Algebra

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From Wikipedia, the free encyclopedia

For the kind of algebraic structure, see Algebra over a field. For other uses, see Algebra (disambiguation).

The quadratic formula expresses the solution of the equation ax² + bx + c = 0, where a is not zero, in terms of its coefficients a, b and c.

Algebra (from Arabic ‏الجبر‎ (al-jabr) 'reunion of broken parts,^[1] bonesetting'^[2]) is the study of variables and the rules for manipulating these variables in formulas;^[3] it is a unifying thread of almost all of mathematics.^[4]

Elementary algebra deals with the manipulation 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 mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some 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 and a Boolean algebra. A mathematician specialized in algebra is called an algebraist.

Etymology

The word algebra comes from the title of a book by Muhammad ibn Musa al-Khwarizmi.^[5]

The word algebra comes from the Arabic: الجبر, romanized: al-jabr, lit. 'reunion of broken parts,^[1] bonesetting^[2]' from the title of the early 9th century book ʿIlm al-jabr wa l-muqābala "The Science of Restoring and Balancing" by the Persian mathematician and astronomer al-Khwarizmi. In his work, the term al-jabr referred to the operation of moving a term from one side of an equation to the other, المقابلة al-muqābala "balancing" referred to adding equal terms to both sides. Shortened to just algeber or algebra in Latin, the word eventually entered the English language during the 15th century, from either Spanish, Italian, or Medieval Latin. It originally referred to the surgical procedure of setting broken or dislocated bones. The mathematical meaning was first recorded (in English) in the 16th century.^[6]

Different meanings of "algebra"

The word "algebra" has several related meanings in mathematics, as a single word or with qualifiers.

• As a single word without an article, "algebra" names a broad part of mathematics.
• As a single word with an article or in the plural, "an algebra" or "algebras" denotes a specific mathematical structure, whose precise definition depends on the context. Usually, the structure has an addition, multiplication, and scalar multiplication (see Algebra over a field). When some authors use the term "algebra", they make a subset of the following additional assumptions: associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word "algebra" refers to a generalization of the above concept, which allows for n-ary operations.
• With a qualifier, there is the same distinction:
  • Without an article, it means a part of algebra, such as linear algebra, elementary algebra (the symbol-manipulation rules taught in elementary courses of mathematics as part of primary and secondary education), or abstract algebra (the study of the algebraic structures for themselves).
  • With an article, it means an instance of some algebraic structure, like a Lie algebra, an associative algebra, or a vertex operator algebra.
  • Sometimes both meanings exist for the same qualifier, as in the sentence: Commutative algebra is the study of commutative rings, which are commutative algebras over the integers.

Algebra as a branch of mathematics

Algebra began with computations similar to those of arithmetic, with letters standing for numbers.^[7] This allowed proofs of properties that are true no matter which numbers are involved. For example, in the quadratic equation

${\displaystyle �{�}^2+��+�=0,}$

${\displaystyle �,�,�}$ can be any numbers whatsoever (except that ${\displaystyle �}$ cannot be ${\displaystyle 0}$), and the quadratic formula can be used to quickly and easily find the values of the unknown quantity ${\displaystyle �}$ 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 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 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 the second half of the 19th century on, many new fields of mathematics appeared, most of which made use 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 Subject Classification^[8] where none of the first level areas (two digit entries) are called algebra. Today algebra includes section 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.