TheInfoList

Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the broad areas of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, together with
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

,
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

and
analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composit ...
. In its most general form, algebra is the study of
mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that ...
s and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It includes everything from elementary equation solving to the study of abstractions such as
groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identi ...
,
rings Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck Ring may also refer to: Sounds * Ri ...
, and
fields File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe'' FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...
. The more basic parts of algebra are called
elementary algebra Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with spec ...
; the more abstract parts are called
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Elementary algebra differs from
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values. For example, in $x + 2 = 5$ the letter $x$ is an unknown, but applying
additive inverse In mathematics, the additive inverse of a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be ...
s can reveal its value: $x=3$. In , the letters $E$ and $m$ are variables, and the letter $c$ is a
constant Constant or The Constant may refer to: Mathematics * Constant (mathematics) In mathematics, the word constant can have multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other Value (mathematics ...
, the speed of light in a vacuum. Algebra gives methods for writing formulas and solving equations that are much clearer and easier than the older method of writing everything out in words. The word ''algebra'' is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an "algebra", and the word is used, for example, in the phrases
linear algebra 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 matrix (mat ...
and
algebraic topology Algebraic topology is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...
. A mathematician who does research in algebra is called an algebraist.

# Etymology

The word ''algebra'' comes from the ar, الجبر, lit=reunion of broken parts, bonesetting, 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 Persian may refer to: * People and things from Iran, historically called ''Persia'' in the English language ** Persians, Persian people, the majority ethnic group in Iran, not to be conflated with the Iranian peoples ** Persian language, an Iranian ...

mathematician and astronomer
al-Khwarizmi Muḥammad ibn Mūsā al-Khwārizmī ( fa, محمد بن موسی خوارزمی, Moḥammad ben Musā Khwārazmi; ), Arabized as al-Khwarizmi and formerly Latinized as ''Algorithmi'', was a Persian polymath A polymath ( el, πολυμα ...

. 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 fifteenth century, from either Spanish, Italian, or
Medieval Latin Medieval Latin was the form of Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share ...
. It originally referred to the surgical procedure of setting broken or dislocated bones. The mathematical meaning was first recorded (in English) in the sixteenth 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 In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear map, bilinear product (mathematics), product. Thus, an algebra is an algebraic structure consisting of a set (mathematics), set to ...
). When some authors use the term "algebra", they make a subset of the following additional assumptions:
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
,
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, unital, and/or finite-dimensional. In
universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spa ...
, 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 matrix (mat ...
,
elementary algebra Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with spec ...
(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 Works * ...
and
secondary education Secondary education covers two phases on the International Standard Classification of Education The International Standard Classification of Education (ISCED) is a statistical framework for organizing information on education Education i ...
), or
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
(the study of the algebraic structures for themselves). ** With an article, it means an instance of some abstract structure, like a
Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, an
associative algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, or a
vertex operator algebra In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven useful ...
. ** Sometimes both meanings exist for the same qualifier, as in the sentence: ''
Commutative algebra Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry ...
is the study of
commutative ring In , a branch of , a commutative ring is a in which the multiplication operation is . The study of commutative rings is called . Complementarily, is the study of s where multiplication is not required to be commutative. Definition and first e ...
s, which are commutative algebras over the integers''.

# Algebra as a branch of mathematics

Algebra began with computations similar to those of
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
, 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 Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...

:$ax^2+bx+c=0,$ $a, b, c$ can be any numbers whatsoever (except that $a$ cannot be $0$), and the
quadratic formula In elementary algebra Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas a ...

