Which Numbers Are Real

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Which Numbers Are Real

Author : Michael Henle
ISBN : 9780883857779
Genre : Mathematics
File Size : 74.84 MB
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Everyone 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.
Category: Mathematics

The Real Numbers

Author : John Stillwell
ISBN : 9783319015774
Genre : Mathematics
File Size : 30.65 MB
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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.
Category: Mathematics

Proceedings Of The Boston Colloquium For The Philosophy Of Science 1964 1966

Author : Robert S. Cohen
ISBN : 9789401035088
Genre : Science
File Size : 84.76 MB
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This third volume of Boston Studies in the Philosophy of Science contains papers which are based upon Colloquia from 1964 to 1966. In most cases, they have been substantially modified subsequent to presentation and discussion. Once again we publish work which goes beyond technical analysis of scientific theories and explanations in order to include philo sophical reflections upon the history of science and also upon the still problematic interactions between metaphysics and science. The philo sophical history of scientific ideas has increasingly been recognized as part of the philosophy of science, and likewise the cultural context of the genesis of such ideas. There is no school or attitude to be taken as de fining the scope or criteria of our Colloquium, and so we seek to under stand both analytic and historical aspects of science. This volume, as the previous two, constitutes a substantial part of our final report to the U. S. National Science Foundation, which has continued its support of the Boston Colloquium for the Philosophy of Science by a grant to Boston University. That report will be concluded by a subse quent volume of these Studies. It is a pleasure to record our thanks to the Foundation for its confidence and funds. We dedicate this book to the memory of Norwood Russell Hanson. During this academic year of 1966-67, this beloved and distinguished American philosopher participated in our Colloquium, and he did so before.
Category: Science

The Real Numbers And Real Analysis

Author : Ethan D. Bloch
ISBN : 9780387721774
Genre : Mathematics
File Size : 31.84 MB
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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.
Category: Mathematics

Precalculus With Limits

Author : Ron Larson
ISBN : 9781337516853
Genre : Mathematics
File Size : 85.58 MB
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Larson's PRECALCULUS WITH LIMITS is known for delivering the same sound, consistently structured explanations and exercises of mathematical concepts as the market-leading PRECALCULUS, with a laser focus on preparing students for calculus. In LIMITS, the author includes a brief algebra review of core precalculus topics along with coverage of analytic geometry in three dimensions and an introduction to concepts covered in calculus. With the Fourth Edition, Larson continues to revolutionize the way students learn material by incorporating more real-world applications, ongoing review, and innovative technology. How Do You See It? exercises give students practice applying the concepts, and new Summarize features, and Checkpoint problems reinforce understanding of the skill sets to help students better prepare for tests. The companion website LarsonPrecalculus.com offers free access to multiple tools and resources to supplement students’ learning. Stepped-out solution videos with instruction are available at CalcView.com for selected exercises throughout the text. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
Category: Mathematics

Fostering Children S Mathematical Power

Author : Arthur Baroody
ISBN : 9781135674052
Genre : Education
File Size : 48.6 MB
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First published in 1998. Routledge is an imprint of Taylor & Francis, an informa company.
Category: Education

Epistemic Foundations Of Fuzziness

Author : Kofi Kissi Dompere
ISBN : 9783540880844
Genre : Computers
File Size : 83.12 MB
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This monograph is a treatment on optimal fuzzy rationality as an enveloping of decision-choice rationalities where limited information, vagueness, ambiguities and inexactness are essential characteristics of our knowledge structure and reasoning processes. The volume is devoted to a unified system of epistemic models and theories of decision-choice behavior under total uncertainties composed of fuzzy and stochastic types. The unified epistemic analysis of decision-choice models and theories begins with the question of how best to integrate vagueness, ambiguities, limited information, subjectivity and approximation into the decision-choice process. The answer to the question leads to the shifting of the classical paradigm of reasoning to fuzzy paradigm. This is followed by discussions and establishment of the epistemic foundations of fuzzy mathematics where the nature and role of information and knowledge are explicated and represented. The epistemic foundation allows total uncertainties that constrain decision-choice activities, knowledge enterprise, logic and mathematical structures as our cognitive instruments to be discussed in reference to the phenomena of fuzzification, defuzzification and fuzzy logic. The discussions on these phenomena lead us to analyze and present models and theories on decision-choice rationality and the needed mathematics for problem formulation, reasoning and computations. The epistemic structures of two number systems made up of classical numbers and fuzzy numbers are discussed in relation to their differences, similarities and relative relevance to decision-choice rationality. The properties of the two number systems lead to the epistemic analysis of two mathematical systems that allow the partition of the mathematical space in support of decision-choice space of knowledge and non-knowledge production into four cognitively separate but interdependent cohorts whose properties are analyzed by the methods and techniques of category theory. The four cohorts are identified as non-fuzzy and non-stochastic, non-fuzzy and stochastic both of which belong to the classical paradigm and classical mathematical space; and fuzzy and non-stochastic, and fuzzy and stochastic cohorts both of which belong to the fuzzy paradigm and fuzzy mathematical space. The differences in the epistemic foundations of the two mathematical systems are discussed. The discussion leads to the establishment of the need for fuzzy mathematics and computing as a new system of reasoning in both exact and inexact sciences. The mathematical structures of the cohorts are imposed on the decision-choice process to allow a grouping of decision-choice models and theories. The corresponding classes of decision-choice theories have the same characteristics as the logico-mathematical cohorts relative to the assumed information-knowledge structures. The four groupings of models and theories on decision-choice activities are then classified as: 1) non-fuzzy and non-stochastic class with exact and full information-knowledge structure (no uncertainty), 2) non-fuzzy and stochastic class with exact and limited information-knowledge structure (stochastic uncertainty), 3) fuzzy and non-stochastic class with full and fuzzy information-knowledge structure (fuzzy uncertainty) and 4) Fuzzy and stochastic class with fuzzy and limited information-knowledge structure (fuzzy and stochastic uncertainties). All these different classes of decision choice problems have their corresponding rationalities which are fully discussed to present a unified logical system of theories on decision-choice process. The volume is concluded with epistemic discussions on the nature of contradictions and paradoxes viewed as logical decision-choice problems in the classical paradigm, and how these contradictions and paradoxes may be resolved through fuzzy paradigm and the methods and techniques of optimal fuzzy decision-choice rationality. The logical problem of sorites paradox with its resolution is given as an example. Interested audience includes those working in the areas of economies, decision-choice theories, philosophy of sciences, epistemology, mathematics, computer science, engineering, cognitive psychology, fuzzy mathematics and mathematics of fuzzy-stochastic processes.
Category: Computers

