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Что (кто) такое algebra - определение

PART OF MATHEMATICS IN WHICH LETTERS AND OTHER SYMBOLS ARE USED TO REPRESENT NUMBERS AND QUANTITIES IN FORMULAE AND EQUATION

Al-Gebra; Stochastic process algebra; Aljebra; Stochastic process algebras; Algebra (mathematics); ALGEBRA; Algerbra; Intermediate algebra; Diagram algebra; Diagrammatic algebra; Cossic art; Rule of Coss; Algebraic processes; Algbra; Alegbra

algebra

Algebra is a type of mathematics in which letters are used to represent possible quantities.

N-UNCOUNT

algebra

<mathematics, logic> 1. A loose term for an {algebraic structure}. 2. A vector space that is also a ring, where the vector space and the ring share the same addition operation and are related in certain other ways. An example algebra is the set of 2x2 matrices with {real numbers} as entries, with the usual operations of addition and matrix multiplication, and the usual scalar multiplication. Another example is the set of all polynomials with real coefficients, with the usual operations. In more detail, we have: (1) an underlying set, (2) a field of scalars, (3) an operation of scalar multiplication, whose input is a scalar and a member of the underlying set and whose output is a member of the underlying set, just as in a vector space, (4) an operation of addition of members of the underlying set, whose input is an ordered pair of such members and whose output is one such member, just as in a vector space or a ring, (5) an operation of multiplication of members of the underlying set, whose input is an ordered pair of such members and whose output is one such member, just as in a ring. This whole thing constitutes an 'algebra' iff: (1) it is a vector space if you discard item (5) and (2) it is a ring if you discard (2) and (3) and (3) for any scalar r and any two members A, B of the underlying set we have r(AB) = (rA)B = A(rB). In other words it doesn't matter whether you multiply members of the algebra first and then multiply by the scalar, or multiply one of them by the scalar first and then multiply the two members of the algebra. Note that the A comes before the B because the multiplication is in some cases not commutative, e.g. the matrix example. Another example (an example of a Banach algebra) is the set of all bounded linear operators on a Hilbert space, with the usual norm. The multiplication is the operation of composition of operators, and the addition and scalar multiplication are just what you would expect. Two other examples are tensor algebras and {Clifford algebras}. [I. N. Herstein, "Topics in Algebra"]. (1999-07-14)

algebra

['ald??br?]

¦ noun the part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations.

?a system of this based on given axioms.

Derivatives

algebraic ald??'bre??k adjective

algebraical adjective

algebraically adverb

algebraist noun

Word History

The word algebra comes from Arabic al-jabr 'the mending of broken parts', entering Middle English, via Italian, Spanish, and medieval Latin, in the sense 'the setting of broken bones'. The modern mathematical sense comes from the title of a book, ?ilm al-jabr wa'l-mu?abala 'the science of restoring what is missing and equating like with like', by the 9th-century Muslim mathematician Abu. Ja?far Muhammad ibn Mu?sa. His nickname, al-?warizmi (literally 'the man from ?warizm', now Khiva in Uzbekistan) is the root of the word algorithm.

Википедия

Algebra

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

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.

https://en.wikipedia.org/wiki/Algebra

Примеры употребления для algebra

1. Students typically take Algebra II in high school after taking a basic algebra course and geometry.

2. For example, all students would be required to take algebra.

3. A working knowledge of algebra would come in handy here.

4. "Should girls be exposing their bellybuttons in Algebra class?

5. I had not felt such confusion as a learner since algebra class, back in ninth grade.