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 , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

s,
vectors Vector may refer to: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
,
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
, and
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s. The structural properties of these non-numerical objects were then abstracted into
algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s such as
groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identi ...
,
rings Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck Ring may also refer to: Sounds * Ri ...
, and
fields File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe'' FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...
. Before the 16th century, mathematics was divided into only two subfields,
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
and
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

. 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 infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...
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 until it 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) is called ''algebra''. Today algebra includes section 08-General algebraic systems, 12- Field theory and
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s, 13-
Commutative algebra Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry ...
, 15-
Linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out se ...
and
multilinear algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
;
matrix theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, 16- Associative rings and algebras, 17-
Nonassociative ring A non-associative algebra (or distributive algebra) is an algebra over a field where the binary operation, binary multiplication operation is not assumed to be associative operation, associative. That is, an algebraic structure ''A'' is a non-assoc ...
s and
algebras In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, 18-
Category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
;
homological algebra Homological algebra is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...
, 19-
K-theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
and 20-
Group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
. Algebra is also used extensively in 11-
Number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

and 14-
Algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

.

# History

## Early history of algebra

The roots of algebra can be traced to the ancient
Babylonians Babylonia () was an ancient Ancient history is the aggregate of past eventsWordNet Search – ...
, who developed an advanced arithmetical system with which they were able to do calculations in an
algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

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+\cdots +a_nx_n+b=0, where x_1, \ldots, x_n are the variable (mathematics), variables (or unknown (mathematics), unknowns), and b, a_1, \ldots, a_n are the coeffi ...

s,
quadratic equation In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...

s, and indeterminate linear equations. By contrast, most
Egyptians Egyptians ( arz, المصريين, ; cop, ⲣⲉⲙⲛ̀ⲭⲏⲙⲓ, remenkhēmi) are an ethnic group of people originating from the country of Egypt Egypt ( ar, مِصر, Miṣr), officially the Arab Republic of Egypt, is a spanning t ...
of this era, as well as
Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...
and
Chinese mathematicsMathematics in China emerged independently by the 11th century BC. The Chinese independently developed a real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies ...
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 The British Museum, in the Bloomsbury Bloomsbury is a district in the West End of London The West End of London (commonly referred to as the West End ...

'', Euclid's ''Elements'', and ''
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 2nd century CE. This book is one of the earliest surv ...
''. 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 an Classical Athens, Athenian philosopher during the Classical Greece, Classical period in Ancient Greece, founder of the Platonist school of thought and the Platoni ...

, Greek mathematics had undergone a drastic change. The Greeks created a
geometric algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the autho ...
Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic language, Coptic: Rakodī; el, Αλεξάνδρεια ''Alexandria'') is the List of cities and towns in Egypt, third-largest city in Egypt after Cairo and Giza, ...

n Greek mathematician and the author of a series of books called ''
Arithmetica ''Arithmetica'' ( grc-gre, Ἀριθμητικά) is an Ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roug ...

''. These texts deal with solving
algebraic equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s, and have led, in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

, to the modern notion of
Diophantine equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. Earlier traditions discussed above had a direct influence on the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī (c. 780–850). He later wrote ''
The Compendious Book on Calculation by Completion and Balancing ''The Compendious Book on Calculation by Completion and Balancing'' ( ar, ٱلْكِتَاب ٱلْمُخْتَصَر فِي حِسَاب ٱلْجَبْر وَٱلْمُقَابَلَة, ''al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqā ...
'', which established algebra as a mathematical discipline that is independent of
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

and
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
. The
Hellenistic The Hellenistic period spans the period of History of the Mediterranean region, Mediterranean history 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 31  ...
mathematicians
Hero of Alexandria Hero of Alexandria (; grc-gre, Ἥρων ὁ Ἀλεξανδρεύς, ''Heron ho Alexandreus'', also known as Heron of Alexandria ; c. 10 AD – c. 70 AD), was a Greek mathematician and engineer who was active in his native city of Alexandria, R ...

and Diophantus as well as
Indian mathematicians The chronology of Indian mathematicians spans from the Indus Valley Civilization oxen for pulling a cart and the presence of the chicken The chicken (''Gallus gallus domesticus''), a subspecies of the red junglefowl, is a type of d ...
such as
Brahmagupta Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, doc ...

