Algebra is a branch of mathematics concerning the study of structure, relation and quantity. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, This article sets out the set-theoretic notion of relation For a more elementary point of view see Binary relations and Triadic relations Quantity is a kind of property which exists as magnitude or multitude The name is derived from the treatise written by the Persian^[1] mathematician, astronomer, astrologer and geographer, Muhammad bin Mūsā al-Khwārizmī titled Kitab al-Jabr wa-l-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing"), which provided symbolic operations for the systematic solution of linear and quadratic equations. layout and formatting it should ensure no clashes with the top of the infobox This is a sub-article of History of science in the Islamic World and Astrology. A geographer is a Scientist whose area of study is Geography, the study of Earth 's physical environment and Human habitat Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala ( Arabic for "The Compendious Book on Calculation by Completion and Balancing" A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. Al-Khwarizimi's book made its way to Europe and was translated into Latin as Liber algebrae et almucabala.

Together with geometry, analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Analysis has its beginnings in the rigorous formulation of Calculus. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Elementary algebra is often part of the curriculum in secondary education and provides an introduction to the basic ideas of algebra, including effects of adding and multiplying numbers, the concept of variables, definition of polynomials, along with factorization and determining their roots. 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 Australia See also Education Addition is the mathematical process of putting things together A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. A variable (ˈvɛərɪəbl is an Attribute of a physical or an abstract System which may change its Value while it is under Observation. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, factorization ( also factorisation in British English) or factoring is the decomposition of an object (for This article is about the zeros of a function which should not be confused with the value at zero.

Algebra is much broader than elementary algebra and can be generalized. In addition to working directly with numbers, algebra covers working with symbols, variables, and set elements. The musical instrument is spelled Cymbal. A symbol is something --- such as an object, Picture, written word a sound a piece In Mathematics, the elements or members of a set (or more generally a class) are all those objects which when collected together make up the Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields. In Mathematics, an operator is a function which operates on (or modifies another function In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division

Contents 1 Classification 2 Elementary algebra 2.1 Polynomials 3 Abstract algebra 3.1 Groups—structures of a set with a single binary operation 3.2 Rings and fields—structures of a set with two particular binary operations, (+) and (×) 4 Objects called algebras 5 History 6 See also 7 References 8 External links

Classification

Algebra may be divided roughly into the following categories:

• Elementary algebra, in which the properties of operations on the real number system are recorded using symbols as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied (note that this usually includes the subject matter of courses called intermediate algebra and college algebra), also called second year and third year algebra;
• Abstract algebra, sometimes also called modern algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated; this includes, among other fields,
• Linear algebra, in which the specific properties of vector spaces are studied (including matrices);
• Universal algebra, in which properties common to all algebraic structures are studied. 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 In Mathematics, the real numbers may be described informally in several different ways A variable (ˈvɛərɪəbl is an Attribute of a physical or an abstract System which may change its Value while it is under Observation. In mathematics the word expression is a term for any well-formed combination of mathematical symbols An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, an axiomatic system is any set of Axioms from which some or all axioms can be used in conjunction to logically derive Theorems Linear algebra is the branch of Mathematics concerned with In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models"
• Algebraic number theory, in which the properties of numbers are studied through algebraic systems. In Mathematics, algebraic number theory is a major branch of Number theory which studies the Algebraic structures related to Algebraic integers Number theory inspired much of the original abstraction in algebra. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes
• Algebraic geometry in its algebraic aspect. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with
• Algebraic combinatorics, in which abstract algebraic methods are used to study combinatorial questions. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects

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 structure (a metric or a topology) which is compatible with the algebraic structure. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of The list includes a number of areas of functional analysis:

Elementary algebra

Main article: Elementary algebra

Elementary algebra is the most basic form of algebra. For functional analysis as used in psychology see the Functional analysis (psychology article In Mathematics, with 2- or 3-dimensional vectors with real -valued entries the idea of the "length" of a vector is intuitive and can easily be extended to In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, especially Functional analysis, a Banach algebra, named after Stefan Banach, is an Associative algebra A over the In Mathematics, especially Functional analysis, a Banach algebra, named after Stefan Banach, is an Associative algebra A over the In Mathematics, a topological algebra A is a Topological vector space with a continuous multiplication \cdot:A\times A \longrightarrow In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the 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 It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone In arithmetic, only numbers and their arithmetical operations (such as +, −, ×, ÷) occur. A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. 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. In Mathematics, the real numbers may be described informally in several different ways
• 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 3x + 1 = 10"). An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent
• It allows the formulation of functional relationships (such as "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. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function ").

