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This articles introduces “algebra” and its applications.
Algebra (from Arabic al-jebr meaning “reunion of broken parts”) is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure mathematics. Algebra has numerous usages in daily life and is commonly taught in public schools.
Elementary algebra, often part of the curriculum in secondary education, introduces the concept of variables representing numbers. Statements based on these variables are manipulated using the rules of operations that apply to numbers, such as addition. This can be done for a variety of reasons, including equation solving. Algebra is much broader than elementary algebra and studies what happens when different rules of operations are used and when operations are devised for things other than numbers. Addition and multiplication can be generalized and their precise definitions lead to structures such as groups, rings and fields, studied in the area of mathematics called abstract algebra.
With Babylonian roots, claims for the 3rd century Greek mathematician Diophantus and Muhammad ibn Musa al-Khwarizmi (from whose name “algorithm’ comes) to the title “father of algebra”, algebra expresses fundamental truths of abstraction that would occur in any culture that reached similar levels of mathematical sophistication as these are requisite for further development. Algebra itself becomes an object of study in modern topics such as universal algebra.
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. This is usually taught at school under the title algebra (or intermediate algebra and college algebra in subsequent years). University-level courses in group theory may also be called elementary 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.
- Algebraic number theory, in which the properties of numbers are studied through algebraic systems. Number theory inspired much of the original abstraction in algebra.
- Algebraic geometry applies abstract algebra to the problems of geometry.
- Algebraic combinatorics, in which abstract algebraic methods are used to study combinatorial questions.