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. Historically, and in current teaching, the study of algebra starts with the solving of equations, such as the quadratic equation above. That is to say, to find all the solutions of the equation. For example, in the quadratic equation a x 2 + b x + c = 0, which satisfy the equation. This allowed proofs of properties that are true no matter which numbers are involved.
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.Īlgebra began with computations similar to those of arithmetic, with letters standing for numbers. With an article, it means an instance of some algebraic structure, like a Lie algebra, an associative algebra, or a vertex operator algebra. 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 a qualifier, there is the same distinction:. In universal algebra, the word "algebra" refers to a generalization of the above concept, which allows for n-ary operations. When some authors use the term "algebra", they make a subset of the following additional assumptions: associative, commutative, unital, and/or finite-dimensional. Usually, the structure has an addition, multiplication, and scalar multiplication (see Algebra over a field). 
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. As a single word without an article, "algebra" names a broad part of mathematics. The word "algebra" has several related meanings in mathematics, as a single word or with qualifiers. The mathematical meaning was first recorded (in English) in the 16th century. It originally referred to the surgical procedure of setting broken or dislocated bones. 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. 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. The word algebra comes from the Arabic: الجبر, romanized: al-jabr, lit.'reunion of broken parts, bonesetting ' from the title of the early 9th century book cIlm al-jabr wa l-muqābala "The Science of Restoring and Balancing" by the Persian mathematician and astronomer al-Khwarizmi. The word algebra comes from the title of a book by Muhammad ibn Musa al-Khwarizmi. A mathematician specialized in algebra is called an algebraist. Sometimes, the same phrase is used for a subarea and its main algebraic structures for example, Boolean algebra and a Boolean algebra. 
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. 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. 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). Abstract algebra is the name given in education to the study of algebraic structures such as groups, rings, and fields. Įlementary algebra deals with the manipulation of variables as if they were numbers (see the image), and is therefore essential in all applications of mathematics.
Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas it is a unifying thread of almost all of mathematics. The quadratic formula expresses the solution of the equation ax 2 + bx + c = 0, where a is not zero, in terms of its coefficients a, b and c.Īlgebra (from Arabic الجبر ( al-jabr) 'reunion of broken parts, bonesetting') is one of the broad areas of mathematics.
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