, continued the traditions of Egypt and Babylon, though Diophantus' ''Arithmetica'' and Brahmagupta's ''
Brāhmasphuṭasiddhānta The ''Brāhmasphuṭasiddhānta'' ("Correctly Established Doctrine of Brahma Brahma ( sa, ब्रह्मा, Brahmā) is one of the Hindu deities, principal deities of Hinduism, though his importance has declined in recent centuries. He i ...
'' 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 Arabic 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
or
zero 0 (zero) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

, thus he had to distinguish several types of equations. In the context where algebra is identified with the
theory of equations {{unreferenced, date=May 2014 In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and math ...
, 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". A debate now exists whether who (in the general sense) is more entitled to be known 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 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 Rhetorical modes (also known as modes of discourse) describe the variety, conventions, and purposes of the major kinds of languag ...
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 Omar Khayyam (; fa, عمر خیّام ; 18 May 1048 – 4 December 1131) was a Persian Persian may refer to: * People and things from Iran, historically called ''Persia'' in the English language ** Persians, Persian people, the majority ethni ...
is credited with identifying the foundations of
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

and found the general geometric solution of the
cubic equation roots A root is the part of a plant that most often lies below the surface of the soil but can also be aerial or aerating, that is, growing up above the ground or especially above water. Root or roots may also refer to: Art, entertainment, a ...

. 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 Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī ( fa, شرف‌الدین مظفر بن محمد بن مظفر توسی; c. 1135 – c. 1213) was an Iranian peoples, Iranian Islamic mathematics, mathematician and Islamic ...
, found algebraic and numerical solutions to various cases of cubic equations. He also developed the concept of a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
. The Indian mathematicians
Mahavira Mahavira ( sa, महावीर:), also known as Vardhamana, was the 24th ''Tirthankara In Jainism Jainism (), traditionally known as ''Jain Dharma'', is an ancient Indian religion and the method of acquiring perfect knowledge ...
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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek ...
,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, solved various cases of cubic, quartic,
quintic In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
and higher-order
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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 A mathematician is someone who uses an extensive knowledge of mathem ...

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 ∑''n''2, ∑''n''3 and used the method of successive approximation to determine square roots.

## Modern history of algebra

François Viète François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Gre ...
's work on
new algebra New is an adjective referring to something recently made, discovered, or created. New or NEW may refer to: Music * New, singer of K-pop group The Boyz Albums and EPs * ''New'' (album), by Paul McCartney, 2013 * ''New'' (EP), by Regurgitator, ...
at the close of the 16th century was an important step towards modern algebra. In 1637,
René Descartes René Descartes ( or ; ; Latinized Latinisation or Latinization can refer to: * Latinisation of names, the practice of rendering a non-Latin name in a Latin style * Latinisation in the Soviet Union, the campaign in the USSR during the 1920s ...

published ''
La Géométrie ''La Géométrie'' was published Publishing is the activity of making information, literature, music, software and other content available to the public for sale or for free. Traditionally, the term refers to the distribution of printed works ...
'', inventing
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

was developed by Japanese mathematician Seki Kōwa in the 17th century, followed independently by
Gottfried Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. ( – 14 November 1716) was a German polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, " ...
ten years later, for the purpose of solving systems of simultaneous linear equations using
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
.
Gabriel Cramer Gabriel Cramer (; 31 July 1704 – 4 January 1752) was a Genevan mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quan ...

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 In Galois theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analys ...
.
Paolo Ruffini Paolo Ruffini (September 22, 1765 – May 10, 1822) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as qua ...

was the first person to develop the theory of
permutation group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, and like his predecessors, also in the context of solving algebraic equations.
Abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
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 (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
, and on constructibility issues.
George Peacock George Peacock FRS (9 April 1791 – 8 November 1858) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

was the founder of axiomatic thinking in arithmetic and algebra.
Augustus De Morgan Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmeti ...

discovered
relation algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 tr ...

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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...

