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MOST IMPORTANT SCIENTIFIC DISCOVERIES
Fundamentals of Algebra. History and essence of scientific discovery
Directory / The most important scientific discoveries It is believed that the Hellenes borrowed the first information on algebra from the Babylonians. The Greek Neoplatonic philosopher Proclus Diadochus noted in his essay: "According to most opinions, geometry was first discovered in Egypt, had its origin in the measurement of areas." The impact of the traditions of Babylonian algebra on the mathematics of ancient Greece and the algebraic school of Islamic countries is emphasized in the History of Mathematics. The creation of the foundations of mathematics in the form that we are accustomed to when studying this science at school fell to the Greeks and dates back to the XNUMXth-XNUMXth centuries BC. Ancient science reached the pinnacle in the works Euclid, Archimedes, Apollonia. A new rise in ancient mathematics in the XNUMXrd century AD is associated with the work of the great mathematician Diophantus. His main work is Arithmetic. Unfortunately, only six books out of thirteen books have survived to our time. Diophantus managed to revive and develop the numerical algebra of the Babylonians, freeing it from the geometric constructions used by the Greeks. Diophantus first appears letter symbolism. He introduced the notation: unknown, square, cube, fourth, fifth, and sixth powers, as well as the first six negative powers. In the History of Mathematics, this is specially noted: “The book of Diophantus testifies to the presence of literal symbolism in him. The significance of this step is enormous. Only on this basis could literal calculus be created, a formula apparatus developed that allows us to replace part of our mental operations with mechanical transformations. However, Diophantus , apparently, did not find followers in this matter either in his era, or much later.Only from the end of the XNUMXth century did intensive development of algebraic symbolism begin in Europe, and the completion of the creation of letter calculus occurred only at the end of the XNUMXth - beginning of the XNUMXth century in the works Vieta и Descartes". "Diophantus," writes V.A. Nikiforovsky, "formulated the rules of algebraic operations with powers of the unknown, corresponding to our multiplication and division of powers with natural exponents, and the rules of multiplication signs. This made it possible to compactly write down polynomials, multiply them, and operate with equations. He He also pointed out the rules for the transfer of negative terms of the equation to another part of it with opposite signs, the mutual annihilation of the same terms in both parts of the equation. Starting from the 595th century, the center of mathematical culture gradually moved to the east - to the Hindus and Arabs. Hindu mathematics was numerical. It is marked by the desire to achieve the rigor of the Hellenes in the proofs and justification of geometry, being content with drawings. The main achievements of the Hindus are that they introduced the numbers, which we call Arabic, and the positional system of notation of numbers, discovered the duality of the roots of the quadratic equation, the two-valuedness of the square root, and introduced negative numbers. The first application of the decimal positional system known to us dates back to 346 - a plate has been preserved on which the number of years XNUMX is written in such a system. The most famous mathematicians of India were Aryabhata (nicknamed "the first", about 500) and Brahmagupta (about 625). Hindus considered numbers without regard to geometry. They extended the rules of action on rational numbers to irrational numbers, making direct calculations on them. Another achievement of the Indians in the improvement of algebraic symbolism is that they introduced the notation for several different unknowns and their powers. Like Diophantus, they were essentially abbreviations of words. Following the Indian mathematicians, the mathematicians of the Near and Middle East began to use the rule of position. A special role in the history of the development of algebra in the first half of the XNUMXth century was played by al-Khwarizmi's treatise in Arabic called "The Book of Restoration and Opposition" (in Arabic - "Kitab al-jabr wal-muqabala"). Later, when translating into Latin, the Arabic title of the treatise was retained. Over time, "al-jabr" was reduced to "algebra". In the treatise, the solution of equations is no longer considered in connection with arithmetic, but as an independent branch of mathematics. An Arabic mathematician shows that unknowns, their squares and free terms of equations are used in algebra. Al-Khwarizmi called the unknown "the root". When solving various types of equations, al-Khwarizmi proposes to transfer the negative terms of the equations from one part to another, calling it restoration. The subtraction of equal terms from both sides of the equation in this case, he calls opposition (wal muqabala). “In his treatise al-Khwarizmi,” Alexander Svechnikov notes, “considers an unknown number as a quantity of a special kind, introduces the term root, calls the free term dirham (as the monetary unit was called at that time). He distributes equations by type, explains how apply the rules of completion and opposition, formulate rules for solving equations of various types. In the manuscripts of al-Khwarizmi, all mathematical expressions and all calculations are written in words, which is why the algebra of that time and later was called rhetorical, that is, verbal. During the period of work on the algebraic treatise, al-Khwarizmi already knew about the numerical algebra of Babylon and other countries of the East. He was familiar with the geometric algebra of the Greeks and the achievements of Indian astronomers and mathematicians. Al-Khwarizmi singled out algebraic material as a special section of mathematics and freed it from geometric interpretation, although in some cases he used geometric proofs. Al-Khwarizmi's algebraic work became a model that was studied and imitated by many later mathematicians. Subsequent algebraic writings and textbooks began to approach modern ones in character. The