LITMIR.BIZ

This book, for the first time, provides laymen and mathematicians alike with a detailed picture of the historical development of one of the most momentous achievements of the human intellect ― the calculus. It describes with accuracy and perspective the long development of both the integral and the differential calculus from their early beginnings in antiquity to their final emancipation in the 19th century from both physical and metaphysical ideas alike and their final elaboration as mathematical abstractions, as we know them today, defined in terms of formal logic by means of the idea of a limit of an infinite sequence.But while the importance of the calculus and mathematical analysis ― the core of modern mathematics ― cannot be overemphasized, the value of this first comprehensive critical history of the calculus goes far beyond the subject matter. This book will fully counteract the impression of laymen, and of many mathematicians, that the great achievements of mathematics were formulated from the beginning in final form. It will give readers a sense of mathematics not as a technique, but as a habit of mind, and serve to bridge the gap between the sciences and the humanities. It will also make abundantly clear the modern understanding of mathematics by showing in detail how the concepts of the calculus gradually changed from the Greek view of the reality and immanence of mathematics to the revised concept of mathematical rigor developed by the great 19th century mathematicians, which held that any premises were valid so long as they were consistent with one another. It will make clear the ideas contributed by Zeno, Plato, Pythagoras, Eudoxus, the Arabic and Scholastic mathematicians, Newton, Leibnitz, Taylor, Descartes, Euler, Lagrange, Cantor, Weierstrass, and many others in the long passage from the Greek «method of exhaustion» and Zeno's paradoxes to the modern concept of the limit independent of sense experience; and illuminate not only the methods of mathematical discovery, but the foundations of mathematical thought as well.

Подробнее

"This book is a radical departure from all previous concepts of advanced calculus," declared the Bulletin of the American Mathematics Society, «and the nature of this departure merits serious study of the book by everyone interested in undergraduate education in mathematics.» Classroom-tested in a Princeton University honors course, it offers students a unified introduction to advanced calculus. Starting with an abstract treatment of vector spaces and linear transforms, the authors introduce a single basic derivative in an invariant form. All other derivatives — gradient, divergent, curl, and exterior — are obtained from it by specialization. The corresponding theory of integration is likewise unified, and the various multiple integral theorems of advanced calculus appear as special cases of a general Stokes formula. The text concludes by applying these concepts to analytic functions of complex variables.

Подробнее

This textbook for advanced courses in group theory focuses on finite groups, with emphasis on the idea of group actions. Early chapters summarize presupposed facts, identify important themes, and establish the notation used throughout the book. Subsequent chapters explore the normal and arithmetical structures of groups as well as applications. Topics include the normal structure of groups: subgroups; homomorphisms and quotients; series; direct products and the structure of finitely generated Abelian groups; and group action on groups. Additional subjects range from the arithmetical structure of groups to classical notions of transfer and splitting by means of group action arguments. More than 675 exercises, many accompanied by hints, illustrate and extend the material.

Подробнее

Group theory represents one of the most fundamental elements of mathematics. Indispensable in nearly every branch of the field, concepts from the theory of groups also have important applications beyond mathematics, in such areas as quantum mechanics and crystallography.Hans J. Zassenhaus, a pioneer in the study of group theory, has designed this useful, well-written, graduate-level text to acquaint the reader with group-theoretic methods and to demonstrate their usefulness as tools in the solution of mathematical and physical problems. Starting with an exposition of the fundamental concepts of group theory, including an investigation of axioms, the calculus of complexes, and a theorem of Frobenius, the author moves on to a detailed investigation of the concept of homomorphic mapping, along with an examination of the structure and construction of composite groups from simple components. The elements of the theory of p-groups receive a coherent treatment, and the volume concludes with an explanation of a method by which solvable factor groups may be split off from a finite group.Many of the proofs in the text are shorter and more transparent than the usual, older ones, and a series of helpful appendixes presents material new to this edition. This material includes an account of the connections between lattice theory and group theory, and many advanced exercises illustrating both lattice-theoretical ideas and the extension of group-theoretical concepts to multiplicative domains.