Polynomials

Main article: Polynomial

A polynomial is an expression that is constructed from one or more variables 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 whole number exponent). In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In mathematics the word expression is a term for any well-formed combination of mathematical symbols A variable (ˈvɛərɪəbl is an Attribute of a physical or an abstract System which may change its Value while it is under Observation. For example, is a polynomial in the single variable x.

An important class of problems in algebra is factorization of polynomials, that is, expressing a given polynomial as a product of other polynomials. In Mathematics, factorization ( also factorisation in British English) or factoring is the decomposition of an object (for The example polynomial above can be factored as A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable. This article is about the zeros of a function which should not be confused with the value at zero.

Abstract algebra

Main article: Abstract algebra

See also: Algebraic structure

Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone A number is an Abstract object, tokens of which are Symbols used in Counting and measuring.

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 elements) selected by property, specific for the set. A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. In Mathematics, the elements or members of a set (or more generally a class) are all those objects which when collected together make up the All collections of the familiar types of numbers are sets. Other examples of sets include the set of all two-by-two matrices, the set of all second-degree polynomials (ax² + bx + c), the set of all two dimensional vectors in the plane, and the various finite groups such as the cyclic groups which are the group of integers modulo n. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, a finite group is a group which has finitely many elements In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers Set theory is a branch of logic and not technically a branch of algebra. Logic is the study of the principles of valid demonstration and Inference.

Binary operations: The notion of addition (+) is abstracted to give a binary operation, * say. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two Addition is the mathematical process of putting things together 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 gives another element in the set; this condition is called closure. In Mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set Addition (+), subtraction (-), multiplication (×), and division (÷) can be binary operations when defined on different sets, as is addition and multiplication of matrices, vectors, and polynomials. Addition is the mathematical process of putting things together Subtraction is one of the four basic Arithmetic operations it is the inverse of Addition, meaning that if we start with any number and add any number and then subtract In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication.

Identity elements: The numbers zero and one are abstracted to give the notion of an identity element for an operation. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that Zero is the identity element for addition and one is the identity element for multiplication. For a general binary operator * the identity element e must satisfy a * e = a and e * a = a. This holds for addition as a + 0 = a and 0 + a = a and multiplication a × 1 = a and 1 × a = a. However, if we take the positive natural numbers and addition, there is no identity element.

Inverse elements: The negative numbers give rise to the concept of inverse elements. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to 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 a * a^-1 = e and a^-1 * a = e.

Associativity: Addition of integers has a property called associativity. In Mathematics, associativity is a property that a Binary operation can have That is, the grouping of the numbers to be added does not affect the sum. For example: (2+3)+4=2+(3+4). In general, this becomes (a * b) * c = a * (b * c). This property is shared by most binary operations, but not subtraction or division or octonion multiplication. In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real

Commutativity: Addition of integers also has a property called commutativity. In Mathematics, commutativity is the ability to change the order of something without changing the end result That is, the order of the numbers to be added does not affect the sum. For example: 2+3=3+2. In general, this becomes a * b = b * a. Only some binary operations have this property. It holds for the integers with addition and multiplication, but it does not hold for matrix multiplication or quaternion multiplication . In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician

Groups—structures of a set with a single binary operation

Main article: Group (mathematics)

See also: Group theory and Examples of groups

Combining the above concepts gives one of the most important structures in mathematics: a group. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. Some elementary examples of groups in Mathematics are given on Group (mathematics. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element 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. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two
• Every element has an inverse: for every member a of S, there exists a member a^-1 such that a * a^-1 and a^-1 * a are both identical to the identity element.
• The operation is associative: if a, b and c are members of S, then (a * b) * c is identical to a * (b * c).

