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Algebra

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   Algebra is a branch of mathematics concerning the study of structure,
   relation and quantity. The name is derived from 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"), which provided symbolic operations for the
   systematic solution of linear and quadratic equations.

   Together with geometry, analysis, and number theory, algebra is one of
   the several 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.

   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.

Classification

   Linear algebra lecture about determinants and inverse matrices.
   Enlarge
   Linear algebra lecture about determinants and inverse matrices.

   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);
     * Abstract algebra, sometimes also called modern algebra, in which
       algebraic structures such as groups, rings and fields are
       axiomatically defined and investigated;
     * 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.

   In advanced studies, axiomatic algebraic systems such as groups, rings,
   fields, and algebras over a field are investigated in the presence of a
   natural geometric structure (a topology) which is compatible with the
   algebraic structure. The list includes a number of areas of functional
   analysis:
     * Normed linear spaces
     * Banach spaces
     * Hilbert spaces
     * Banach algebras
     * Normed algebras
     * Topological algebras
     * Topological groups

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. Although in arithmetic, only
   numbers and their arithmetical operations (such as +, −, ×, ÷) occur,
   in algebra, numbers are often denoted by symbols (such as a, x, 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.
     * 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 the
       function is performed on.").

Abstract algebra

   Abstract algebra extends the familiar concepts found in elementary
   algebra and arithmetic of numbers to more general concepts.

   Sets: Rather than just considering the different types of numbers,
   abstract algebra deals with the more general concept of sets: a
   collection of objects called elements. All 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 (ax^2 +
   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.

   Binary operations: The notion of addition (+) is abstracted to give a
   binary operation, * say. For two elements a and b in a set S a*b gives
   another element in the set, (technically this condition is called
   closure). Addition (+), subtraction (-), multiplication (×), and
   division (÷) are all binary operations as is addition and
   multiplication of matrices, vectors, and polynomials.

   Identity elements: The numbers zero and one are abstracted to give the
   notion of an identity element. 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. 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. 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.

   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.

Groups

   Combining the above concepts gives one of the most important structures
   in mathematics: a group. A group is a combination of a set S and a
   binary operation '*' with the following properties:
     * The operation is closed: if a and b are members of S, then so is a
       * b.

                In fact, it is redundant to mention this property, for
                every binary operation must be closed. So, the statement
                "a group is a combination of a set S and a binary
                operation '*'" is already saying that the operation is
                closed. However, closure is frequently emphasized
                repeating it as a group property.

     * 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.

   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.

   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.
   Examples
   Set: 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}
   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 a \begin{matrix} \frac{1}{a} \end{matrix} a
   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.

   All groups are monoids, and all monoids are semigroups.

Rings and fields—structures with two binary operations

   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.

   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 +.

   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.

   The integers are an example of a ring. The integers have additional
   properties which make it an integral domain.

   A field is a ring with the additional property that all the elements
   excluding 0 form an Abelian group under ×. The multiplicative (×)
   identity is written as 1 and the multiplicative inverse of a is written
   as a^-1.

   The rational numbers, real number and complex numbers are all examples
   of fields.

Algebras

   The word algebra is also used for various algebraic structures:
     * Algebra over a field
     * Algebra over a set
     * Boolean algebra
     * F-algebra and F-coalgebra in category theory
     * Sigma-algebra

History

   Hellenistic mathematician Euclid details geometrical algebra in
   Elements.
   Enlarge
   Hellenistic mathematician Euclid details geometrical algebra in
   Elements.

   The origins of algebra can be traced to the ancient Babylonians, 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.

   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 Muslim mathematician Muhammad
   ibn Mūsā al-khwārizmī in 820. 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 from Diophantus. 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. While 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. Another Persian mathematician Omar
   Khayyam developed algebraic geometry and found the general geometric
   solution of the cubic equation. The Indian mathematicians Mahavira and
   Bhaskara, and the Chinese mathematician Zhu Shijie, solved various
   cubic, quartic, quintic and higher-order polynomial equations.

   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.

   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 and in the Bakhshali
       Manuscript; and
     * Symbolic algebra, which sees its culmination in the work of
       Leibniz.

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

   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.
     * 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 ax^2 = c and ax^2 + 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: Hellenized Egyptian mathematician Hero of Alexandria,
       treats algebraic equations in three volumes of mathematics.
     * Circa 200: Hellenized Babylonian mathematician 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, gives integral
       solutions of simultaneous indeterminate linear equations, and
       describes a differential equation.
     * 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. He discovers that quadratic
       equations have two roots, including both negative as well as
       irrational roots.
     * 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, x^2,
       x^3, ... and 1/x, 1/x^2, 1/x^3, ... 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 quadratic equations with more than one
       unknown, various cubic, quartic and higher-order polynomial
       equations, Pell's equation, the general indeterminate quadratic
       equation, as well as indeterminate cubic, quartic and higher-order
       equations.
     * 1150: Bhaskara, in his Siddhanta Shiromani, solves differential
       equations.
     * 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 the
       solution of transcendental equations by iteration, iterative
       methods for the solution of non-linear equations, and solutions of
       differential equations.
     * 1515: Scipione del Ferro solves a cubic such that the quadratic
       term is missing.
     * 1535: Nicolo Fontana Tartaglia solves a cubic such that the linear
       term is missing.
     * 1545: Girolamo Cardano publishes Ars magna -The great art which
       gives solutions for a variety of cubics as well as Ludovico
       Ferrari's solution of a special quartic equation.
     * 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 posthumus 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.
     * 1683: Japanese mathematician Kowa Seki, in his Method of solving
       the dissimulated problems, discovers the determinant, discriminant,
       and Bernoulli numbers.
     * 1685: Kowa Seki solves the general cubic equation, as well as some
       quartic and quintic equations.
     * 1693: Leibniz solves systems of simultaneous linear equations using
       matrices and determinants.
     * 1750: Gabriel Cramer, in his treatise Introduction to the analysis
       of algebraic curves, states Cramer's rule and studies algebraic
       curves, matrices and determinants.
     * 1830: Galois theory is developed by Évariste Galois in his work on
       abstract algebra.

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