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Author: Brian Clegg
Publisher: Macmillan
Published: 2016-12-06
Total Pages: 303
ISBN-13: 1250081041
DOWNLOAD EBOOKPresents an accessible, in-depth look at the history of numbers and their applications in life and science, from math's surreal presence in the virtual world to the debates about the role of math in science.
Author: Michael Henle
Publisher: American Mathematical Soc.
Published: 2012-12-31
Total Pages: 231
ISBN-13: 1614441073
DOWNLOAD EBOOKEveryone knows the real numbers, those fundamental quantities that make possible all of mathematics from high school algebra and Euclidean geometry through the Calculus and beyond; and also serve as the basis for measurement in science, industry, and ordinary life. This book surveys alternative real number systems: systems that generalize and extend the real numbers yet stay close to these properties that make the reals central to mathematics. Alternative real numbers include many different kinds of numbers, for example multidimensional numbers (the complex numbers, the quaternions and others), infinitely small and infinitely large numbers (the hyperreal numbers and the surreal numbers), and numbers that represent positions in games (the surreal numbers). Each system has a well-developed theory, including applications to other areas of mathematics and science, such as physics, the theory of games, multi-dimensional geometry, and formal logic. They are all active areas of current mathematical research and each has unique features, in particular, characteristic methods of proof and implications for the philosophy of mathematics, both highlighted in this book. Alternative real number systems illuminate the central, unifying role of the real numbers and include some exciting and eccentric parts of mathematics. Which Numbers Are Real? Will be of interest to anyone with an interest in numbers, but specifically to upper-level undergraduates, graduate students, and professional mathematicians, particularly college mathematics teachers.
Author: Jean E. Cunningham
Publisher: Jcc Press
Published: 2017-09-30
Total Pages: 200
ISBN-13: 9780999380109
DOWNLOAD EBOOKHow management accounting evolved with Lean principles.
Author: John Stillwell
Publisher: Springer Science & Business Media
Published: 2013-10-16
Total Pages: 253
ISBN-13: 331901577X
DOWNLOAD EBOOKWhile 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.
Author: Jonathan Borwein
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 433
ISBN-13: 1461585104
DOWNLOAD EBOOKHow do we recognize that the number . 93371663 . . . is actually 2 IoglQ(e + 7r)/2 ? Gauss observed that the number 1. 85407467 . . . is (essentially) a rational value of an elliptic integral-an observation that was critical in the development of nineteenth century analysis. How do we decide that such a number is actually a special value of a familiar function without the tools Gauss had at his disposal, which were, presumably, phenomenal insight and a prodigious memory? Part of the answer, we hope, lies in this volume. This book is structured like a reverse telephone book, or more accurately, like a reverse handbook of special function values. It is a list of just over 100,000 eight-digit real numbers in the interval [0,1) that arise as the first eight digits of special values of familiar functions. It is designed for people, like ourselves, who encounter various numbers computationally and want to know if these numbers have some simple form. This is not a particularly well-defined endeavor-every eight-digit number is rational and this is not interesting. However, the chances of an eight digit number agreeing with a small rational, say with numerator and denominator less than twenty-five, is small. Thus the list is comprised primarily of special function evaluations at various algebraic and simple transcendental values. The exact numbers included are described below. Each entry consists of the first eight digits after the decimal point of the number in question.
Author: Ethan D. Bloch
Publisher: Springer Science & Business Media
Published: 2011-05-27
Total Pages: 577
ISBN-13: 0387721762
DOWNLOAD EBOOKThis 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.
Author: John Harrison
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 193
ISBN-13: 1447115910
DOWNLOAD EBOOKThis book discusses the use of the real numbers in theorem proving. Typ ically, theorem provers only support a few 'discrete' datatypes such as the natural numbers. However the availability of the real numbers opens up many interesting and important application areas, such as the verification of float ing point hardware and hybrid systems. It also allows the formalization of many more branches of classical mathematics, which is particularly relevant for attempts to inject more rigour into computer algebra systems. Our work is conducted in a version of the HOL theorem prover. We de scribe the rigorous definitional construction of the real numbers, using a new version of Cantor's method, and the formalization of a significant portion of real analysis. We also describe an advanced derived decision procedure for the 'Tarski subset' of real algebra as well as some more modest but practically useful tools for automating explicit calculations and routine linear arithmetic reasoning. Finally, we consider in more detail two interesting application areas. We discuss the desirability of combining the rigour of theorem provers with the power and convenience of computer algebra systems, and explain a method we have used in practice to achieve this. We then move on to the verification of floating point hardware. After a careful discussion of possible correctness specifications, we report on two case studies, one involving a transcendental function.
Author: Stewart Shapiro
Publisher: OUP Oxford
Published: 2000-07-13
Total Pages: 323
ISBN-13: 0192893068
DOWNLOAD EBOOKThinking about Mathematics covers the range of philosophical issues and positions concerning mathematics. The text describes the questions about mathematics that motivated philosophers throughout history and covers historical figures such as Plato, Aristotle, Kant, and Mill. It also presents the major positions and arguments concerning mathematics throughout the twentieth century, bringing the reader up to the present positions and battle lines.
Author: Claude W. Burrill
Publisher:
Published: 1967
Total Pages: 182
ISBN-13:
DOWNLOAD EBOOKAuthor: Milo Beckman
Publisher: Penguin UK
Published: 2021-01-07
Total Pages: 224
ISBN-13: 0241507588
DOWNLOAD EBOOK'The whizz-kid making maths supercool. . . A brilliant book that takes everything we know (and fear) about maths out of the equation - starting with numbers' The Times 'A cheerful, chatty, and charming trip through the world of mathematics. . . Everyone should read this delightful book' Ian Stewart, author of Do Dice Play God? The only numbers in this book are the page numbers. The three main branches of abstract math - topology, analysis, and algebra - turn out to be surprisingly easy to grasp. Or at least, they are when our guide is a math prodigy. With forthright wit and warm charm, Milo Beckman upends the conventional approach to mathematics, inviting us to think creatively about shape and dimension, the infinite and the infinitesimal, symmetries, proofs, and all how all these concepts fit together. Why is there a million dollar prize for counting shapes? Is anything bigger than infinity? And how is the 'truth' of mathematics actually decided? A vivid and wholly original guide to the math that makes the world tick and the planets revolve, Math Without Numbers makes human and understandable the elevated and hypothetical, allowing us to clearly see abstract math for what it is: bizarre, beautiful, and head-scratchingly wonderful.
