Algebra

 Algebra, from the Arabic word al-jabr, meaning "reunion of broken parts," is the study of variables and the rules for manipulating these variables in formulas; it is the unifying thread of almost all of mathematics.

Elementary algebra deals with the manioulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields. Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of nathematics that belong to algebra, some having "algebra" in their name, such as communitative algebra, and some have not, such as Galois theory.

The word algebra is not only used for naming an area of mathematics and some subareas; it is also used for naming some sorts of algebraic structures, such as an algebra over a field, commonly called an algebra. Sometimes, the same phrase is used for a subarea and its main algebraic structures, for example Boolean algebra. A mathematician that specializes in algebra is called an algebraist.

Algebra began with computations similar to those of number theory, with letters standing as numbers. This allowed proofs of properties that are true no matter which numbers are involved.

Historically, and in current teaching, the study of algebra starts with the solving of equations. Then more general questions, such as "does an equation have a solution?", "how many solutions does an equation have?", "what can be said about the nature of the solutions?" are considered. These questions led extending algebra to non-numerical objects, such as permutations, vectors, matrices, and polynomials. The structural properties of those non-numerical objects were then formalized in algebraic structures.

Before the 16th century, mathematics was divided into only two subfields: arithmetic and geometry. Even though some methods, which has been developed much earlier, may be considered nowadays as algebra, the emergence of algebra and, soon thereafter, infinitesimal calculus as subfields of mathematics only dates from the 16th or 17th century. From the second half of the 19th century on, many new fields of mathematics appeared, most of which made use of arithmetic and geometry, and almost all of which used algebra.

Today, algebra has grown considerably and inclydes many branches of mathematics, as can be seen in the Mathematics Subject Classification were none of the first level areas (two digit areas) are called algebra.

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