By Israel Kleiner

ISBN-10: 0817682678

ISBN-13: 9780817682675

ISBN-10: 0817682686

ISBN-13: 9780817682682

This publication contains 5 components. the 1st 3 comprise ten old essays on vital subject matters: quantity thought, calculus/analysis, and facts, respectively. half 4 bargains with a number of traditionally orientated classes, and half 5 offers biographies of 5 mathematicians—Dedekind, Euler, Gauss, Hilbert, and Weierstrass—who performed significant roles within the old occasions defined within the first 4 elements of the work.

Excursions within the background of Mathematics was once written with numerous targets in brain: to arouse arithmetic lecturers’ curiosity within the background in their topic; to inspire arithmetic lecturers with no less than a few wisdom of the heritage of arithmetic to supply classes with a powerful historic part; and to supply an ancient point of view on a few easy issues taught in arithmetic courses.

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Example text

This can be rephrased to say that there are infinitely many primes in the arithmetic sequence 2n C 1 (n D 0, 1, 2, 3, . . ). In 1837, Dirichlet proved a grand generalization of this result by showing that any arithmetic sequence an C b (n D 0, 1, 2, 3, . . ), with a and b relatively prime, contains infinitely many primes. n/=n (s is nD1 a real number greater than 1, and the “Dirichlet character” X is a function that associates with each integer relatively prime to n an nth root of 1, and satisfies certain properties).

It led to the rise of a new area of study – analytic number theory (see Sect. 8). As Euler put it [25, p. 176]: One may see how closely and wonderfully infinitesimal analysis is related . . to the theory of numbers, however repugnant the latter may seem to that higher kind of calculus. x/ of degree three or four (their graphs are called elliptic curves; see Sect. 9 and [9, 25]). More broadly, building bridges between different, seemingly unrelated, areas of mathematics is an important and powerful idea, for it brings to bear the tools of one field in the service of the other.

W. Scharlau and H. Opolka. From Fermat to Minkowski: Lectures on the Theory of Numbers and its Historical Development, Springer-Verlag, 1985. 23. S. Singh, Fermat’s Enigma: The Quest to Solve the World’s Greatest Mathematical Problem, Penguin, 1997. 24. J. Stillwell, Elements of Number Theory, Springer, 2003. 25. A. Weil, Number Theory: An Approach through History, Birkh¨auser, 1984. 26. B. H. Yandell, The Honors Class: Hilbert’s Problems and their Solvers, A K Peters, 2002. 27. G. M. Ziegler, The great prime-number record races, Notices of the Amer.

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Excursions in the history of mathematics by Israel Kleiner


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