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

# Areas of mathematics with the word algebra in their name

Some areas of mathematics that fall under the classification abstract 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 matrix (mat ...
is one example. Others do not:
group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
,
ring theory In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
, 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 some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with spec ...
, the part of algebra that is usually taught in elementary courses of mathematics. *
Abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, in which
algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s such as
groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identi ...
,
rings Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck Ring may also refer to: Sounds * Ri ...
and
fields File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe'' FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...
are
axiomatically An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or ...
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 matrix (mat ...
, 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+\cdots +a_nx_n+b=0, where x_1, \ldots, x_n are the variable (mathematics), variables (or unknown (mathematics), unknowns), and b, a_1, \ldots, a_n are the coeffi ...

s,
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s and
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
are studied. *
Boolean algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, a branch of algebra abstracting the computation with the
truth value In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...
s ''false'' and ''true''. *
Commutative algebra Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry ...
, the study of
commutative ring In , a branch of , a commutative ring is a in which the multiplication operation is . The study of commutative rings is called . Complementarily, is the study of s where multiplication is not required to be commutative. Definition and first e ...
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 expression (mathematics), ...
, the implementation of algebraic methods as
algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

s and
computer program In imperative programming, a computer program is a sequence of instructions in a programming language that a computer can execute or interpret. In declarative programming, a ''computer program'' is a Set (mathematics), set of instructions. A comp ...
s. *
Homological algebra Homological algebra is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...
, the study of algebraic structures that are fundamental to study
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s. *
Universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spa ...
, in which properties common to all algebraic structures are studied. *
Algebraic number theory Algebraic number theory is a branch of number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is th ...
, in which the properties of numbers are studied from an algebraic point of view. *
Algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

, a branch of geometry, in its primitive form specifying curves and surfaces as solutions of polynomial equations. *
Algebraic combinatorics Algebraic combinatorics is an area of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (m ...
, in which algebraic methods are used to study combinatorial questions. *
Relational algebra In database theory, relational algebra is a theory that uses algebraic structures with a well-founded semantics for modeling data, and defining queries on it. The theory has been introduced by Edgar F. Codd. The main application of relational ...
: a set of
finitary relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s that is closed under certain operators. Many mathematical structures are called algebras: *
Algebra over a field In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear map, bilinear product (mathematics), product. Thus, an algebra is an algebraic structure consisting of a set (mathematics), set to ...
or more generally
algebra over a ring In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
.
Many classes of algebras over a field or over a ring have a specific name: **
Associative algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
**
Non-associative algebra A non-associative algebra (or distributive algebra) is an algebra over a field where the binary operation, binary multiplication operation is not assumed to be associative operation, associative. That is, an algebraic structure ''A'' is a non-ass ...
**
Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
**
Hopf algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

**
C*-algebra In mathematics, specifically in functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a commo ...
**
Symmetric algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
**
Exterior algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
**
Tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra over a field, algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', i ...
* In
measure theory Measure is a fundamental concept of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contai ...
, ** Sigma-algebra ** Algebra over a set * In category theory ** F-algebra and F-coalgebra ** T-algebra * In logic, ** Relation algebra, a residuated Boolean algebra expanded with an involution called converse. ** Boolean algebra (structure), Boolean algebra, a complemented lattice, complemented distributive lattice. ** Heyting algebra

# 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 (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
beyond the basic principles of
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
. In arithmetic, only numbers and their arithmetical operations (such as +, −, ×, ÷) occur. In algebra, numbers are often represented by symbols called variable (mathematics), variables (such as ''a'', ''n'', ''x'', ''y'' or ''z''). This is useful because: * It allows the general formulation of arithmetical laws (such as ''a'' + ''b'' = ''b'' + ''a'' for all ''a'' and ''b''), and thus is the first step to a systematic exploration of the properties of the real number, real number system. * It allows the reference to "unknown" numbers, the formulation of equations 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 function (mathematics), functional relationships. (For instance, "If you sell ''x'' tickets, then your profit will be 3''x'' − 10 dollars, or ''f''(''x'') = 3''x'' − 10, where ''f'' is the function, and ''x'' is the number to which the function is applied".)

