Algebra

: This article is about the branch of mathematics. For other uses of the term algebra from the Greek words gean ebrisko ,= AL-GEBRA (calculations in earth surface) and Arabic AL 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 operations 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), ring (mathematics) and field (mathematics). Along with geometry and mathematical analysis, 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 equation, quadratic equation, and indeterminate equations more than 3,000 years ago.

*Circa s, although in a strictly geometrical fashion.

*Circa , The Nine Chapters of Mathematical Art .

*Circa treats algebraic equations in three volumes of mathematics.

*Circa , a work featuring solutions of algebraic equations and on the theory of numbers.

Indian mathematicians, 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 as well as irrational roots. They treated indeterminate quadratic equations.

The word algebra itself is derived from the name of the treatise first written by 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 .

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 number are recorded using symbols as place holders to denote mathematical constant 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), ring (mathematics) and field (mathematics)s are axiomatization defined and investigated;

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

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 geometry 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=

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 . [http://www.math.umd.edu/~czorn/hist_algebra.pdf]

Computer algebra system

[http://www.ucs.louisiana.edu/~sxw8045/history.htm Highlights in the history of algebra]

=External links=

*[http://www.mathleague.com/help/algebra/algebra.htm Explanation of Basic Topics] *[http://www.sparknotes.com/math/#algebra1 Sparknotes Review of Algebra I and II]