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Together with geometry, analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics. 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. For other uses, see Geometry (disambiguation). ... 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 of problems that arise from their study. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. ... Secondary education - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ... In mathematics, multiplication is an elementary arithmetic operation. ... For other uses, see Number (disambiguation). ... In computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ...

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. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields. A symbol or (in many senses) token is a representation of something — an idea, object, concept, quality, etc. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, an element (also called a member) is an object contained in a set (or more generally a class). ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

Algebra may be divided roughly into the following categories:

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. The list includes a number of areas of functional analysis: Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... In computer science and mathematics, a variable (pronounced ) (sometimes called an object or identifier in computer science) is a symbolic representation used to denote a quantity or expression. ... A mathematical expression is a string of symbols which describes (or expresses) a (potential or actual) computation using operators and operands. ... An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ... This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, axiomatization is the process of defining the basic axiomatic systems from which mathematical theories can be derived. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... 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, any abstract quantities that can be added and multiplied. ... Universal algebra (sometimes called General algebra) is the field of mathematics that studies the ideas common to all algebraic structures. ... This article or section does not cite its references or sources. ... 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 of problems that arise from their study. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ... Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ... For other uses, see Geometry (disambiguation). ... In mathematics a metric or distance function is a function which defines a distance between elements of a set. ... For other uses, see Topology (disambiguation). ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...

In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ... In mathematics, Banach spaces (pronounced ), named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ... In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ... Wikipedia does not yet have an article with this exact name. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G Ã— G â†’ G and the inverse operation G â†’ G are continuous maps. ...

## Elementary algebra

Main article: 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 beyond the basic principles of arithmetic. In arithmetic, only numbers and their arithmetical operations (such as +, −, ×, ÷) occur. In algebra, numbers are often denoted by symbols (such as a, x, or y). This is useful because: Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ... For other uses, see Number (disambiguation). ...

• It allows the general formulation of arithmetical laws (such as a + b = b + a for all a and b), and thus is the first step to a systematic exploration of the properties of the real number system.
• It allows the reference to "unknown" numbers, the formulation of equations and the study of how to solve these (for instance, "Find a number x such that 3x + 1 = 10").
• 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.").

In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ... This article is about functions in mathematics. ...

### 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). For example, $x^2 + 2x -3,$ is a polynomial in the single variable x. In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In mathematics, the word expression is a very general term for any well-formed combination of mathematical symbols. ... In computer science and mathematics, a variable (pronounced ) (sometimes called an object or identifier in computer science) is a symbolic representation used to denote a quantity or expression. ...

An important class of problems in algebra is factorization of polynomials, that is, expressing a given polynomial as a product of other polynomials. The example polynomial above can be factored as $(x-1)(x+3),!.$ A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ...

## Abstract algebra

Main article: Abstract algebra

Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ... For other uses, see Number (disambiguation). ...

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. 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. Set theory is a branch of logic and not technically a branch of algebra. In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... For other uses, see Number (disambiguation). ... In mathematics, an element (also called a member) is an object contained in a set (or more generally a class). ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... This article is about vectors that have a particular relation to the spatial coordinates. ... 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 element g (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value â€” the modulus. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Logic (from Classical Greek Î»ÏŒÎ³Î¿Ï‚ logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ...

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. 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. For other uses, see identity (disambiguation). ...

Inverse elements: The negative numbers give rise to the concept of inverse elements. For addition, the inverse of a is -a, and for multiplication the inverse is 1/a. A general inverse element a-1 must satisfy the property that a * a-1 = e and a-1 * a = e. In mathematics, the idea of inverse element generalises the concepts of negation, in relation to addition, and reciprocal, in relation to multiplication. ...

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: (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, associativity is a property that a binary operation can have. ... In mathematics, the octonions are a nonassociative extension of the quaternions. ...

Commutativity: Addition of integers also has a property called commutativity. 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 . Mathematical meaning A map or binary operation is said to be commutative when, for any x in A and any y in B . ... In mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix. ... Graphical representation of quaternion units product as 90Â°-rotation in 4D-space, ij = k, ji = -k, ij = -ji This page describes the mathematical entity. ...

### Groups – structures of a set with a single binary operation

Main article: Group (mathematics)

Combining the above concepts gives one of the most important structures in mathematics: a group. A group is a combination of a set S and a single binary operation '*', defined in any way you choose, but with the following properties: This picture illustrates how the hours on a clock form a group under modular addition. ... Group theory is that branch of mathematics concerned with the study of groups. ... Some elementary examples of groups in mathematics are given on Group (mathematics). ... This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ...

• An identity element e exists, such that for every member a of S, e * a and a * e are both identical to a.
• Every element has an inverse: for every member a of S, there exists a member a-1 such that a * a-1 and a-1 * a are both identical to the identity element.
• The operation is associative: if a, b and c are members of S, then (a * b) * c is identical to a * (b * c).

If a group is also commutative - 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. Example showing the commutativity of addition (3 + 2 = 2 + 3) For other uses, see Commute (disambiguation). ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ...

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. 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. In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...

