Algebra

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 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 groups, rings and fields. Along with geometry and 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, quadratic, and indeterminate equations more than 3,000 years ago.

• Circa 300 BC: Greek mathematician Euclid, who taught and died at Alexandria in Egypt, in Book 2 of his Elements addresses quadratic equations, although in a strictly geometrical fashion.
• Circa 100 BC: algebraic equations are treated in the Chinese mathematics book Jiuzhang suanshu, The Nine Chapters of Mathematical Art.
• Circa 150 AD: Greek mathematician Hero of Alexandria treats algebraic equations in three volumes of mathematics.
• Circa 200 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.
• 476 AD Indian mathematician, Aryabhata obtains whole number solutions to linear equations by a method equivalent to 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 early Indians recognized that quadratic equations have two roots, and included negative as well as irrational roots. They treated indeterminate quadratic equations.
• 820 AD The word algebra is derived from the name of the treatise first written by Persian mathematician Khwarizmi titled: Al-Jabr wa-al-Muqabilah meaning The book of summary concerning calculating by transposition and reduction. The word al-jabr means "reunion". Much of Khwarizmi's works on reduction was included in the book and added to many methods we have in Algebra now.
• 1202 AD Algebra was introduced to Europe largely through the work of Leonardo Fibonacci of Pisa in his work Liber Abaci .

Classification

Algebra may be roughly divided 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 like 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:

• 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

Wikibooks has more about this subject:

Algebra

• 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". [1]
• Computer algebra system
• Highlights in the history of algebra

External links

• Explanation of Basic Topics
• Sparknotes' Review of Algebra I and II
• Understanding Algebra. An online algebra text by James W. Brennan.
• Algebra--the basic ideas     First of 6 parts in a short course on basic algebra at the high school level.