algebraic treatise of al-Khwarizmi served as the beginning of the creation of the science of algebra. He was among the first works on mathematics translated into Latin. At that time in Europe all scientific works were written and printed in Latin. When solving a problem, the main thing is understanding the content of the problem, the ability to express it in the language of algebra. Simply put, write down the condition of the problem using symbols - mathematical signs. Diophantus, as already mentioned, gave the concept of an algebraic equation, written in symbols, but very far from modern ones. François Viet was the first to designate with letters not only unknowns, but also given quantities. Thus, he managed to introduce into science the great idea of the possibility of performing algebraic transformations on symbols, that is, to introduce the concept of a mathematical formula. In this way, he made a decisive contribution to the creation of literal algebra, which completed the development of Renaissance mathematics and paved the way for the appearance of the results of Fermat, Descartes, and Newton. François Viet (1540-1603) was born in the south of France in the small town of Fantinay-le-Comte. Vieta's father was a prosecutor. According to tradition, the son chose the profession of his father and became a lawyer after graduating from the University of Poitou. In 1560, the twenty-year-old lawyer began his career in his native city, but three years later he moved to the service of the noble Huguenot de Partenay family. He became the secretary of the owner of the house and the teacher of his daughter, twelve-year-old Catherine. It was teaching that aroused in the young lawyer an interest in mathematics. In 1671, Viète entered the civil service, becoming an advisor to the Parliament and then an advisor to King Henry III of France. In 1580, Henry III appointed Vieta to the important state post of racketmaster, which gave the right to control the implementation of orders in the country on behalf of the king and to suspend the orders of large feudal lords. While in public service, Viet remained a scientist. He became famous for being able to decipher the intercepted correspondence of the King of Spain with his representatives in the Netherlands, thanks to which the King of France was fully aware of the actions of his opponents. In 1584, at the insistence of the Guises, Vieta was removed from office and expelled from Paris. It was during this period that the peak of his work falls. Having received unexpected leisure, the scientist set as his goal the creation of a comprehensive mathematics that allows solving any problems. He developed the conviction that "there must be a general, still unknown science, embracing both the witty inventions of the latest algebraists, and the deep geometric research of the ancients." Vieta outlined the program of his research and listed the treatises, united by a common idea and written in the mathematical language of the new alphabetic algebra, in the famous "Introduction to Analytical Art" published in 1591. The enumeration went in the order in which these works were to be published in order to form a single whole - a new direction in science. Unfortunately, a single whole did not work out. The treatises were published in a completely random order, and many saw the light only after Vieta's death. One of the treatises was not found at all. However, the main idea of the scientist was remarkably successful: the transformation of algebra into a powerful mathematical calculus began. The very name "algebra" Vieta in his writings replaced the words "analytical art". He wrote in a letter to de Partenay: "All mathematicians knew that under algebra and almukabala ... incomparable treasures were hidden, but they did not know how to find them. Problems that they considered the most difficult are quite easily solved by dozens with the help of our art ... " Viet called the basis of his approach species logistics. Following the example of the ancients, he clearly distinguished between numbers, magnitudes and relations, collecting them into a certain system of "species". This system included, for example, variables, their roots, squares, cubes, square-squares, etc., as well as many scalars, which corresponded to real dimensions - length, area or volume. For these species, Viet gave special symbols, designating them in capital letters of the Latin alphabet. Vowels were used for unknown quantities, consonants were used for variables. Viet showed that, by operating with symbols, it is possible to obtain a result that is applicable to any relevant quantities, that is, to solve the problem in a general form. This marked the beginning of a radical change in the development of algebra: literal calculus became possible. Demonstrating the power of his method, the scientist brought in his works a stock of formulas that could be used to solve specific problems. Of the action signs, he used "+" and "-", the radical sign and the horizontal bar for division. The work was denoted by the word "in". Viet was the first to use brackets, which, however, he did not have the form of brackets, but lines over a polynomial. But he did not use many of the signs introduced before him. So, a square, a cube, etc. denoted by words or the first letters of words. The symbolism of Vieta made it possible both to solve specific problems and to find general patterns, fully substantiating them. Thus, algebra became an independent branch of mathematics, independent of geometry. "This innovation, and especially the use of literal coefficients, marked the beginning of a fundamental change in the development of algebra: only now has algebraic calculus become possible as a system of formulas, as an operational algorithm." The symbolism of Vieta was subsequently followed by Pierre de Fermat. A further significant improvement in algebraic symbolism belongs to Descartes. Rene Descartes introduced lowercase letters of the Latin alphabet to denote coefficients. To designate unknowns, he used the last letters of the same alphabet. This innovation was widely adopted in the works of mathematicians and, with minor changes, has survived to this day. Author: Samin D.K.
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