Theory of equations

I. Scan the article and say:

- on how many problems the interest in the theory oequations was concentrated and what were they

- who worked on the fundamental theorem

- what Gauss's demonstration of the fundamental theorem initiated

- what the efforts of Lagrange, Vandermonde, and Waring illustrate

After the dramatic successes of Niccolò Fontana Tartaglia and Lodovico Ferrari in the 16th century the theory of equations developed slowly, as problems resisted solution by known techniques. In the later 18th century the subject experienced an infusion of new ideas. Interest was concentrated on two problems. The first was to establish the existence of a root of the general polynomial equation of degree n. The second was to express the roots as algebraic functions of the coefficients, or to show why it was not in general possible to do so.

The proposition that the general polynomial with real coefficients has a root of the form a + b √(-1) became known later as the fundamental theorem of algebra. By 1742 Euler recognized that roots appear in conjugate pairs; if a + b √(−1) is a root, then so is a − b √(-1) . Thus, if a + b √(-1) is a root of f(x) = 0, then f(x) = (x² - 2ax - a² - b²)g(x). The fundamental theorem was therefore equivalent to asserting that a polynomial may be decomposed into linear and quadratic factors. This result was of considerable importance for the theory of integration, since by the method of partial fractions it ensured that a rational function, the quotient of two polynomials, could always be integrated in terms of algebraic and elementary transcendental functions.

Although d'Alembert, Euler, and Lagrange worked on the fundamental theorem, the first successful proof was developed by Carl Friedrich Gauss in his doctoral dissertation of 1799. Earlier researchers had investigated special cases or had concentrated on showing that all possible roots were of the form a ± b √(-1) . Gauss tackled the problem of existence directly. Expressing the unknown in terms of the polar variables r and θ, he showed that a solution of the polynomial would lie at the intersection of two curves of the form T(r, θ) = 0 and U(r, θ) = 0. By a careful and rigorous investigation he proved that the two curves intersect.

Gauss's demonstration of the fundamental theorem initiated a new approach to the question of mathematical existence. In the 18th century mathematicians were interested in the nature of particular analytic processes or the form that given solutions should take. Mathematical entities were regarded as things that were given, not as things whose existence needed to be established. Because analysis was applied in geometry and mechanics, the formalism seemed to possess a clear interpretation that obviated any need to consider questions of existence. Gauss's demonstration was the beginning of a change of attitude in mathematics, of a shift to the rigorous, internal development of the subject.

The problem of expressing the roots of a polynomial as functions of the coefficients was addressed by several mathematicians independently around 1770. The Cambridge mathematician Edward Waring published treatises in 1762 and 1770 on the theory of equations. In 1770 Lagrange presented a long expository memoir on the subject to the Berlin Academy, and in 1771 Alexandre Vandermonde submitted a paper to the French Academy of Sciences. Although the ideas of the three men were related, Lagrange's memoir was the most extensive and most influential historically.

Lagrange presented a detailed analysis of the solution by radicals of second-, third-, and fourth-degree equations and investigated why these solutions failed when the degree was greater than or equal to five. He introduced the novel idea of considering functions of the roots and examining the values they assumed as the roots were permuted. He was able to show that the solution of an equation depends on the construction of a second resolvent equation, but he was unable to provide a general procedure for solving the resolvent when the degree of the original equation was greater than four. Although his theory left the subject in an unfinished condition, it provided a solid basis for future work. The search for a general solution to the polynomial equation would provide the greatest single impetus for the transformation of algebra in the 19th century.

The efforts of Lagrange, Vandermonde, and Waring illustrate how difficult it was to develop new concepts in algebra. The history of the theory of equations belies the view that mathematics is subject to an almost automatic technical development. Much of the later algebraic work would be devoted to devising terminology, concepts, and methods necessary to advance the subject.

II. Read the passage again and say if the statements are true or false:

1. In the late 18th century the theory of equations developed very rapidly.

2. The assertion about decomposion of a polynomial into linear and quadratic factors was important for the theory of integration.

3. The fundamental theorem was first successfully proved by Carl Friedrich Gauss.

4. Mathematics shifted to the rigorous, internal development due to Gauss.

5. Lagrange's memoir on the problem of expressing the roots of a polynomial as functions of the coefficients was the most extensive but least influential historically.

6. Lagrange's theory left the subject in a finished condition and provided a solid basis for future work.

III. In what meanings are the words "demonstration" and "memoir" used in the sentences? Translate them.

1. Gauss's demonstration of the fundamental theorem initiated a new approach to the question of mathematical existence.

2. In 1770 Lagrange presented a long expository memoir on the subject to the Berlin Academy…

IV. How do you understand the sentence? Provide your explanations.

…mathematics is subject to an almost automatic technical development.

V. Suggest the Russian counterparts for the words and word combinations:

projective geometry, probability theory, mature branches of mathematics, a real coefficient, the fundamental theorem of algebra, a conjugate pair, linear and quadratic factors, the theory of integration, a partial fraction, a rational function, an elementary transcendental function, to develop a proof, to tackle a problem, a mathematical entity, a demonstration of a theorem, to concentrate on, a solution of an equation, to depend on, a resolvent equation, to provide a solid basis for, a polynomial equation, to develop new concepts in algebra, to belie a view, to advance the subject.

VI. Translate the article at sight.

Most of the powerful abstract mathematical theories in use today originated in the 19th century, so that any historical account of the period should be supplemented by reference to detailed treatments of these topics. Moreover, mathematics grew so much during this period that any account must necessarily be selective. Nonetheless, some broad features stand out. The growth of mathematics as a profession was accompanied by a sharpening division between mathematics and the physical sciences, and contact between the two subjects takes place today across a clear professional boundary. One result of this separation has been that mathematics, no longer able to rely on its scientific import for its validity, developed markedly higher standards of rigour. It was also freed to develop in directions that had little to do with applicability. Some of these pure creations have turned out to be surprisingly applicable, while the attention to rigour has led to a wholly novel conception of the nature of mathematics and logic. Moreover, many outstanding questions in mathematics yielded to the more conceptual approaches that came into vogue.

Date: 2015-01-29; view: 1616