Algebra

Algebra from bonesetting' is one of a broad areas of mathematics. Roughly speaking, algebra is the explore of mathematical symbols as well as the rules for manipulating these symbols in formulas; it is a unifying thread of most all of mathematics.

Elementary algebra deals with the manipulation of variables as if they were numbers see the image, in addition to is therefore essential in all applications of mathematics. Abstract algebra is the hold given in education to the examine of algebraic structures such(a) as groups, rings, & fields. Linear algebra, which deals with linear equations and linear mappings, is used for sophisticated presentations of geometry, and has numerous practical a formal request to be considered for a position or to be allowed to do or have something. in weather forecasting, for example. There are numerous areas of mathematics that belong to algebra, some having "algebra" in their name, such(a) as commutative algebra and some not, such as Galois theory.

The word algebra is non 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 and a Boolean algebra. A mathematician specialized in algebra is called an algebraist.

Algebra as a branch of mathematics

Algebra began with computations similar to those of arithmetic, with letters standing for numbers. This gives proofs of properties that are true no matter which numbers are involved. For example, in the quadratic equation

can be all numbers whatsoever except that cannot be , and the quadratic formula can be used to quickly and easily find the values of the unknown quantity which satisfy the equation. That is to say, to find all the solutions of the equation.

Historically, and in current teaching, the study of algebra starts with the solving of equations, such as the quadratic equation above. Then more general questions, such as "does an equation relieve oneself a solution?", "how many solutions does an equation have?", "what can be said approximately the types 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 these non-numerical objects were then formalized into algebraic structures such as groups, rings, and fields.

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

Today, algebra has grown considerably and includes many branches of mathematics, as can be seen in the Mathematics identified Classification where none of the number one level areas two digit entries are called algebra. Today algebra includes module 08-General algebraic systems, 12-Field theory and polynomials, 13-Commutative algebra, 15-Linear and multilinear algebra; matrix theory, 16-Associative rings and algebras, 17-Nonassociative rings and algebras, 18-Category theory; homological algebra, 19-K-theory and 20-Group theory. Algebra is also used extensively in 11-Number theory and 14-Algebraic geometry.