessarily unique, if it exists. 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. 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''−1. A general two-sided inverse element ''a''−1 satisfies the property that ''a'' ∗ ''a''−1 = ''e'' and ''a''−1 ∗ ''a'' = ''e'', where ''e'' is the identity element.
Associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
: 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: . 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.
Commutativity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
: 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 In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the ...
and quaternion multiplication are both non-commutative.

## Groups

Combining the above concepts gives one of the most important structures in mathematics: a group. A group is a combination of a set ''S'' and a single
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...
∗, 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''−1 such that ''a'' ∗ ''a''−1 and ''a''−1 ∗ ''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
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
– 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. 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 non-zero
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rati ...
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 , since ''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 , which is not an integer. The theory of groups is studied in
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen a ...
. A major result of this theory is the
classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else ...
, mostly published between about 1955 and 1983, which separates the
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...
s into roughly 30 basic types. Semi-groups, quasi-groups, and
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids a ...
s are
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
s similar to groups, but with less constraints on the operation. They comprise a set and a closed binary operation but do not necessarily satisfy the other conditions. A semi-group has an ''associative'' binary operation but might not have an identity element. A
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids a ...
is a semi-group which does have an identity but might not have an inverse for every element. A quasi-group 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. All groups are monoids, and all monoids are semi-groups.

## Rings and fields

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 rings and fields. A 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 an 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''.
Distributivity In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic ...
generalises the ''distributive law'' for numbers. For the integers and and × is said to be ''distributive'' over +. The integers are an example of a ring. The integers have additional properties which make it an
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...
. A 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''−1. The rational numbers, the real numbers and the complex numbers are all examples of fields.

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