Algebra

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The algebra is one of the basic branches of mathematics; it deals with the properties of arithmetic operations. In the vernacular algebra is often used as the computation with unknowns in equations, (for example, x 1 = 2); the unknown is (or unknowns ) represented by letters. As the founder of algebra applies the Greek Diophantus of Alexandria, who lived sometime between 100 BC and 350 AD. His 13 volumes of comprehensive work Arithmetica is the oldest extant in which the algebraic method (ie, calculating with letters) will be used.

• 4.1 The History
• 4.2 Textbooks

History

Word history

The first representation of the algebraic method can be found in Arithmetica, a teaching and task book of Diophantus of Alexandria, whose time of origin is dated to the 1st century BC, according to other sources to the 4th century AD. Another representation of the algebra is the Aryabhattiya, a mathematical textbook of the Indian mathematician Aryabhata in the 5th century; the methodology used Bijaganitam was called. From the 9th century took over and then refined scholars from the Arab -speaking world, this method, which they al - ǧabr ( from Arabic: . " The complete " / " setup " ) called. The term is from the title of computing textbook Al- Kitaab al - Mukhtasar fī Hisab al - jabr wa - ʾ l - muqabala ( "This concise book about the calculation method by supplementing and balancing ", dating back to 825 ) of the Persian mathematician and polymath al - Khwarizmi taken, who worked in Baghdad in the 9th century. Four centuries after the publication of the book appeared his Latin translation Ludus algebrae almucgrabalaeque. From " al - ǧabr " is today's word " algebra " developed.

Time of the Babylonians

Already 2000 years before our era the ancient Babylonians were able systems of equations of the form

Equivalent to a quadratic equation of the form are to be solved. Such equations can have irrational numbers as solutions. However, the Babylonians were not interested in exact solutions, but calculated, usually by means of linear interpolation, approximate solutions. Also, the Babylonians dealt not with negative numbers. One of the most famous clay tablets of the Babylonians is Plimpton 322, which was created 1900-1600 BC. It lists Pythagorean triples, which means that the Babylonians knew the significance of these figures already 1000 years before Pythagoras.

Time of the Egyptians

The Babylonian algebra was more advanced than the Egyptian algebra the same time. While the Babylonians themselves dealing with quadratic equations, the Egyptians studied mainly linear equations.

The Rhind papyrus, one of the main sources of today's knowledge of the mathematics of ancient Egypt, was translated around 1650 BC by Ahmes from an older plant. In the papyrus to linear equations of the form and wherein A, B, and C are known and x is unknown is achieved by geometrical methods.

Time of the Greeks

Just as the Egyptians and Babylonians also studied the ancient Greeks algebraic equations. However, they were not only interested in practical issues, but saw especially in the early stages geometrical issues as a central part of their area of ​​philosophy. This was the beginning of algebra and geometry and mathematics as a science. The terms of algebraic equations represented by the Greeks pages, mostly stretching, geometric objects. Using methods of construction with ruler and compass, they determined solutions of certain algebraic equations. As the ancient Greek algebra was therefore justified by the geometry, it is called the geometric algebra. Recently, however, this interpretation is controversial.

The second volume, written by Euclid Elements contains a set of algebraic statements that were formulated in the language of geometry. Euclid discussed in, inter alia, the elements of the theory of surface application, going back to the Altpythagoreer. With this method one can solve certain linear and quadratic equations with one variable from the perspective of modern algebra. In the tenth book of Euclid's Elements handed a proof of the irrationality of the square root of 2 irrational proportions were already the Pythagoreans ( off of their number concept ) are known which had proved Euclid's theorem already in general form.

Diophantus of Alexandria, who probably lived around the year 250 AD, is considered the greatest algebraists of antiquity. His first and most important work, the Arithmetica, originally consisted of thirteen individual books, of which only six have survived. With this work, he solved the arithmetic and algebra in terms of considering positive, rational solutions to problems entirely by the geometry from. Also differed mathematics of Diophantus of the Babylonians, for he was primarily interested in exkaten and not approximate solutions.

Classical and modern algebra

The algebra are divided with respect to their origin in the classical and the modern algebra. Methods of algebra that have been developed until the 19th century, called ' classical algebra '. In it, we examine algebraic equations

On properties of its solutions. Important statements in classical algebra are proved by Gauss fundamental theorem of algebra, which states that an algebraic equation of order n has exactly n solutions, and the set of Abel, which states that it 5 for an algebraic equation -degree in general no solution formula is similar to the PQ formula.

Around 1830 developed Évariste Galois ( 1811-1832 ), the Galois theory. This can be understood as the beginning of modern algebra. Since that time, the algebra developed away from the theory of algebraic equations through to group and ring theory.

Using the example of the great Fermat's theorem one can see, however, that can not be the classic and the modern algebra clearly separate. The assumption that the algebraic equation has no solution with integer for, was formulated in the 17th century by Pierre de Fermat. The question about solutions of the equation contained in the guess is a typical question of classical algebra or the written during this time number theory. However, the presumption only in 1995 ( by Andrew Wiles and Richard Taylor) could be proved with more modern methods of algebraic geometry and algebraic number theory.

Algebra as a branch of mathematics: Definition and Structure

The content and methods of algebra have so expanded greatly throughout history that it has become difficult to specify the concept of algebra in a succinct definition. Below are some branches of algebra and some adjacent to the algebra, other subregions are mentioned. However, these are not sharply distinguished from each other.

• The elementary algebra is the algebra in the sense of school mathematics. It includes the calculation rules of natural, whole, broken and real numbers, dealing with expressions containing variables, and ways to solve simple algebraic equations.
• The abstract algebra is a fundamental discipline of modern mathematics. It deals with special algebraic structures such as groups, rings, bodies and their linkage.
• The linear algebra deals with the solution of linear systems of equations, the study of vector spaces and the determination of eigenvalues ​​; it forms the basis for the analytical geometry.
• The multilinear algebra studied in contrast to the tensor algebraic properties of tensors and other multi- linear maps.
• The commutative algebra deals with commutative rings and their ideals, modules and algebras and is closely linked to the algebraic geometry.
• The real algebra studied algebraic number fields, in which an arrangement can be defined. Next on positive polynomials are studied.
• The computer algebra deals with the symbolic manipulation of algebraic expressions. One focus is the exact arithmetic of whole, rational and algebraic numbers and with polynomials over these number spaces. On the theoretical side, this subarea is allocated to the search for efficient algorithms as well as the determination of the complexity of these algorithms. On the practical side, a variety of computer algebra systems have been developed that enable the computer-assisted manipulation of algebraic expressions.
• The universal or general algebra considered in general algebraic structures.
• Algebraic geometry investigated zeros of systems of algebraic equations.
• The algebraic number theory examines questions of number theory with the help of methods of algebra.
• The homological algebra contains methods that originally issues of the topology were repaid as part of algebraic topology to algebraic situations.