# Algebra - Elementary Algebra, Applications, Graphing Algebraic Equations, Linear Algebra, Matrix Algebra, Abstract Algebra

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rules mathematical sets generalization

Algebra is often referred to as a generalization of **arithmetic**. As such, it is a collection of rules: rules for translating words into the symbolic notation of **mathematics**, rules for formulating mathematical statements using symbolic notation, and rules for rewriting mathematical statements in a manner that leaves their truth unchanged.

The power of elementary algebra, which grew out of a desire to solve problems in arithmetic, stems from its use of variables to represent numbers. This allows the generalization of rules to whole sets of numbers. For example, the solution to a problem may be the **variable** x or a rule such as ab=ba can be stated for all numbers represented by the variables a and b.

Elementary algebra is concerned with expressing problems in terms of mathematical symbols and establishing general rules for the combination and manipulation of those symbols. There is another type of algebra, however, called abstract algebra, which is a further generalization of elementary algebra, and often bears little resemblance to arithmetic. Abstract algebra begins with a few basic assumptions about sets whose elements can be combined under one or more binary operations, and derives theorems that apply to all sets, satisfying the initial assumptions.

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Algebra was popularized in the early ninth century by al-Khowarizmi, an Arab mathematician, and the author of the first algebra book, Al-jabr wa'l Muqabalah, from which the English word algebra is derived. An influential book in its day, it remained the standard text in algebra for a long time. The title translates roughly to "restoring and balancing," referring to the primary…

Applications of algebra are found everywhere. The principles of algebra are applied in all branches of mathematics, for instance, calculus, geometry, and topology. They are applied every day by men and women working in all types of business. As a typical example of applying algebraic methods, consider the following problem. A painter is given the job of whitewashing three billboards along the high…

The methods of algebra are extended to geometry, and vice versa, by graphing. The value of graphing is two-fold. It can be used to describe geometric figures using the language of algebra, and it can be used to depict geometrically the algebraic relationship between two variables. For example, suppose that Fred is twice the age of his sister Irma. Since Irma's age is unknown, Fred's …

Abstract algebra represents a further generalization of elementary algebra. By defining such constructs as groups, based on a set of initial assumptions, called axioms, provides theorems that apply to all sets satisfying the abstract algebra axioms. A group is a set of elements together with a binary operation that satisfies three axioms. Because the binary operation in question may be any of a nu…

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