Подробнее

Much in our daily lives is defined in numerical terms-from the moment we wake in the morning and look at the clock to dialing a phone or paying a bill. But what exactly is a number? When did man begin to count and record numbers? Who made the first calculating machine-and when? At what point did people first think of solving problems by equations? These and many other questions about numbers are answered in this engrossing, clearly written book.Written for general readers by a teacher of mathematics, the jargon-free text traces the evolution of counting systems, examines important milestones, investigates numbers, words, and symbols used around the world, and identifies common roots. The dawn of numerals is also covered, as are fractions, addition, subtraction, multiplication, division, arithmetic symbols, the origins of infinite cardinal arithmetic, symbols for the unknown, the status of zero, numbers and religious belief, recreational math, algebra, the use of calculators — from the abacus to the computer — and a host of other topics.This entertaining and authoritative book will not only provide general readers with a clearer understanding of numbers and counting systems but will also serve teachers as a useful resource. «The success of Flegg's lively exposition and the care he gives to his surprisingly exciting topic recommend this book to every library.» — Choice.

Подробнее

This enlightening survey of mathematical concept formation holds a natural appeal to philosophically minded readers, and no formal training in mathematics is necessary to appreciate its clear exposition of mathematic fundamentals. Rather than a system of theorems with completely developed proofs or examples of applications, readers will encounter a coherent presentation of mathematical ideas that begins with the natural numbers and basic laws of arithmetic and progresses to the problems of the real-number continuum and concepts of the calculus.Contents include examinations of the various types of numbers and a criticism of the extension of numbers; arithmetic, geometry, and the rigorous construction of the theory of integers; the rational numbers, the foundation of the arithmetic of natural numbers, and the rigorous construction of elementary arithmetic. Advanced topics encompass the principle of complete induction; the limit and point of accumulation; operating with sequences and differential quotient; remarkable curves; real numbers and ultrareal numbers; and complex and hypercomplex numbers.In issues of mathematical philosophy, the author explores basic theoretical differences that have been a source of debate among the most prominent scholars and on which contemporary mathematicians remain divided. «With exceptional clarity, but with no evasion of essential ideas, the author outlines the fundamental structure of mathematics.» — Carl B. Boyer, Brooklyn College. 27 figures. Index.

Подробнее

No mathematical background is necessary to appreciate this classic of probability theory, which remains unsurpassed in its clarity, readability, and sheer charm. Its author, British logician John Venn (1834-1923), popularized the famous Venn Diagrams that are commonly used for teaching elementary mathematics. In The Logic of Chance, he employs the same directness that makes his diagrams so effective.The three-part treatment commences with an overview of the physical foundations of the science of probability, including surveys of the arrangement and formation of the series of probability; the origin or process of causation of the series; how to discover and prove the series; and the conception of randomness. The second part examines the logical superstructure on the basis of physical foundations, encompassing the measurement of belief; the rules of inference in probability; the rule of succession; induction; chance, causation, and design; material and formal logic; modality; and fallacies. The final section explores various applications of the theory of probability, including such intriguing aspects as insurance and gambling, the credibility of extraordinary stories, and approximating the truth by means of the theory of averages.

Подробнее

This lighthearted work uses a variety of practical applications and puzzles to take a look at today's mathematical trends. In nine chapters, Professor Pedoe covers mathematical games, chance and choice, automatic thinking, and more.

Подробнее

This highly technical introduction to formal languages in computer science covers all areas of mainstream formal language theory, including such topics as operations on languages, context-sensitive languages, automata, decidability, syntax analysis, derivation languages, and more. Geared toward advanced undergraduates and graduate students, the treatment examines mathematical topics related to mathematical logic, set theory, and linguistics. All subjects are integral to the theory of computation.Numerous worked examples appear throughout the book, and end-of-chapter exercises enable readers to apply theory and methods to real-life problems. Elegant mathematical proofs are provided for almost all theorems.

Подробнее

This classic study notes the first appearance of a mathematical symbol and its origin, the competition it encountered, its spread among writers in different countries, its rise to popularity, its eventual decline or ultimate survival. The author’s coverage of obsolete notations — and what we can learn from them — is as comprehensive as those which have survived and still enjoy favor. Originally published in 1929 in a two-volume edition, this monumental work is presented here in one volume.

Подробнее