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. 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. Group theory is that branch of mathematics concerned with the study of groups. ... The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the 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, ..., n} where is a natural number. ... In mathematics, a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself. ...

 Set: Operation Closed Examples Natural numbers $mathbb{N}$ Integers $mathbb{Z}$ Rational numbers $mathbb{Q}$ (also real $mathbb{R}$ and complex $mathbb{C}$ numbers) Integers mod 3: {0,1,2} + × (w/o zero) + × (w/o zero) + − × (w/o zero) ÷ (w/o zero) + × (w/o zero) Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Identity 0 1 0 1 0 NA 1 NA 0 1 Inverse NA NA -a NA -a NA $begin{matrix} frac{1}{a} end{matrix}$ 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 ($mathbb{Z}_2$)

Semigroups, quasigroups, and monoids are structures similar to groups, but more general. They comprise a set and a closed binary operation, but do not necessarily satisfy the other conditions. A semigroup has an associative binary operation, but might not have an identity element. A monoid is a semigroup which does have an identity but might not have an inverse for every element. 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. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... Please refer to Real vs. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ... In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ... In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ... In mathematics, a semigroup is an algebraic structure consisting of a set S closed under an associative binary operation. ... In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics, a semigroup is an algebraic structure consisting of a set S closed under an associative binary operation. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ...

All groups are monoids, and all monoids are semigroups.

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

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

Groups just have one binary operation. To fully explain the behaviour of the different types of numbers, structures with two operators need to be studied. The most important of these are rings, and fields. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... 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 a branch of mathematics which studies the properties of fields. ... Field theory is the branch of mathematics in which fields are studied. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

Distributivity generalised the distributive law for numbers, and specifies the order in which the operators should be applied, (called the precedence). 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 +. In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... In arithmetic and algebra, when a number or expression is both preceded and followed by a binary operation, a rule is required for which operation should be applied first. ...

A ring has two binary operations (+) and (×), with × distributive over +. Under the first operator (+) it forms an Abelian group. Under the second operator (×) it is associative, but it does not need to have 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. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

The integers are an example of a ring. The integers have additional properties which make it an integral domain. In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 â‰  1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ...

A field is a ring with the additional property that all the elements excluding 0 form an Abelian group under ×. The multiplicative (×) identity is written as 1 and the multiplicative inverse of a is written as a-1. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

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: In abstract algebra, an algebraic structure consists of a set together with a collection of operations or relations defined on it which satisfy certain axioms. ...

In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R. In this article, all rings and algebras are assumed to be unital and associative. ... In mathematics a field of sets is a pair where is a set and is an algebra over i. ... In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ... In mathematics, specifically in category theory, an -coalgebra for an endofunctor is an object of together with a -morphism . In this sense F-coalgebras are dual to F-algebras. ... In mathematics, specifically in category theory, an -coalgebra for an endofunctor is an object of together with a -morphism . In this sense F-coalgebras are dual to F-algebras. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... In category theory, a monad or triple is a type of functor, together with two associated natural transformations. ...

## History

Main article: History 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. 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. 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. 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. 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: Algebra History of Mathematics History of Algebra ^ (Hayashi 2005, p. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... A stamp issued September 6, 1983 in the Soviet Union, commemorating al-KhwÄrizmÄ«s (approximate) 1200th anniversary. ... A page from the book (Arabic for The Compendious Book on Calculation by Completion and Balancing), also known under a shorter name spelled as Hisab al-jabr wâ€™al-muqabala, Kitab al-Jabr wa-l-Muqabala and other transliterations) is a mathematical book written approximately 820 AD by the Persian... Babylonian clay tablet YBC 7289 with annotations. ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ... Graph sample of 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 is an equation for which there is an infinite set of solutions â€“ for example, 2x = y. ... The 1st millennium BC encompasses the Iron Age and sees the rise of successive empires. ... For other uses, see Geometry (disambiguation). ... The Rhind Mathematical Papyrus ( papyrus British Museum 10057 and pBM 10058), is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. ... The Sulba Sutras or Sulva Sutras are a text of Vedic mathematics. ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... The Nine Chapters on the Mathematical Art (ä¹ç« ç®—è¡“) is a Chinese mathematics book, probably composed in the 1st century AD, but perhaps as early as 200 BC. This book is the earliest surviving mathematical text from China that has come down to us by being copied by scribes and (centuries later...

Later, the Indian mathematicians developed algebraic methods to a high degree of sophistication. Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Brahmagupta was the first to solve equations using general methods. He solved the linear indeterminate equations, quadratic equations, second order indeterminate equations and equations with multiple variable. This article is under construction. ... Brahmagupta (à¤¬à¥à¤°à¤¹à¥à¤®à¤—à¥à¤ªà¥à¤¤) ( ) (589â€“668) was an Indian mathematician and astronomer. ...

