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For the novel by Iain M. Banks, see The Algebraist. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Much early work in algebra, as the Arabic origin of its name suggests, was done in the Middle East, by Persian. In E = mc. 2, the letters E. Algebra gives methods for solving equations and expressing formulas that are much easier (for those who know how to use them) than the older method of writing everything out in words. The word algebra is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an . The word entered the English language during the fifteenth 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 the sixteenth century. Usually the structure has an addition, multiplication, and a scalar multiplication (see Algebra over a field). When some authors use the term . In universal algebra, the word . For example, in the quadratic equationax. Then the structural properties of these non- numerical objects were abstracted to define algebraic structures such as groups, rings, and fields. Before the 1. 6th 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 1. From the second half of 1. Today, algebra has grown until it includes many branches of mathematics, as can be seen in the Mathematics Subject Classification. Today algebra includes section 0. General algebraic systems, 1. Field theory and polynomials, 1. Commutative algebra, 1. Linear and multilinear algebra; matrix theory, 1. Associative rings and algebras, 1. Nonassociative rings and algebras, 1. Category theory; homological algebra, 1. K- theory and 2. 0- Group theory. Algebra is also used extensively in 1. Number theory and 1. Algebraic geometry. History. Early history of algebra. The roots of algebra can be traced to the ancient Babylonians. The Babylonians developed formulas to calculate solutions for problems typically solved today by using linear equations, quadratic equations, and indeterminate linear equations. By contrast, most Egyptians of this era, as well as Greek and Chinese mathematics in the 1st millennium BC, usually solved such equations by geometric methods, such as those described in the Rhind Mathematical Papyrus, Euclid's Elements, and The Nine Chapters on the Mathematical Art. The geometric work of the Greeks, typified in the Elements, provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations, although this would not be realized until mathematics developed in medieval Islam. The Greeks created a geometric algebra where terms were represented by sides of geometric objects, usually lines, that had letters associated with them. These texts deal with solving algebraic equations. He later wrote The Compendious Book on Calculation by Completion and Balancing, which established algebra as a mathematical discipline that is independent of geometry and arithmetic. Later, Persian and Arabic mathematicians developed algebraic methods to a much higher degree of sophistication. Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Al- Khwarizmi's contribution was fundamental. He solved linear and quadratic equations without algebraic symbolism, negative numbers or zero, thus he had to distinguish several types of equations. He also studied an equation for its own sake and . Yet another Persian mathematician, Sharaf al- D. In the 1. 3th century, the solution of a cubic equation by Fibonacci is representative of the beginning of a revival in European algebra. As the Islamic world was declining, the European world was ascending. And it is here that algebra was further developed. History of algebra. Fran. Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid- 1. The idea of a determinant was developed by Japanese mathematician. Seki K. Gabriel Cramer also did some work on matrices and determinants in the 1. Permutations were studied by Joseph- Louis Lagrange in his 1. R. Paolo Ruffini was the first person to develop the theory of permutation groups, and like his predecessors, also in the context of solving algebraic equations. Abstract algebra was developed in the 1. Galois theory, and on constructibility issues. Augustus De Morgan discovered relation algebra in his Syllabus of a Proposed System of Logic. Josiah Willard Gibbs developed an algebra of vectors in three- dimensional space, and Arthur Cayley developed an algebra of matrices (this is a noncommutative algebra). Others do not: group theory, ring theory, and field theory are examples. In this section, we list some areas of mathematics with the word . It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. In arithmetic, only numbers and their arithmetical operations (such as +, . In algebra, numbers are often represented by symbols called variables (such as a, n, x, y or z). This is useful because: It allows the general formulation of arithmetical laws (such as a + b = b + a for all a and b), and thus is the first step to a systematic exploration of the properties of the real number system. It allows the reference to . This step leads to the conclusion that it is not the nature of the specific numbers that allows us to solve it, but that of the operations involved.)It allows the formulation of functional relationships. For example, x. 2 + 2x . A polynomial expression is an expression that may be rewritten as a polynomial, by using commutativity, associativity and distributivity of addition and multiplication. A polynomial function is a function that is defined by a polynomial, or, equivalently, by a polynomial expression. The two preceding examples define the same polynomial function. Two important and related problems in algebra are the factorization of polynomials, that is, expressing a given polynomial as a product of other polynomials that can not be factored any further, and the computation of polynomial greatest common divisors. The example polynomial above can be factored as (x . A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable. Education. It has been suggested that elementary algebra should be taught to students as young as eleven years old. Here are listed fundamental concepts in abstract algebra. Sets: Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of sets: a collection of all objects (called elements) selected by property specific for the set. All collections of the familiar types of numbers are sets. Other examples of sets include the set of all two- by- two matrices, the set of all second- degree polynomials (ax. Set theory is a branch of logic and not technically a branch of algebra. Binary operations: The notion of addition (+) is abstracted to give a binary operation, . The notion of binary operation is meaningless without the set on which the operation is defined. For two elements a and b in a set S, a . Addition (+), subtraction (. Zero is the identity element for addition and one is the identity element for multiplication. For a general binary operator . This holds for addition as a + 0 = a and 0 + a = a and multiplication a . Not all sets and operator combinations have an identity element; for example, the set of positive natural numbers (1, 2, 3, ..) has no identity element for addition. Inverse elements: The negative numbers give rise to the concept of inverse elements. For addition, the inverse of a is written . A general two- sided inverse element a. That is, the grouping of the numbers to be added does not affect the sum. For example: (2 + 3) + 4 = 2 + (3 + 4). In general, this becomes (a . This property is shared by most binary operations, but not subtraction or division or octonion multiplication. Commutativity: Addition and multiplication of real numbers are both commutative. That is, the order of the numbers does not affect the result. For example: 2 + 3 = 3 + 2. In general, this becomes a . This property does not hold for all binary operations. For example, matrix multiplication and quaternion multiplication are both non- commutative. Groups. Combining the above concepts gives one of the most important structures in mathematics: a group. A group is a combination of a set S and a single binary operation . In this group, the identity element is 0 and the inverse of any element a is its negation, . The associativity requirement is met, because for any integers a, b and c, (a + b) + c = a + (b + c)The nonzero rational numbers form a group under multiplication. Here, the identity element is 1, since 1 . The inverse of a is 1/a, since a . This is because, in general, the multiplicative inverse of an integer is not an integer. For example, 4 is an integer, but its multiplicative inverse is . A major result in this theory is the classification of finite simple groups, mostly published between about 1. Semigroups, quasigroups, and monoids are structures similar to groups, but more general.
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