Algebra

# Algebra

Overview
Algebra is the branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

concerning the study of the rules of operations
Operation (mathematics)
The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....

and relations
Relation (mathematics)
In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...

, and the constructions and concepts arising from them, including terms
Term (mathematics)
A term is a mathematical expression which may form a separable part of an equation, a series, or another expression.-Definition:In elementary mathematics, a term is either a single number or variable, or the product of several numbers or variables separated from another term by a + or - sign in an...

, polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

s, equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...

s and algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...

s. Together with geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

, topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

, combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

, and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, algebra is one of the main branches of pure mathematics
Pure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...

.

Elementary algebra
Elementary algebra
Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. It is typically taught in secondary school under the term algebra. The major difference between algebra and...

, often part of the curriculum in secondary education
Secondary education
Secondary education is the stage of education following primary education. Secondary education includes the final stage of compulsory education and in many countries it is entirely compulsory. The next stage of education is usually college or university...

, introduces the concept of variables
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

representing number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....

s. Statements based on these variables are manipulated using the rules of operations that apply to numbers, such as addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....

.
Discussion

Encyclopedia
Algebra is the branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

concerning the study of the rules of operations
Operation (mathematics)
The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....

and relations
Relation (mathematics)
In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...

, and the constructions and concepts arising from them, including terms
Term (mathematics)
A term is a mathematical expression which may form a separable part of an equation, a series, or another expression.-Definition:In elementary mathematics, a term is either a single number or variable, or the product of several numbers or variables separated from another term by a + or - sign in an...

, polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

s, equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...

s and algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...

s. Together with geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

, topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

, combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

, and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, algebra is one of the main branches of pure mathematics
Pure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...

.

Elementary algebra
Elementary algebra
Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. It is typically taught in secondary school under the term algebra. The major difference between algebra and...

, often part of the curriculum in secondary education
Secondary education
Secondary education is the stage of education following primary education. Secondary education includes the final stage of compulsory education and in many countries it is entirely compulsory. The next stage of education is usually college or university...

, introduces the concept of variables
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

representing number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....

s. Statements based on these variables are manipulated using the rules of operations that apply to numbers, such as addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....

. This can be done for a variety of reasons, including equation solving
Equation solving
In mathematics, to solve an equation is to find what values fulfill a condition stated in the form of an equation . These expressions contain one or more unknowns, which are free variables for which values are sought that cause the condition to be fulfilled...

. Algebra is much broader than elementary algebra and studies what happens when different rules of operations are used and when operations are devised for things other than numbers. Addition and multiplication
Multiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....

can be generalized and their precise definitions lead to structures
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...

such as groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

, rings
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

and fields
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

, studied in the area of mathematics called abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

.

## History

By the time of Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...

, Greek mathematics
Greek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...

had undergone a drastic change. The Greeks
Ancient Greece
Ancient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...

created a geometric algebra
Geometric algebra
Geometric algebra , together with the associated Geometric calculus, provides a comprehensive alternative approach to the algebraic representation of classical, computational and relativistic geometry. GA now finds application in all of physics, in graphics and in robotics...

where terms were represented by sides of geometric objects, usually lines, that had letters associated with them. Diophantus
Diophantus
Diophantus of Alexandria , sometimes called "the father of algebra", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations, many of which are now lost...

(3rd century AD), sometimes called "the father of algebra", was an Alexandria
Alexandria
Alexandria is the second-largest city of Egypt, with a population of 4.1 million, extending about along the coast of the Mediterranean Sea in the north central part of the country; it is also the largest city lying directly on the Mediterranean coast. It is Egypt's largest seaport, serving...

n Greek mathematician
Greek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...

and the author of a series of books called Arithmetica
Arithmetica
Arithmetica 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 equations and indeterminate equations.Equations in the book are called Diophantine equations...

. These texts deal with solving algebraic equations.

