algebra

the mathematics of generalized arithmetical operations

This article is about the branch of mathematics. For other uses of the term see algebra (disambiguation).''Algebra is a branch of mathematics which studies structure and quantity. It may be roughly characterized as a generalization and abstraction of arithmetic, in which operatoroperations are performed on symbols rather than numbers. It includes elementary algebra, taught to high school students, as well as abstract algebra which covers such structures as group (mathematics)groups, ring (mathematics)rings and field (mathematics)fields. Along with geometry and mathematical analysisanalysis, it is one of the three main branches of mathematics.

History - The origins of algebra can be traced to the cultures of the ancient Egyptians and Babylonians who used an early type of algebra to solve linear equationlinear, quadratic equationquadratic, and indeterminate equations more than 3,000 years ago.

Circa 300 BC: GreeksGreek mathematician Euclid in Book 2 of his Euclid's Elements''Elements'' addresses quadratic equations, although in a strictly geometrical fashion.

Circa 100 BC: algebraic equations are treated in the ChinaChinese mathematics book Jiuzhang suanshu, ''The Nine Chapters of Mathematical Art''.

Circa 150150 AD: Greek mathematician Hero of Alexandria treats algebraic equations in three volumes of mathematics.

Circa 200200 AD: Greek mathematician Diophantus, often referred to as the "father of algebra", writes his famous ''Arithmetica'', a work featuring solutions of algebraic equations and on the theory of numbers.Indian mathematiciansIndian mathematician, Aryabhata (476 AD) obtained whole number solutions to linear equations by a method equivalent to the modern one. Bhaskara II (1114 AD), who wrote the text ''Bijaganita'' (algebra), was the first to recognize that a positive number has two square roots. The Hindus recognized that quadratic equations have two roots, and included Negative and non-negative numbersnegative as well as irrational roots. They treated indeterminate quadratic equations. The word ''algebra'' is derived from the name of the treatise first written by PersiansPersian mathematician Al-Khwarizmi Khwarazmi in 820 AD titled: ''Jabr o MuqabelehKetāb e mukhtasar dar Hesab o Jabr o Mughabeleh'' meaning ''The book of summary concerning calculating by transposition and reduction''. The word ''al-jabr'' (from which ''algebra'' is derived) means ''"reunion"'', ''"connection"'' or ''"completion"''.The name given by the Persian/Iranian scientist Khwarazmi. It entered English via Spanish LanguageSpanish.Algebra was introduced to Europe largely through the work of Leonardo Fibonacci of Pisa in his work ''Liber Abaci'' in 1202.

Classification - Algebra may be roughly divided into the following categories:

elementary algebra, in which the properties of operations on the real numberreal number system are recorded using symbols as "place holders" to denote mathematical constantconstants 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 group (mathematics)groups, ring (mathematics)rings and field (mathematics)fields are axiomatizationaxiomatically defined and investigated;

linear algebra, in which the specific properties of vector spaces are studied (including matrix (mathematics)matrices);

universal algebra, in which properties common to all algebraic structures are studied.In advanced studies, axiomatic algebraic systems like groups, rings, fields, and algebras over a field are investigated in the presence of a natural geometrygeometric structure (a topology) which is compatible with the algebraic structure. The list includes

Normed linear spaces

Banach spaces

Hilbert spaces

Banach algebras

Normed algebras

Topological algebras

Topological groups

Algebras - The word ''algebra'' is also used for various algebraic structures:

algebra over a field

algebra over a set

Boolean algebra

sigma-algebra

F-algebra and F-coalgebra in category theory

References -

Ziauddin Sardar, Jerry Ravetz, and Borin Van Loon, ''Introducing Mathematics'' (Totem Books, 1999).

Donald R. Hill, ''Islamic Science and Engineering'' (Edinburgh University Press, 1994).

George Gheverghese Joseph, ''The Crest of the Peacock : The Non-European Roots of Mathematics'' (Princeton University Press, 2000).

See also - book

Fundamental theorem of algebra (which is really a theorem of mathematical analysis, not of algebra)

Diophantus, "father of algebra"

Mohammed al-Khwarizmi, also known as "father of Algebra". math.umd.edu

Computer algebra system

ucs.louisiana.edu - Highlights in the history of algebra

External links -

mathleague.com - Explanation of Basic Topics

sparknotes.com - Sparknotes' Review of Algebra I and II

jamesbrennan.org - ''Understanding Algebra.'' An online algebra text by James W. !Brennan.Category:AlgebraCatego ry:Arabic? !wordsaf:Algebrafa:جبرbg:А� �гебраbn:বীজগণ� ��তca:Àlgebraco:Algebracs:A lgebrada:Algebrade:Algebraet:A lgebraes:Álgebraeo:Algebroeu: Algebrafr:Algèbreko:대수학 io:Algebroid:Aljabaris:Algebra it:Algebrahe:אלגברהla:Al gebralt:Algebramk:Алгебр аms:Algebranl:Algebraja:代� �学no:Algebrapl:Algebrapt:Ál gebraru:Алгебраsco:Alge brasimple:Algebrask:Algebrasr: Алгебраfi:Algebrasv:Alg ebratl:Aldyebravi:Đại? !sốtr:Cebiruk:Алгебраz h:代数

All Elementary Mathematics
Web high mathematical school. All sections of curriculum of elementary mathematics. Arithmetic, algebra, geometry, trigonometry,functions and graphs, analysis. Theory and solving problems. Versions of examination tests. Online consulting. Preparatory to universities and colleges.
http://www.bymath.com/

Algebra Help Software
Bagatrix designs basic math, pre-algebra, and algebra software that solves your homework problems with step-by-step help.
http://www.bagatrix.com/

The MathRoom.ca
Online math lessons and tutoring for High School and College math students
http://www.the-mathroom.ca/

Bluebit Software
Bluebit software offers mathematical software components and libraries.
http://www.bluebit.gr/

Interactive Mathematics Miscellany and Puzzles
A nearly encyclopedic collection of K12 math topics and problems, more than 700 of which are accompanied by interactive Java illustrations. Many problems come with multiple solutions and many theorems with multiple proofs.
http://www.cut-the-knot.org/

<a href=http://www.saroalioto.it/>Mathematics for the medium school</a>
Arithmetic, geometry and algebra with excel, sciences, maps, diagrams, hypertexts, play
http://www.saroalioto.it/

UbiMathPoint
Math resources for K12 and teachers. Papers, tests on line and exercises with solutions.
http://www.pernigo.com/math/default.aspx

The Algebra Page
Includes lessons, calculators and worksheets.
http://www.algebrahelp.com/

Algebra Homework Help, Online Solvers
Interactive homework problems. Topics include Pre-Algebra, linear Algebra and other college Algebra.
http://www.algebra.com/

S.O.S. Math
Contains tutorials covering algebra, geometry, calculus, differential equations, matrices, and complex variables. Reviews the most important results, techniques and formulas. Presented in worksheet format and require active participation. Includes practice quizzes and forum board.
http://www.sosmath.com/

Northern Illinois University
Department of Mathematical Sciences
http://www.math.niu.edu/