If a group is also commutative - 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. In Mathematics, commutativity is the ability to change the order of something without changing the end result An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the

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 numbers form a group under multiplication. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions 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 1/4, which is not an integer.

The theory of groups is studied in group theory. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. A major result in this theory is the classification of finite simple groups, mostly published between about 1955 and 1983, which is thought to classify all of the finite simple groups into roughly 30 basic types. The classification of the finite simple groups, also called the enormous theorem is believed to classify all finite simple groups. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning

Examples
Set:	Natural numbers		Integers		Rational numbers (also real and complex numbers)				Integers mod 3: {0,1,2}
Operation	+	× (w/o zero)	+	× (w/o zero)	+	−	× (w/o zero)	÷ (w/o zero)	+	× (w/o zero)
Closed	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes
Identity	0	1	0	1	0	NA	1	NA	0	1
Inverse	NA	NA	-a	NA	-a	NA		NA	0,2,1, respectively	NA, 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	monoid	monoid	Abelian group	monoid	Abelian group	quasigroup	Abelian group	quasigroup	Abelian group	Abelian group ()

Semigroups, quasigroups, and monoids are structures similar to groups, but more general. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, especially in Abstract algebra, a quasigroup is an Algebraic structure resembling a group in the sense that " division An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, especially in Abstract algebra, a quasigroup is an Algebraic structure resembling a group in the sense that " division An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, a semigroup is an Algebraic structure consisting of a nonempty set S together with an Associative Binary operation In Mathematics, especially in Abstract algebra, a quasigroup is an Algebraic structure resembling a group in the sense that " division In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation They comprise a set and a closed binary operation, but do not necessarily satisfy the other conditions. A semigroup has an associative binary operation, but might not have an identity element. In Mathematics, a semigroup is an Algebraic structure consisting of a nonempty set S together with an Associative Binary operation A monoid is a semigroup which does have an identity but might not have an inverse for every element. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation A quasigroup 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. In Mathematics, especially in Abstract algebra, a quasigroup is an Algebraic structure resembling a group in the sense that " division

All groups are monoids, and all monoids are semigroups.

Rings and fields—structures of a set with two particular binary operations, (+) and (×)

Main articles: ring (mathematics) and field (mathematics)

See also: Ring theory, Glossary of ring theory, Field theory (mathematics), and glossary of field theory

Groups just have one binary operation. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those Ring theory is the branch of Mathematics in which rings are studied that is structures supporting both an Addition and a Multiplication operation Field theory is the branch of Mathematics in which fields are studied To fully explain the behaviour of the different types of numbers, structures with two operators need to be studied. The most important of these are rings, and fields. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division

Distributivity generalised the distributive law for numbers, and specifies the order in which the operators should be applied, (called the precedence). In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law In Algebra and Computer programming, when a number or expression is both preceded and followed by a Binary operation, a rule is required for which operation For the integers (a + b) × c = a×c+ b×c and c × (a + b) = c×a + c×b, and × is said to be distributive over +.

A ring has two binary operations (+) and (×), with × distributive over +. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real 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 allowed. The additive (+) identity element is written as 0 and the additive inverse of a is written as −a.

The integers are an example of a ring. The integers have additional properties which make it an integral domain. In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such

A field is a ring with the additional property that all the elements excluding 0 form an Abelian group under ×. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division 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 or more generally Algebra over a ring
• Algebra over a set
• Boolean algebra
• F-algebra and F-coalgebra in category theory
• Sigma-algebra
• T-Algebras of monads. In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with 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 In Mathematics a field of sets is a pair \langle X \mathcal{F} \rangle where X is a set and \mathcal{F} is an algebra In Abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. In Mathematics, specifically in Category theory, an F-algebra for an Endofunctor F: \mathbf{C}\longrightarrow \mathbf{C} In Mathematics, specifically in Category theory, an F-coalgebra for an Endofunctor F: \mathbf{C}\longrightarrow \mathbf{C} In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a σ-algebra (or sigma-algebra) ( Sigma is a Greek letter upper case Σ lower case σ over a set X is a nonempty In Category theory, a monad or triple is an (endo- Functor, together with two associated Natural transformations They are important in the theory