While the word algebra comes from the Arabic language
Arabic language
Arabic is a name applied to the descendants of the Classical Arabic language of the 6th century AD, used most prominently in the Quran, the Islamic Holy Book...

( "restoration") and much of its methods from Arabic/Islamic mathematics
Islamic mathematics
In the history of mathematics, mathematics in medieval Islam, often termed Islamic mathematics or Arabic mathematics, covers the body of mathematics preserved and developed under the Islamic civilization between circa 622 and 1600...

, its roots can be traced to earlier traditions, which had a direct influence on Muhammad ibn Mūsā al-Khwārizmī
'There is some confusion in the literature on whether al-Khwārizmī's full name is ' or '. Ibn Khaldun notes in his encyclopedic work: "The first who wrote upon this branch was Abu ʿAbdallah al-Khowarizmi, after whom came Abu Kamil Shojaʿ ibn Aslam." . 'There is some confusion in the literature on...

(c. 780–850). He later wrote The Compendious Book on Calculation by Completion and Balancing
The Compendious Book on Calculation by Completion and Balancing
, also known under a shorter name spelled as Hisab al-jabr w’al-muqabala, Kitab al-Jabr wa-l-Muqabala and other transliterations) is a mathematical book written in Arabic in approximately AD 820 by the Persian (Arabic for "The Compendious Book on Calculation by Completion and Balancing", in...

, which established algebra as a mathematical discipline that is independent of geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

and arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

.

The roots of algebra can be traced to the ancient Babylonians
Babylonian mathematics
Babylonian mathematics refers to any mathematics of the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited...

, who developed an advanced arithmetical system with which they were able to do calculations in an algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

ic fashion. The Babylonians developed formulas to calculate solutions for problems typically solved today by using linear equation
Linear equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....

In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...

s, and indeterminate linear equations
Indeterminate equation
An indeterminate equation, in mathematics, is an equation for which there is an infinite set of solutions; for example, 2x = y is a simple indeterminate equation. Indeterminate equations cannot be directly solved from the given information...

. By contrast, most Egyptians
Egyptian mathematics
Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt from ca. 3000 BC to ca. 300 BC.-Overview:Written evidence of the use of mathematics dates back to at least 3000 BC with the ivory labels found at Tomb Uj at Abydos. These labels appear to have been used as tags for...

of this era, as well as Greek
Greek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...

and Chinese
Chinese mathematics
Mathematics in China emerged independently by the 11th century BC. The Chinese independently developed very large and negative numbers, decimals, a place value decimal system, a binary system, algebra, geometry, and trigonometry....

mathematicians in the 1st millennium BC
1st millennium BC
The 1st millennium BC encompasses the Iron Age and sees the rise of many successive empires, and spanned from 1000 BC to 1 BC.The Neo-Assyrian Empire, followed by the Achaemenids. In Greece, Classical Antiquity begins with the colonization of Magna Graecia and peaks with the rise of Hellenism. The...

, usually solved such equations by geometric methods, such as those described in the Rhind Mathematical Papyrus
Rhind Mathematical Papyrus
The Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. It dates to around 1650 BC...

, Euclid's Elements
Euclid's Elements
Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...

, and The Nine Chapters on the Mathematical Art
The Nine Chapters on the Mathematical Art
The Nine Chapters on the Mathematical Art is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 1st century CE...

. 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, though this would not be realized until the medieval Muslim mathematicians.

