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.
Other articles related to "algebra, algebras":
... In mathematics, a topological algebra A over a topological field K is a topological vector space together with a continuous multiplication that makes it an algebra over K ... A unital associative topological algebra is a topological ring ... An example of a topological algebra is the algebra C of continuous real-valued functions on the closed unit interval, or more generally any Banach algebra ...
... A ring has two binary operations (+) and (×), with × distributive over + ... Under the first operator (+) it forms an abelian group ...
... was first defined in 1844 by German mathematician Hermann Grassmann in exterior algebra as the result of the exterior product of two vectors ... William Kingdon Clifford in 1888 added the geometric product to Grassmann's algebra, incorporating the ideas of both Hamilton and Grassmann, and founded Clifford algebra, that ... Today the bivector is largely studied as a topic in geometric algebra, a Clifford algebra over real or complex vector spaces with a nondegenerate quadratic form ...
... such as lists and trees, can be obtained as initial algebras of specific endofunctors ... While there may be several initial algebras for a given endofunctor, they are unique up to isomorphism, which informally means that the "observable" properties of a data structure can be ... one function, they give , which makes this an F-algebra for the endofunctor F sending to ...
... Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings ... Both algebraic geometry and algebraic number theory build on commutative algebra ...
Famous quotes containing the word algebra:
“Poetry has become the higher algebra of metaphors.”
—José Ortega Y Gasset (18831955)