Algebra is related to mathematics, but for historical reasons, the word "algebra" has three meanings as a bare word, depending on the context. The word also constitutes various terms in mathematics, showing more variation in the meaning. This article gives a broad overview of them, including the history.
As a single word, "algebra" can mean:
- Use of letters and symbols to represent values and their relations, especially for solving equations. This is also called "Elementary algebra". Historically, this was the meaning in pure mathematics too, like seen in "fundamental theorem of algebra", but not now.
- In modern pure mathematics,
- a major branch of mathematics which studies relations and operations. It's sometimes called abstract algebra, or "modern algebra" to distinguish it from elementary algebra.
- a mathematical structure as a "linear" ring, is also called "algebra," or sometimes "algebra over a field", to distinguish it from its generalizations.
The adjective "algebraic" usually means relation to abstract algebra, as in "algebraic structure". But in some cases it refers to equation solving, reflecting the evolution of the field.
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 as a branch of mathematics is much broader than elementary algebra, studying what happens, beyond arithmetics of normal numbers, when different rules of operations and relations are used. It leads to constructions and concepts arising from them, including terms, polynomials, equations. When the rules of addition and multiplication are generalized, their precise definitions lead to the notions of algebraic structures such as groups, rings and fields, studied in the area of mathematics called abstract algebra.
Algebra is one of the main branches of pure mathematics, together with geometry, analysis, topology, combinatorics, and number theory.
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