The Hellenistic
Hellenistic civilization
Hellenistic civilization represents the zenith of Greek influence in the ancient world from 323 BCE to about 146 BCE...

mathematicians Hero of Alexandria
Hero of Alexandria
Hero of Alexandria was an ancient Greek mathematician and engineerEnc. Britannica 2007, "Heron of Alexandria" who was active in his native city of Alexandria, Roman Egypt...

and Diophantus
Diophantus
Diophantus of Alexandria , sometimes called "the father of algebra", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations, many of which are now lost...

as well as Indian mathematicians
Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...

such as Brahmagupta
Brahmagupta
Brahmagupta was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy. His best known work is the Brāhmasphuṭasiddhānta , written in 628 in Bhinmal...

continued the traditions of Egypt and Babylon, though Diophantus' Arithmetica
Arithmetica
Arithmetica 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 equations and indeterminate equations.Equations in the book are called Diophantine equations...

and Brahmagupta's Brahmasphutasiddhanta
Brahmasphutasiddhanta
The main work of Brahmagupta, Brāhmasphuṭasiddhānta , written c.628, contains ideas including a good understanding of the mathematical role of zero, rules for manipulating both negative and positive numbers, a method for computing square roots, methods of solving linear and some quadratic...

are on a higher level. For example, the first complete arithmetic solution (including zero and negative solutions) to quadratic equation
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...

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 was the first to solve equations using general methods. He solved the linear indeterminate equations, quadratic equations, second order indeterminate equations and equations with multiple variables.

The Greek
Greeks
The Greeks, also known as the Hellenes , are a nation and ethnic group native to Greece, Cyprus and neighboring regions. They also form a significant diaspora, with Greek communities established around the world....

mathematician Diophantus
Diophantus
Diophantus of Alexandria , sometimes called "the father of algebra", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations, many of which are now lost...

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. 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
Reduction (mathematics)
In mathematics, reduction refers to the rewriting of an expression into a simpler form. For example, the process of rewriting a fraction into one with the smallest whole-number denominator possible is called "reducing a fraction"...

" 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, 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 problem
Problem
A problem is an obstacle, impediment, difficulty or challenge, or any situation that invites resolution; the resolution of which is recognized as a solution or contribution toward a known purpose or goal...

s to be resolved, but an exposition
Expository writing
Expository writing is a type of writing where the purpose is to inform, explain, describe, or define the author's subject to the reader. Expository text is meant to deposit information and is the most frequently used type of writing by students in colleges and universities. A well-written...

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."

The Persian mathematician Omar Khayyam
Omar Khayyám
Omar Khayyám was aPersian polymath: philosopher, mathematician, astronomer and poet. He also wrote treatises on mechanics, geography, mineralogy, music, climatology and theology....

is credited with identifying the foundations of algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

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. He also developed the concept of a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

. The Indian mathematicians Mahavira
Mahavira (mathematician)
Mahavira was a 9th-century Indian Jain mathematician from Gulbarga who asserted that the square root of a negative number did not exist. He gave the sum of a series whose terms are squares of an arithmetical progression and empirical rules for area and perimeter of an ellipse. He was patronised by...

and Bhaskara II, the Persian mathematician Al-Karaji
Al-Karaji
' was a 10th century Persian Muslim mathematician and engineer. His three major works are Al-Badi' fi'l-hisab , Al-Fakhri fi'l-jabr wa'l-muqabala , and Al-Kafi fi'l-hisab .Because al-Karaji's original works in Arabic are lost, it is not...

, and the Chinese mathematician Zhu Shijie
Zhu Shijie
Zhu Shijie , courtesy name Hanqing , pseudonym Songting , was one of the greatest Chinese mathematicians lived during the Yuan Dynasty....

, solved various cases of cubic, quartic, quintic
Quintic equation
In mathematics, a quintic function is a function of the formg=ax^5+bx^4+cx^3+dx^2+ex+f,\,where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero...

and higher-order polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

equations using numerical methods. In the 13th century, the solution of a cubic equation by Fibonacci
Fibonacci
Leonardo Pisano Bigollo also known as Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or, most commonly, simply Fibonacci, was an Italian mathematician, considered by some "the most talented western mathematician of the Middle Ages."Fibonacci is best known to the modern...

is representative of the beginning of a revival in European algebra. As the Islamic world was declining, the European world was ascending. And it is here that algebra was further developed.

François Viète
François Viète
François Viète , Seigneur de la Bigotière, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations...

