Students today require not just sound mathematical abilities, but also problem solving and reasoning skills. Contemporary's Real Numbers series helps student learn to reason and to solve problems that they encounter in daily life. Real Numbers uses a visual approach so students become competent in a broad range of math topics.
Students today require not just sound mathematical abilities, but also problem solving and reasoning skills. Contemporary's Real Numbers series helps student learn to reason and to solve problems that they encounter in daily life. Real Numbers uses a visual approach so students become competent in a broad range of math topics.
While most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. Analysis needs the real numbers to model the line, and to support the concepts of continuity and measure. But these seemingly simple requirements lead to deep issues of set theory—uncountability, the axiom of choice, and large cardinals. In fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself. By focusing on the set-theoretic aspects of analysis, this text makes the best of two worlds: it combines a down-to-earth introduction to set theory with an exposition of the essence of analysis—the study of infinite processes on the real numbers. It is intended for senior undergraduates, but it will also be attractive to graduate students and professional mathematicians who, until now, have been content to "assume" the real numbers. Its prerequisites are calculus and basic mathematics. Mathematical history is woven into the text, explaining how the concepts of real number and infinity developed to meet the needs of analysis from ancient times to the late twentieth century. This rich presentation of history, along with a background of proofs, examples, exercises, and explanatory remarks, will help motivate the reader. The material covered includes classic topics from both set theory and real analysis courses, such as countable and uncountable sets, countable ordinals, the continuum problem, the Cantor–Schröder–Bernstein theorem, continuous functions, uniform convergence, Zorn's lemma, Borel sets, Baire functions, Lebesgue measure, and Riemann integrable functions.
This comprehensive volume is perfect for students who are interested in higher-level study of numbers and measurements. The book delves into the history of mathematical reasoning and the progression of numerical thought. Readers will learn how our world is shaped by the number and measurement systems that have arisen over time. They will also engage in the history of the development of number and measurement systems and the biographies of some of the greatest mathematical minds throughout history. This is a perfect volume for anyone interested in higher-level math and the stories behind it.
Communication and, indeed, our comprehension of the world in general are largely ordered by the number and measurement systems that have arisen over time. This book delves into the history of mathematical reasoning and the progression of numerical thought around the world. With detailed biographies of seminal thinkers and theorists, readers develop a sophisticated understanding of some of the most fundamental arithmetical concepts as well as the individuals who established them.
This text is a rigorous, detailed introduction to real analysis that presents the fundamentals with clear exposition and carefully written definitions, theorems, and proofs. It is organized in a distinctive, flexible way that would make it equally appropriate to undergraduate mathematics majors who want to continue in mathematics, and to future mathematics teachers who want to understand the theory behind calculus. The Real Numbers and Real Analysis will serve as an excellent one-semester text for undergraduates majoring in mathematics, and for students in mathematics education who want a thorough understanding of the theory behind the real number system and calculus.
Introducing graduate students and researchers to mathematical physics, this book discusses two recent developments: the demonstration that causality can be defined on discrete space-times; and Sewell's measurement theory, in which the wave packet is reduced without recourse to the observer's conscious ego, nonlinearities or interaction with the rest of the universe. The definition of causality on a discrete space-time assumes that space-time is made up of geometrical points. Using Sewell's measurement theory, the author concludes that the notion of geometrical points is as meaningful in quantum mechanics as it is in classical mechanics, and that it is impossible to tell whether the differential calculus is a discovery or an invention. Providing a mathematical discourse on the relation between theoretical and experimental physics, the book gives detailed accounts of the mathematically difficult measurement theories of von Neumann and Sewell.