History

Main article: History of algebra

See also: Timeline of algebra

The origins of algebra can be traced to the ancient Babylonians,^[2] who developed an advanced arithmetical system with which they were able to do calculations in an algebraic fashion. Elementary algebra is the branch of Mathematics that deals with solving for the Operands of Arithmetic Equations. A timeline of key algebraic developments are as follows Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala ( Arabic for "The Compendious Book on Calculation by Completion and Balancing" Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (ancient Iraq) from the days of the early Sumerians to the fall of Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone With the use of this system they were able to apply formulas and calculate solutions for unknown values for a class of problems typically solved today by using linear equations, quadratic equations, and indeterminate linear equations. A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. 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 By contrast, most Egyptians of this era, and most Indian, Greek and Chinese mathematicians in the first millennium BC, usually solved such equations by geometric methods, such as those described in the Rhind Mathematical Papyrus, Sulba Sutras, Euclid's Elements, and The Nine Chapters on the Mathematical Art. Egyptian mathematics refers to the style and methods of Mathematics performed in Ancient Egypt. Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. Greek mathematics, as that term is used in this article is the Mathematics written in Greek, developed from the 6th century BC to the 5th century Mathematics in China emerged independently by the 11th century BC The 1st millennium BC encompasses the Iron Age and sees the rise of successive empires Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position The Rhind Mathematical Papyrus (RMP (also designated as papyrus British Museum 10057 and pBM 10058 is named after Alexander Henry Rhind, a Scottish The Shulba Sutras or Śulbasūtras ( Sanskrit śulba: "string cord rope" are Sutra texts belonging to the Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek The Nine Chapters on the Mathematical Art ( is a Chinese Mathematics book composed by several generations of scholars from the 10th&ndash2nd century BC and 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.

Later, the Indian mathematicians developed algebraic methods to a high degree of sophistication. Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Brahmagupta was the first to solve equations using general methods. Brahmagupta ( (598–668 was an Indian mathematician and astronomer. He solved the linear indeterminate equations, quadratic equations, second order indeterminate equations and equations with multiple variable.

The word "algebra" is named after the Arabic word "al-jabr" from the title of the book al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala, meaning The book of Summary Concerning Calculating by Transposition and Reduction, a book written by the Persian mathematician Muhammad ibn Mūsā al-khwārizmī in 820. Arabic (ar الْعَرَبيّة (informally ar عَرَبيْ) in terms of the number of speakers is the largest living member of the Semitic language Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala ( Arabic for "The Compendious Book on Calculation by Completion and Balancing" The word Al-Jabr means "reunion". The Hellenistic mathematician Diophantus has traditionally been known as "the father of algebra" but debate now exists as to whether or not Al-Khwarizmi should take that title. Diophantus of Alexandria ( Greek: b between 200 and 214 d between 284 and 298 AD sometimes called "the father of Algebra " a title some claim should ^[3] Those who support Al-Khwarizmi point to the fact that much of his work on reduction is still in use today and that he gave an exhaustive explanation of solving quadratic equations. In Mathematics, reduction refers to the rewriting of an expression into a simpler form Those who support Diophantus point to the fact that the algebra found in Al-Jabr is more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical. Diophantus of Alexandria ( Greek: b between 200 and 214 d between 284 and 298 AD sometimes called "the father of Algebra " a title some claim should ^[4] Another Persian mathematician, Omar Khayyam, developed algebraic geometry and found the general geometric solution of the cubic equation. For the Thoroughbred racehorse see Omar Khayyam (horse Ghiyās od-Dīn Abol-Fath Omār ibn Ebrāhīm Khayyām Neyshābūri (غیاث الدین Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. The Indian mathematicians Mahavira and Bhaskara II, and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quartic, quintic and higher-order polynomial equations. Mahavira was a 9th century Indian Mathematician from Gulbarga who asserted that the Square root of a Negative number did Bhaskara (1114 &ndash 1185 also known as Bhaskara II and Bhaskara Achārya ("Bhaskara the teacher" was an Indian mathematician Zhu Shijie ( fl 13th century) Courtesy name Hanqing (汉卿 Pseudonym Songting (松庭 was one of the greatest Chinese In Mathematics, a quartic equation is one which can be expressed as a Quartic function equalling zero In Mathematics, a quintic equation is a Polynomial Equation of degree five In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations

Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century. The idea of a determinant was developed by Japanese mathematician Kowa Seki in the 17th century, followed by Gottfried Leibniz ten years later, for the purpose of solving systems of simultaneous linear equations using matrices. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In the History of mathematics, Japanese mathematics or wasan (和算 denotes a genuinely distinct kind of mathematics developed in Japan during the or (born 1637/1642? – October 24, 1708) was a Japanese Mathematician who created a new algebraic notation system and laid In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Gabriel Cramer also did some work on matrices and determinants in the 18th century. Gabriel Cramer ( July 31, 1704 - January 4, 1752) was a Swiss Mathematician, born in Geneva. Abstract algebra was developed in the 19th century, initially focusing on what is now called Galois theory, and on constructibility issues. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory A point in the Euclidean plane is a constructible point if given a fixed Coordinate system (or a fixed Line segment of unit Length

The stages of the development of symbolic algebra are roughly as follows:

• Rhetorical algebra, which was developed by the Babylonians and remained dominant up to the 16th century;
• Geometric constructive algebra, which was emphasised by the Vedic Indian and classical Greek mathematicians;
• Syncopated algebra, as developed by Diophantus, Brahmagupta and the Bakhshali Manuscript; and
• Symbolic algebra, which was initiated by Abū al-Hasan ibn Alī al-Qalasādī^[5] and sees its culmination in the work of Gottfried Leibniz. The Vedic Period (or Vedic Age) is the period in the History of India during which the Vedas, the oldest sacred texts of Hinduism, were being Diophantus of Alexandria ( Greek: b between 200 and 214 d between 284 and 298 AD sometimes called "the father of Algebra " a title some claim should Brahmagupta ( (598–668 was an Indian mathematician and astronomer. The Bakhshali Manuscript is a mathematical manuscript written on birch bark which was found near the village of Bakhshali in 1881 in what was Abū al-Hasan ibn ʿAlī al-Qalaṣādī (1412 in Baza, Spain &ndash 1486 in Béja, Tunisia) was an Arab Muslim mathematician

A timeline of key algebraic developments are as follows:

• Circa 1800 BC: The Old Babylonian Strassburg tablet seeks the solution of a quadratic elliptic equation. The Chronology of the first dynasty of Babylonia is debated as there is a Babylonian King List A and a Babylonian King List B.
• Circa 1600 BC: The Plimpton 322 tablet gives a table of Pythagorean triples in Babylonian Cuneiform script. Of the approximately half million Babylonian Clay tablets excavated since the beginning of the 19th century several thousand are of a mathematical nature A Pythagorean triple consists of three positive Integers a, b, and c, such that a 2 +  b 2 =  Babylonia was an Amorite state in lower Mesopotamia (modern southern Iraq) with Babylon as its capital
• Circa 800 BC: Indian mathematician Baudhayana, in his Baudhayana Sulba Sutra, discovers Pythagorean triples algebraically, finds geometric solutions of linear equations and quadratic equations of the forms ax² = c and ax² + bx = c, and finds two sets of positive integral solutions to a set of simultaneous Diophantine equations. Baudhāyana, (fl ca 800 BCE was an Indian mathematician whowas most likely also a priest The Shulba Sutras or Śulbasūtras ( Sanskrit śulba: "string cord rope" are Sutra texts belonging to the In Mathematics, a Diophantine equation is an indeterminate Polynomial Equation that allows the variables to be Integers only
• Circa 600 BC: Indian mathematician Apastamba, in his Apastamba Sulba Sutra, solves the general linear equation and uses simultaneous Diophantine equations with up to five unknowns.
• Circa 300 BC: In Book II of his Elements, Euclid gives a geometric construction with Euclidean tools for the solution of the quadratic equation for positive real roots. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry The construction is due to the Pythagorean School of geometry.
• Circa 300 BC: A geometric construction for the solution of the cubic is sought (doubling the cube problem). It is now well known that the general cubic has no such solution using Euclidean tools. Pentagon constructgif|thumb|right|Construction of a regular pentagon]] Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or Angles
• Circa 100 BC: Algebraic equations are treated in the Chinese mathematics book Jiuzhang suanshu (The Nine Chapters on the Mathematical Art), which contains solutions of linear equations solved using the rule of double false position, geometric solutions of quadratic equations, and the solutions of matrices equivalent to the modern method, to solve systems of simultaneous linear equations. The Nine Chapters on the Mathematical Art ( is a Chinese Mathematics book composed by several generations of scholars from the 10th&ndash2nd century BC and In Numerical analysis, the false position method or regula falsi method is a Root-finding algorithm that combines features from the Bisection method
• Circa 100 BC: The Bakhshali Manuscript written in ancient India uses a form of algebraic notation using letters of the alphabet and other signs, and contains cubic and quartic equations, algebraic solutions of linear equations with up to five unknowns, the general algebraic formula for the quadratic equation, and solutions of indeterminate quadratic equations and simultaneous equations. Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. Middle kingdoms of India refers to the political entities in India from the 2nd century BC since the decline of the Maurya Empire, and the corresponding A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable
• Circa 150 AD: Hero of Alexandria treats algebraic equations in three volumes of mathematics. Hero (or Heron) of Alexandria ( Ήρων ο Αλεξανδρεύς) (c
• Circa 200: Diophantus, who lived in Egypt and is often considered the "father of algebra", writes his famous Arithmetica, a work featuring solutions of algebraic equations and on the theory of numbers. Diophantus of Alexandria ( Greek: b between 200 and 214 d between 284 and 298 AD sometimes called "the father of Algebra " a title some claim should Arithmetica is an ancient Greek text on Mathematics written by the Mathematician Diophantus in the 3rd century CE.
• 499: Indian mathematician Aryabhata, in his treatise Aryabhatiya, obtains whole-number solutions to linear equations by a method equivalent to the modern one, describes the general integral solution of the indeterminate linear equation and gives integral solutions of simultaneous indeterminate linear equations. Āryabhaṭa ( Devanāgarī: आर्यभट (AD 476 &ndash 550 is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics
• Circa 625: Chinese mathematician Wang Xiaotong finds numerical solutions of cubic equations.
• 628: Indian mathematician Brahmagupta, in his treatise Brahma Sputa Siddhanta, invents the chakravala method of solving indeterminate quadratic equations, including Pell's equation, and gives rules for solving linear and quadratic equations. Brahmagupta ( (598–668 was an Indian mathematician and astronomer. The chakravala method is a cyclic Algorithm to solve indeterminate Quadratic equations including Pell's equation. Pell's equation is any Diophantine equation of the form x^2-ny^2=1\ where n is a nonsquare integer and x
• 820: The word algebra is derived from operations described in the treatise written by the Persian mathematician Muḥammad ibn Mūsā al-Ḵwārizmī titled Al-Kitab al-Jabr wa-l-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing") on the systematic solution of linear and quadratic equations. layout and formatting it should ensure no clashes with the top of the infobox A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. Al-Khwarizmi is often considered as the "father of algebra", much of whose works on reduction was included in the book and added to many methods we have in algebra now.
• Circa 850: Persian mathematician al-Mahani conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. layout and formatting it should ensure no clashes with the top of the infobox Abu-Abdullah Muhammad ibn Īsa Māhānī, was a Persian mathematician and astronomer from Mahan, Kerman, Persia. Doubling the cube (also known as The Delian Problem) is one of the three most famous geometric problems unsolvable by Compass and straightedge construction
• Circa 850: Indian mathematician Mahavira solves various quadratic, cubic, quartic, quintic and higher-order equations, as well as indeterminate quadratic, cubic and higher-order equations. Mahavira was a 9th century Indian Mathematician from Gulbarga who asserted that the Square root of a Negative number did