’s work at the close of the 16th century marks the start of the classical discipline of algebra. In 1637, René Descartes
René Descartes
René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...

published La Géométrie
La Géométrie
La Géométrie was published in 1637 as an appendix to Discours de la méthode , written by René Descartes. In the Discourse, he presents his method for obtaining clarity on any subject...

, inventing analytic geometry
Analytic geometry
Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...

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
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

was developed by Japanese mathematician
Japanese mathematics
denotes a distinct kind of mathematics which was developed in Japan during the Edo Period . The term wasan, from wa and san , was coined in the 1870s and employed to distinguish native Japanese mathematics theory from Western mathematics .In the history of mathematics, the development of wasan...

Kowa Seki in the 17th century, followed independently by Gottfried Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

ten years later, for the purpose of solving systems of simultaneous linear equations using matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

. Gabriel Cramer
Gabriel Cramer
Gabriel Cramer was a Swiss mathematician, born in Geneva. He showed promise in mathematics from an early age. At 18 he received his doctorate and at 20 he was co-chair of mathematics.In 1728 he proposed a solution to the St...

also did some work on matrices and determinants in the 18th century. Permutations were studied by Joseph Lagrange
Joseph Lagrange
Count Joseph Lagrange was a French soldier who rose through the ranks and gained promotion to the rank of general officer during the French Revolutionary Wars, subsequently pursuing a successful career during the Napoleonic Wars and winning promotion to the top military rank of General of Division....

in his 1770 paper Réflexions sur la résolution algébrique des équations devoted to solutions of algebraic equations, in which he introduced Lagrange resolvents. Paolo Ruffini
Paolo Ruffini
Paolo Ruffini was an Italian mathematician and philosopher.By 1788 he had earned university degrees in philosophy, medicine/surgery, and mathematics...

was the first person to develop the theory of permutation group
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 ; the relationship is often written as...

s, and like his predecessors, also in the context of solving algebraic equations.

Abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

was developed in the 19th century, initially focusing on what is now called Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...

, and on constructibility
Constructible number
A point in the Euclidean plane is a constructible point if, given a fixed coordinate system , the point can be constructed with unruled straightedge and compass...

issues. The "modern algebra" has deep nineteenth-century roots in the work, for example, of Richard Dedekind
Richard Dedekind
Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...

and Leopold Kronecker
Leopold Kronecker
Leopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...

and profound interconnections with other branches of mathematics such as algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...

and algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

. George Peacock
George Peacock
George Peacock was an English mathematician.-Life:Peacock was born on 9 April 1791 at Thornton Hall, Denton, near Darlington, County Durham. His father, the Rev. Thomas Peacock, was a clergyman of the Church of England, incumbent and for 50 years curate of the parish of Denton, where he also kept...

was the founder of axiomatic thinking in arithmetic and algebra. Augustus De Morgan
Augustus De Morgan
Augustus De Morgan was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. The crater De Morgan on the Moon is named after him....

discovered relation algebra
Relation algebra
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation...

in his Syllabus of a Proposed System of Logic. Josiah Willard Gibbs
Josiah Willard Gibbs
Josiah Willard Gibbs was an American theoretical physicist, chemist, and mathematician. He devised much of the theoretical foundation for chemical thermodynamics as well as physical chemistry. As a mathematician, he invented vector analysis . Yale University awarded Gibbs the first American Ph.D...

developed an algebra of vectors in three-dimensional space, and Arthur Cayley
Arthur Cayley
Arthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....

developed an algebra of matrices (this is a noncommutative algebra).

## Classification

Algebra may be divided roughly into the following categories:
• Elementary algebra
Elementary algebra
Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. It is typically taught in secondary school under the term algebra. The major difference between algebra and...

