User:Phlsph7/Algebra - Lead

Algebra is the branch of mathematics that studies algebraic systems and the manipulation of equations within those systems. It is a generalization of arithmetic that includes the use of variables besides regular numbers and algebraic operations other than the standard arithmetic operations like addition and multiplication.

Elementary algebra is the main form of algebra taught in school and examines mathematical statements using variables for unspecified values. It seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field investigating variables that appear in several linear equations, so-called systems of linear equations. It tries to discover the values that solve all equations at the same time.

Abstract algebra studies algebraic structures, which consist of a set of mathematical objects together with one or several binary operations defined on that set. It is a generalization of elementary and linear algebra since it allows mathematical objects other than numbers and non-arithmetic operations. It distinguishes between different types of algebraic structures, such as groups, rings, and fields, based on the number of operations they use and the laws they follow. Universal algebra constitutes a further level of generalization that is not limited to binary operations and investigates more abstract patterns that characterize algebraic structures.

Algebraic methods were first studied in the ancient period to solve specific problems in fields like geometry. Subsequent mathematicians examined general techniques to solve equations independent of their specific applications. They relied on verbal descriptions of problems and solutions until the 16th and 17th centuries, when a rigorous mathematical formalism was developed. In the mid-19th century, the scope of algebra broadened beyond a theory of equations to cover diverse types of algebraic operations and algebraic structures.

Algebra is relevant to many branches of mathematics, like geometry, topology, number theory, and calculus, and other fields of inquiry, like logic and the empirical sciences.