## Polynomials

A polynomial is an expression (mathematics), expression that is the sum of a finite number of non-zero Summand, terms, each term consisting of the product of a constant and a finite number of Variable (mathematics), variables raised to whole number powers. For example, ''x''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, 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 root of a function, 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 is started in ninth grade.

# Abstract algebra

Abstract algebra extends the familiar concepts found in elementary algebra and
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
of numbers to more general concepts. Here are the listed fundamental concepts in abstract algebra. Set (mathematics), Sets: Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of ''sets'': a collection of all objects (called Element (mathematics), elements) selected by property specific for the set. All collections of the familiar types of numbers are sets. Other examples of sets include the set of all two-by-two Matrix (mathematics), matrices, the set of all second-degree polynomials (''ax''2 + ''bx'' + ''c''), the set of all two dimensional Vector (geometric), vectors in the plane, and the various finite groups such as the cyclic groups, which are the groups of integers modular arithmetic, modulo ''n''. Set theory is a branch of logic and not technically a branch of algebra. Binary operations: The notion of addition (+) is abstracted to give a ''binary operation'', ∗ say. The notion of binary operation is meaningless without the set on which the operation is defined. For two elements ''a'' and ''b'' in a set ''S'', ''a'' ∗ ''b'' is another element in the set; this condition is called Closure (mathematics), closure. Addition (+), subtraction (−), multiplication (×), and Division (mathematics), division (÷) can be binary operations when defined on different sets, as are addition and multiplication of matrices, vectors, and polynomials. Identity elements: The numbers zero and one are abstracted to give the notion of an ''identity element'' for an operation. Zero is the identity element for addition and one is the identity element for multiplication. For a general binary operator ∗ the identity element ''e'' must satisfy ''a'' ∗ ''e'' = ''a'' and ''e'' ∗ ''a'' = ''a'', 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: 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. Commutative operation, Commutativity: Addition and multiplication of real numbers are both commutative. That is, the order of the numbers does not affect the result. For example: 2 + 3 = 3 + 2. In general, this becomes ''a'' ∗ ''b'' = ''b'' ∗ ''a''. This property does not hold for all binary operations. For example, matrix multiplication and Quaternion, quaternion multiplication are both non-commutative.

## Groups

Combining the above concepts gives one of the most important structures in mathematics: a group (mathematics), group. A group is a combination of a set ''S'' and a single binary operation ∗, defined in any way you choose, but with the following properties: * An identity element ''e'' exists, such that for every member ''a'' of ''S'', ''e'' ∗ ''a'' and ''a'' ∗ ''e'' are both identical to ''a''. * Every element has an inverse: for every member ''a'' of ''S'', there exists a member ''a''−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 commutativity, commutative – that is, for any two members ''a'' and ''b'' of ''S'', ''a'' ∗ ''b'' is identical to ''b'' ∗ ''a'' – then the group is said to be Abelian group, abelian. For example, the set of integers under the operation of addition is a group. In this group, the identity element is 0 and the inverse of any element ''a'' is its negation, −''a''. The associativity requirement is met, because for any integers ''a'', ''b'' and ''c'', (''a'' + ''b'') + ''c'' = ''a'' + (''b'' + ''c'') The non-zero rational numbers 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
. A major result in this theory is the classification of finite simple groups, mostly published between about 1955 and 1983, which separates the finite set, finite simple groups into roughly 30 basic types. Semi-groups, quasi-groups, and monoids structure similar to groups, but more general. 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 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 Ring (mathematics), rings and Field (mathematics), fields. A Ring (mathematics), ring has two binary operations (+) and (×), with × distributive over +. Under the first operator (+) it forms an ''abelian group''. Under the second operator (×) it is associative, but it does not need to have 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 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. A Field (mathematics), field is a ''ring'' with the additional property that all the elements excluding 0 form an ''abelian group'' under ×. The multiplicative (×) identity is written as 1 and the multiplicative inverse of ''a'' is written as ''a''−1. The rational numbers, the real numbers and the complex numbers are all examples of fields.

* Outline of algebra * Outline of linear algebra * Algebra tile

* * *

* * * * * * * *