, in which the properties of operations on the real number system
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

are recorded using symbols as "place holders" to denote constants
Constant (mathematics)
In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition to variable In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition...

and variables
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

, and the rules governing mathematical expressions and equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...

s involving these symbols are studied. This is usually taught at school under the title algebra (or intermediate algebra and college algebra in subsequent years). University-level courses in group theory may also be called elementary algebra.
• Abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, sometimes also called modern algebra, in which algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...

s such as groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

, rings
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

and fields
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

are axiomatically defined and investigated.
• Linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

, in which the specific properties of vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

s are studied (including matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

);
• Universal algebra
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....

, in which properties common to all algebraic structures are studied.
• Algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...

, in which the properties of numbers are studied through algebraic systems. Number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

inspired much of the original abstraction in algebra.
• Algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

applies abstract algebra to the problems of geometry.
• Algebraic combinatorics
Algebraic combinatorics
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra....

, in which abstract algebraic methods are used to study combinatorial questions.

In some directions of advanced study, axiomatic algebraic systems such as groups, rings, fields, and algebras over a field are investigated in the presence of a geometric
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

structure (a metric
Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...

or a topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

) which is compatible with the algebraic structure. The list includes a number of areas of functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

:

## 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
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

beyond the basic principles of arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

. In arithmetic, only number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....

s and their arithmetical operations (such as +, −, ×, ÷) occur. In algebra, numbers are often denoted by symbols (such as a, x, or y). This is useful because:
• It allows the general formulation of arithmetical laws (such as a + b = b + a for all a and b), and thus is the first step to a systematic exploration of the properties of the real number system
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

.
• It allows the reference to "unknown" numbers, the formulation of equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...

s and the study of how to solve these. (For instance, "Find a number x such that 3x + 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.)
• It allows the formulation of functional
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

relationships. (For instance, "If you sell x tickets, then your profit will be 3x − 10 dollars, or f(x) = 3x − 10, where f is the function, and x is the number to which the function is applied.")

### Polynomials

A polynomial is an expression
Expression (mathematics)
In mathematics, an expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Symbols can designate numbers , variables, operations, functions, and other mathematical symbols, as well as punctuation, symbols of grouping, and other syntactic...

that is constructed from one or more variables
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

and constants, using only the operations of addition, subtraction, and multiplication (where repeated multiplication of the same variable is standardly denoted as exponentiation with a constant nonnegative integer exponent). For example, x2 + 2x − 3 is a polynomial in the single variable x.

An important class of problems in algebra is factorization
Factorization
In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original...

of polynomials, that is, expressing a given polynomial as a product of other polynomials. The example polynomial above can be factored as (x − 1)(x + 3). A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable.

## Abstract algebra

Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

of number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....

s to more general concepts.

Sets: Rather than just considering the different types of number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....

s, abstract algebra deals with the more general concept of sets: a collection of all objects (called 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 matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

, the set of all second-degree polynomials (ax2 + bx + c), the set of all two dimensional vectors in the plane, and the various finite groups such as the cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

s which are the group of integers modulo
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....

n. Set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

is a branch of logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

and not technically a branch of algebra.

Binary operation
Binary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....

s
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....

(+) 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, ab is another element in the set; this condition is called closure
Closure (mathematics)
In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...

Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....

(+), subtraction
Subtraction
In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...

(-), multiplication
Multiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....

(×), and division
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...

(÷) can be binary operations when defined on different sets, as is addition and multiplication of matrices, vectors, and polynomials.

Identity element
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...

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 ae = a and ea = a. This holds for addition as a + 0 = a and 0 + a = a and multiplication a × 1 = a and 1 × a = a. Not all set and operator combinations have an identity element; for example, the positive natural numbers (1, 2, 3, ...) have no identity element for addition.

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

Associativity
Associativity
In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...

: 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 (ab) ∗ c = a ∗ (bc). This property is shared by most binary operations, but not subtraction or division or octonion multiplication.

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 ab = ba. This property does not hold for all binary operations. For example, matrix multiplication
Matrix multiplication
In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...

and quaternion multiplication
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...

are both non-commutative.