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. Gabriel Cramer also did some work on matrices and determinants in the 18th century. Abstract algebra was developed in the 19th century, initially focusing on what is now called Galois theory, and on constructibility issues. In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... Japanese mathematics or wasan (和算) is a kind of mathematics developed in Japan during Edo Period (1603-1867) based on Chinese mathematics. ... Kowa Seki (Seki Takakazu, 関 孝和) (1642? – October 24, 1708) was a Japanese mathematician. ... Leibniz redirects here. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... Gabriel Cramer Gabriel Cramer (July 31, 1704 - January 4, 1752) was a Swiss mathematician, born in Geneva. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In mathematics, more specifically in abstract algebra, Galois theory, named after Ã‰variste Galois, provides a connection between field theory and group 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), one can construct the point with unruled straightedge and compass. ...

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.
Cover of the 1621 edition of Diophantus's Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.

• Circa 1800 BC: The Old Babylonian Strassburg tablet seeks the solution of a quadratic elliptic equation.
• Circa 1600 BC: The Plimpton 322 tablet gives a table of Pythagorean triples in Babylonian Cuneiform script.
• 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.
• 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. 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.
• 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.
• 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.
• Circa 150 AD: Hero of Alexandria treats algebraic equations in three volumes of mathematics.
• 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.
• 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.
• 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.
• 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. 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.
• 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.
• 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. He replaces geometrical operations of algebra with modern arithmetical operations, and defines the monomials x, x2, x3, ... 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.
• 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.
• 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.
• 1202: Algebra is introduced to Europe largely through the work of Leonardo Fibonacci of Pisa in his work Liber Abaci.
• 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.
• Circa 1400: Indian mathematician Madhava of Sangamagramma finds iterative methods for approximate solution of non-linear equations.
• Circa 1450: Arab mathematician Abū al-Hasan ibn Alī al-Qalasādī took "the first steps toward the introduction of algebraic symbolism." He represented mathematical symbols using characters from the Arabic alphabet.[5]
• 1535: Nicolo Fontana Tartaglia and others mathematicians in Italy independently solved the general cubic equation.[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.
• 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.
• 1631: Thomas Harriot in a posthumous publication uses exponential notation and is the first to use symbols to indicate "less than" and "greater than".
• 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.[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.
• 1824: Niels Henrik Abel proved that the general quintic equation is insoluble by radicals.[6]
• 1832: Galois theory is developed by Évariste Galois in his work on abstract algebra.[6]

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Image File history File links Wikibooks-logo-en. ... Wikibooks logo Wikibooks, previously called Wikimedia Free Textbook Project and Wikimedia-Textbooks, is a wiki for the creation of books. ... Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 151 languages. ... Algebra is one of the main branches of mathematics, and concerns the study of structure, relation and quantity. ... These list of mathematics articles pages collect pointers to all articles related to mathematics. ... In mathematics, the fundamental theorem of algebra states that every complex polynomial in one variable and of degree  â‰¥  has some complex root. ... A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. ... In arithmetic and algebra, when a number or expression is both preceded and followed by a binary operation, a rule is required for which operation should be applied first. ...

## Notes

1. ^ See the references Toomer, Hogendijk 1998, Oaks cited in the article on al-Khwārizmī.
2. ^ Struik, Dirk J. (1987). A Concise History of Mathematics. New York: Dover Publications.
3. ^ Carl B. Boyer, A History of Mathematics, Second Edition (Wiley, 1991), pages 178, 181
4. ^ Carl B. Boyer, A History of Mathematics, Second Edition (Wiley, 1991), page 228
5. ^ a b O'Connor, John J. & Robertson, Edmund F., “Abu'l Hasan ibn Ali al Qalasadi”, MacTutor History of Mathematics archive
6. ^ a b c d Stewart, Ian, Galois Theory, Third Edition (Chapman & Hall/CRC Mathematics, 2004).
7. ^ O'Connor, John J., and Edmund F. Robertson. Takakazu Seki Kowa. MacTutor History of Mathematics archive.

al-KhwÄrizmÄ« redirects here. ... The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ...

## 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).
• John J O'Connor and Edmund F Robertson, MacTutor History of Mathematics archive (University of St Andrews, 2005).
• 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

It has been suggested that Penguin Modern Poets, Penguin Great Ideas be merged into this article or section. ... The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ... St Marys College Bute Medical School St Leonards College[5][6] Affiliations 1994 Group Website http://www. ... Euler redirects here. ...

Results from FactBites:

 PlanetMath: algebra (201 words) This is version 10 of algebra, born on 2001-10-19, modified 2006-07-31. Object id is 353, canonical name is Algebra. While this entry may be technically correct, I still woulden't know what algebra was by reading it ;) May I suggest a short history of the invention of algebra, as well something to to tune of "a system to manipulate mathematical formulas"
 Highlights in the History of Algebra (1896 words) The development of algebraic notation progressed through three stages: the rhetorical (or verbal) stage, the syncopated stage (in which abbreviated words were used), and the symbolic stage with which we are all familiar. Renaissance mathematics was to be characterized by the rise of algebra. This advance freed algebra from the consideration of particular equations and thus allowed a great increase in generality and opened the possibility for studying the relationship between the coefficients of an equation an the roots of the equation ("theory of equations").
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