algebra

Algebra is a branch of mathematics that studies symbols and the rules for manipulating those symbols. It generalizes arithmetic by focusing on relationships and structures rather than particular numbers, allowing the expression of general rules and patterns.

Historically, algebra originated in ancient civilizations and was advanced in the Islamic Golden Age by scholars such as al-Khwarizmi; the term algebra derives from al-jabr, in his 9th-century treatise. The subject was further developed in Europe during the Renaissance and early modern period by mathematicians such as Viète, Descartes, and Euler, culminating in the modern abstract algebra of 19th and 20th centuries.

Branches include elementary algebra, which deals with manipulating symbols, solving equations, and understanding polynomials; linear algebra, which studies vectors, vector spaces, and linear transformations; abstract algebra, which studies algebraic structures such as groups, rings, fields, and modules; and related fields like boolean algebra and computational algebra.

Core concepts include variables, constants, expressions, equations, identities, polynomials, functions, and systems of equations. Methods range from factoring and solving linear and polynomial equations to more advanced procedures such as Gaussian elimination and, in the abstract setting, the study of homomorphisms and structure-preserving maps.

Applications appear across science and engineering, in computer science for algorithms and cryptography, and in economics and statistics for modeling. Algebra provides a framework for describing and analyzing quantities and their relationships in a precise and general way.