### Groups

Combining the above concepts gives one of the most important structures in mathematics: a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

. A group is a combination of a set S and a single binary operation
Binary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....

∗, defined in any way you choose, but with the following properties:
• An identity element e exists, such that for every member a of S, ea and ae are both identical to a.
• Every element has an inverse: for every member a of S, there exists a member a−1 such that aa−1 and a−1a are both identical to the identity element.
• The operation is associative: if a, b and c are members of S, then (ab) ∗ c is identical to a ∗ (bc).

If a group is also commutative
Commutativity
In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...

—that is, for any two members a and b of S, ab is identical to ba—then the group is said to be abelian
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

.

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

The nonzero rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

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

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

The theory of groups is studied in group theory
Group theory
In mathematics and 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 as groups endowed with additional operations and...

. A major result in this theory is the classification of finite simple groups
Classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...

, mostly published between about 1955 and 1983, which is thought to classify all of the finite simple group
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...

s into roughly 30 basic types.
 Set: Operation Closed Examples Natural numbers N Integers Z Rational numbers Q (also real R and complex C numbers) Integers moduloModular arithmeticIn mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus.... 3: Z3 = {0, 1, 2} + × (w/o zero) + × (w/o zero) + − × (w/o zero) ÷ (w/o zero) + × (w/o zero) Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Identity 0 1 0 1 0 N/A 1 N/A 0 1 Inverse N/A N/A −a N/A −a N/A 1/a N/A 0, 2, 1, respectively N/A, 1, 2, respectively Associative Yes Yes Yes Yes Yes No Yes No Yes Yes Commutative Yes Yes Yes Yes Yes No Yes No Yes Yes Structure monoidMonoidIn abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for... monoidMonoidIn abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for... abelian groupAbelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers... monoidMonoidIn abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for... abelian groupAbelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers... quasigroupQuasigroupIn mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible... abelian groupAbelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers... quasigroupQuasigroupIn mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible... abelian groupAbelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers... abelian groupAbelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers... (Z2)

Semigroup
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...

s, quasigroup
Quasigroup
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible...

s, and monoid
Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...

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
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...

has an associative binary operation, but might not have an identity element. A monoid
Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...

is a semigroup which does have an identity but might not have an inverse for every element. A quasigroup
Quasigroup
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible...

satisfies a requirement that any element can be turned into any other by a unique pre- or post-operation; however the binary operation might not be associative.

All groups are monoids, and all monoids are semigroups.

### Rings and fields

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
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

, and fields
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

.

A ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

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.

Distributivity
Distributivity
In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...

generalises the distributive law for numbers, and specifies the order in which the operators should be applied, (called the precedence
Order of operations
In mathematics and computer programming, the order of operations is a rule used to clarify unambiguously which procedures should be performed first in a given mathematical expression....

). 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.

A field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

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.

## Objects called algebras

The word algebra is also used for various algebraic structures:
• Algebra over a field
Algebra over a field
In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...

or more generally Algebra over a ring
Algebra (ring theory)
In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R....

• Algebra over a set
• Boolean algebra
• Heyting algebra
Heyting algebra
In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b...

• F-algebra
F-algebra
In mathematics, specifically in category theory, an F-algebra is a structure defined according to a functor F. F-algebras can be used to represent data structures used in programming, such as lists and trees...

and F-coalgebra
F-coalgebra
In mathematics, specifically in category theory, an F-coalgebra is a structure defined according to a functor F. For both algebra and coalgebra, a functor is a convenient and general way of organizing a signature...

in category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

• Relational algebra
Relational algebra
Relational algebra, an offshoot of first-order logic , deals with a set of finitary relations that is closed under certain operators. These operators operate on one or more relations to yield a relation...

• Sigma-algebra
Sigma-algebra
In mathematics, a σ-algebra is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures; specifically, the collection of sets over which a measure is defined is a σ-algebra...