• Circa 990: Persian Abu Bakr al-Karaji, in his treatise al-Fakhri, further develops algebra by extending Al-Khwarizmi's methodology to incorporate integral powers and integral roots of unknown quantities. layout and formatting it should ensure no clashes with the top of the infobox (or) (c 953 in Karaj or Karkh &ndash c 1029 was a 10th century Persian Muslim mathematician and engineer. He replaces geometrical operations of algebra with modern arithmetical operations, and defines the monomials x, x², x³, . In Mathematics, the word monomial means two different things in the context of Polynomials The first meaning is a product of powers of Variables . . and 1/x, 1/x², 1/x³, . . . and gives rules for the products of any two of these.
• Circa 1050: Chinese mathematician Jia Xian finds numerical solutions of polynomial equations. Jia Xian (贾宪 was a Chinese mathematician of the Song Dynasty, first half of 11th century
• 1072: Persian mathematician Omar Khayyam develops algebraic geometry and, in the Treatise on Demonstration of Problems of Algebra, gives a complete classification of cubic equations with general geometric solutions found by means of intersecting conic sections. layout and formatting it should ensure no clashes with the top of the infobox For the Thoroughbred racehorse see Omar Khayyam (horse Ghiyās od-Dīn Abol-Fath Omār ibn Ebrāhīm Khayyām Neyshābūri (غیاث الدین
• 1114: Indian mathematician Bhaskara, in his Bijaganita (Algebra), recognizes that a positive number has both a positive and negative square root, and solves various cubic, quartic and higher-order polynomial equations, as well as the general quadratic indeterminant equation. In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose
• 1202: Algebra is introduced to Europe largely through the work of Leonardo Fibonacci of Pisa in his work Liber Abaci. Leonardo of Pisa (c 1170 – c 1250 also known as Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or most commonly simply Fibonacci Pisa is a city in Tuscany, central Italy, on the right bank of the mouth of the Arno River on the Ligurian Sea. Liber Abaci (1202 also spelled as Liber Abbaci) is an historic book on Arithmetic by Leonardo of Pisa known later by his nickname Fibonacci
• Circa 1300: Chinese mathematician Zhu Shijie deals with polynomial algebra, solves quadratic equations, simultaneous equations and equations with up to four unknowns, and numerically solves some quartic, quintic and higher-order polynomial equations. Zhu Shijie ( fl 13th century) Courtesy name Hanqing (汉卿 Pseudonym Songting (松庭 was one of the greatest Chinese In Mathematics, the symmetric algebra S ( V) (also denoted Sym ( V) on a Vector space V over a field In Mathematics, a quintic equation is a Polynomial Equation of degree five
• Circa 1400: Indian mathematician Madhava of Sangamagramma finds iterative methods for approximate solution of non-linear equations. Mādhava of Sangamagrama (born as Irinjaatappilly Madhavan Namboodiri) (c In Computational mathematics, an iterative method attempts to solve a problem (for example an equation or system of equations by finding successive Approximations
• Circa 1450: Arab mathematician Abū al-Hasan ibn Alī al-Qalasādī took "the first steps toward the introduction of algebraic symbolism. Abū al-Hasan ibn ʿAlī al-Qalaṣādī (1412 in Baza, Spain &ndash 1486 in Béja, Tunisia) was an Arab Muslim mathematician See also Table of mathematical symbols Mathematical notation is used in Mathematics, and throughout the Physical sciences, Engineering " He represented mathematical symbols using characters from the Arabic alphabet. This is a listing of common symbols found within all branches of the science of Mathematics. The Arabic alphabet is the script used for writing several languages of Asia and Africa such as Arabic, Persian, and Urdu. ^[5]
• 1535: Nicolo Fontana Tartaglia and others mathematicians in Italy independently solved the general cubic equation. Niccolò Fontana Tartaglia (1499/1500 Brescia, Italy &ndash December 13, 1557, Venice, Italy was a Mathematician ^[6]
• 1545: Girolamo Cardano publishes Ars magna -The great art which gives Fontana's solution to the general quartic equation. ^[6]
• 1572: Rafael Bombelli recognizes the complex roots of the cubic and improves current notation. Rafael Bombelli (1526–1572 was an Italian Mathematician. Born in Bologna, he is the author of a treatise on Algebra and is a central figure
• 1591: Francois Viete develops improved symbolic notation for various powers of an unknown and uses vowels for unknowns and consonants for constants in In artem analyticam isagoge. François Viète (or Vieta) seigneur de la Bigotière ( 1540 - February 13, 1603) generally known as Franciscus Vieta,
• 1631: Thomas Harriot in a posthumous publication uses exponential notation and is the first to use symbols to indicate "less than" and "greater than". Thomas Harriot ( c 1560 – 2 July 1621) was an English astronomer, Mathematician, Ethnographer, and Translator
• 1682: Gottfried Wilhelm Leibniz develops his notion of symbolic manipulation with formal rules which he calls characteristica generalis.
• 1680s: Japanese mathematician Kowa Seki, in his Method of solving the dissimulated problems, discovers the determinant, and Bernoulli numbers. or (born 1637/1642? – October 24, 1708) was a Japanese Mathematician who created a new algebraic notation system and laid In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In Mathematics, the Bernoulli numbers are a Sequence of Rational numbers with deep connections to Number theory. ^[7]
• 1750: Gabriel Cramer, in his treatise Introduction to the analysis of algebraic curves, states Cramer's rule and studies algebraic curves, matrices and determinants. Gabriel Cramer ( July 31, 1704 - January 4, 1752) was a Swiss Mathematician, born in Geneva. Cramer's rule is a Theorem in Linear algebra, which gives the solution of a System of linear equations in terms of Determinants It is named after In Algebraic geometry, an algebraic curve is an Algebraic variety of dimension one
• 1824: Niels Henrik Abel proved that the general quintic equation is insoluble by radicals. Niels Henrik Abel (August 5 1802 &ndash April 6 1829 was a noted Norwegian Mathematician who proved the impossibility of solving the Quintic equation ^[6]
• 1832: Galois theory is developed by Évariste Galois in his work on abstract algebra. ^[6]

