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Fundamental arithmetic operations support virtually all of the engineering, scientific, and financial computations required for practical applications, from cryptography, to financial planning, to rocket science. This comprehensive reference provides researchers with the thorough understanding of number representations that is a necessary foundation for designing efficient arithmetic algorithms. Using the elementary foundations of radix number systems as a basis for arithmetic, the authors develop and compare alternative algorithms for the fundamental operations of addition, multiplication, division, and square root with precisely defined roundings. Various finite precision number systems are investigated, with the focus on comparative analysis of practically efficient algorithms for closed arithmetic operations over these systems. Each chapter begins with an introduction to its contents and ends with bibliographic notes and an extensive bibliography. The book may also be used for graduate teaching: problems and exercises are scattered throughout the text and a solutions manual is available for instructors.

Arithmetic --- Foundations --- Foundations of arithmetic --- Mathematics --- Foundations. --- Philosophy --- Arithmetic - Foundations

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Arithmetic --- Foundations. --- -Foundations --- Foundations of arithmetic --- Foundations --- Mathematics --- Philosophy --- Arithmetic - Foundations.

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The fourth edition of a classic book on logic has been thoroughly revised by the author's son. It is a fundamental guide to modern mathematical logic and to the construction of mathematical theories. The first half covers the elements of logic, and the second half covers the applications of logic in theory building.

Mathematics --- Arithmetic --- Foundations of arithmetic --- Logic of mathematics --- Mathematics, Logic of --- Philosophy. --- Foundations. --- Philosophy --- Mathematics - Philosophy --- Arithmetic - Foundations

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The first complete English translation of a groundbreaking work. An ambitious account of the relation of mathematics to logic. Includes a foreword by Crispin Wright, translators' Introduction, and an appendix on Frege's logic by Roy T. Cook. The German philosopher and mathematician Gottlob Frege (1848-1925) was the father of analytic philosophy and to all intents and purposes the inventor of modern logic. Basic Laws of Arithmetic, originally published in German in two volumes (1893, 1903), is Freges magnum opus. It was to be the pinnacle of Freges lifes work. It represents the final stage of his logicist project the idea that arithmetic and analysis are reducible to logic and contains his mature philosophy of mathematics and logic. The aim of Basic Laws of Arithmetic is to demonstrate the logical nature of mathematical theorems by providing gapless proofs in Frege's formal system using only basic laws of logic, logical inference, and explicit definitions. The work contains a philosophical foreword, an introduction to Frege's logic, a derivation of arithmetic from this logic, a critique of contemporary approaches to the real numbers, and the beginnings of a logicist treatment of real analysis. As is well-known, a letter received from Bertrand Russell shortly before the publication of the second volume made Frege realise that his basic law V, governing the identity of value-ranges, leads into inconsistency. Frege discusses a revision to basic law V written in response to Russells letter in an afterword to volume II. The continuing importance of Basic Laws of Arithmetic lies not only in its bearing on issues in the foundations of mathematics and logic but in its model of philosophical inquiry. Frege's ability to locate the essential questions, his integration of logical and philosophical analysis, and his rigorous approach to criticism and argument in general are vividly in evidence in this, his most ambitious work. Philip Ebert and Marcus Rossberg present the first full English translation of both volumes of Freges major work preserving the original formalism and pagination. The edition contains a foreword by Crispin Wright and an extensive appendix providing an introduction to Frege's formal system by Roy T. Cook. Readership: Scholars and advanced students in philosophy of logic, philosophy of mathematics, and early analytic philosophy --

Arithmetic --- Logic, Symbolic and mathematical. --- Mathematics. --- Arithmétique --- Logique symbolique et mathématique --- Mathématiques --- Foundations. --- Fondements --- Arithmétique --- Logique symbolique et mathématique --- Mathématiques --- Logic, symbolic and mathematical --- Foundations --- Arithmetic - Foundations

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Provability, Computability and Reflection

Recursion theory --- Arithmetic --- Number theory --- Théorie de la récursivité --- Arithmétique --- Théorie des nombres --- Foundations --- Fondements --- EPUB-LIV-FT ELSEVIER-B --- Arithmetic -- Foundations. --- Logic, Symbolic and mathematical -- Periodicals. --- Logic, Symbolic and mathematical. --- Number theory. --- Recursion theory. --- Foundations.