See also

• List of basic algebra topics
• List of mathematics articles
• Fundamental theorem of algebra
• Computer algebra system
• Order of operations

References

• Donald R. Hill, Islamic Science and Engineering (Edinburgh University Press, 1994).
• Ziauddin Sardar, Jerry Ravetz, and Borin Van Loon, Introducing Mathematics (Totem Books, 1999).
• George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics (Penguin Books, 2000). Penguin Books is a British Publisher founded in 1935 by Allen Lane.
• John J O'Connor and Edmund F Robertson, MacTutor History of Mathematics archive (University of St Andrews, 2005). The MacTutor History of Mathematics archive is an award-winning website maintained by John J The University of St Andrews is the oldest University in Scotland and third oldest in the English-speaking world, having been founded between
• I. N. Herstein: Topics in Algebra. ISBN 0-471-02371-X
• R. B. J. T. Allenby: Rings, Fields and Groups. ISBN 0-340-54440-6
• L. Euler: Elements of Algebra, ISBN 978-1-89961-873-6

External links

• 4000 Years of Algebra, lecture by Robin Wilson, at Gresham College, 17th October 2007 (available for MP3 and MP4 download, as well as a text file). Gresham College is an unusual institution of higher learning off Holborn in central London.
• ExampleProblems.com Example problems and solutions from basic and abstract algebra.
• Algebra entry at the Stanford Encyclopedia of Philosophy by Vaughan Pratt

The Stanford Encyclopedia of Philosophy (SEP is a freely-accessible Online encyclopedia of Philosophy maintained by Stanford University.

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