Let S Get Real Or Let S Not Play

Author : Mahan Khalsa
ISBN : 144063291X
Genre : Business & Economics
File Size : 38.45 MB
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The new way to transform a sales culture with clarity, authenticity, and emotional intelligence. Too often, the sales process is all about fear. Customers are afraid that they will be talked into making a mistake; salespeople dread being unable to close the deal and make their quotas. No one is happy. Mahan Khalsa and Randy Illig offer a better way. Salespeople, they argue, do best when they focus 100 percent on helping clients succeed. When customers are successful, both buyer and seller win. When they aren't, both lose. It's no longer sufficient to get clients to buy; a salesperson must also help the client reduce costs, increase revenues, and improve productivity, quality, and customer satisfaction. This book shares the unique FranklinCovey Sales Performance Group methodology that will help readers: · Start new business from scratch in a way both salespeople and clients can feel good about · Ask hard questions in a soft way · Close the deal by opening mindsClose the deal by opening minds From the Hardcover edition.
Category: Business & Economics

Number Shape Symmetry

Author : Diane L. Herrmann
ISBN : 9781466554658
Genre : Mathematics
File Size : 39.7 MB
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Through a careful treatment of number theory and geometry, Number, Shape, & Symmetry: An Introduction to Number Theory, Geometry, and Group Theory helps readers understand serious mathematical ideas and proofs. Classroom-tested, the book draws on the authors’ successful work with undergraduate students at the University of Chicago, seventh to tenth grade mathematically talented students in the University of Chicago’s Young Scholars Program, and elementary public school teachers in the Seminars for Endorsement in Science and Mathematics Education (SESAME). The first half of the book focuses on number theory, beginning with the rules of arithmetic (axioms for the integers). The authors then present all the basic ideas and applications of divisibility, primes, and modular arithmetic. They also introduce the abstract notion of a group and include numerous examples. The final topics on number theory consist of rational numbers, real numbers, and ideas about infinity. Moving on to geometry, the text covers polygons and polyhedra, including the construction of regular polygons and regular polyhedra. It studies tessellation by looking at patterns in the plane, especially those made by regular polygons or sets of regular polygons. The text also determines the symmetry groups of these figures and patterns, demonstrating how groups arise in both geometry and number theory. The book is suitable for pre-service or in-service training for elementary school teachers, general education mathematics or math for liberal arts undergraduate-level courses, and enrichment activities for high school students or math clubs.
Category: Mathematics

Algorithmic Learning Theory

Author : Ming Li
ISBN : 3540635777
Genre : Computers
File Size : 76.93 MB
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This book constitutes the strictly refereed post-workshop proceedings of the Second International Workshop on Database Issues for Data Visualization, held in conjunction with the IEEE Visualization '95 conference in Atlanta, Georgia, in October 1995. Besides 13 revised full papers, the book presents three workshop subgroup reports summarizing the contents of the book as well as the state-of-the-art in the areas of scientific data modelling, supporting interactive database exploration, and visualization related metadata. The volume provides a snapshop of current research in the area and surveys the problems that must be addressed now and in the future towards the integration of database management systems and data visualization.
Category: Computers

Principles Of Real Analysis

Author : S. C. Malik
ISBN : 9788122422771
Genre : Functions of real variables
File Size : 57.23 MB
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Category: Functions of real variables


Author : Albrecht Beutelspacher
ISBN : 9780486810461
Genre : Mathematics
File Size : 44.21 MB
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Posing the question "What exactly is a number?" a distinguished German mathematician presents this intriguing and accessible survey. Albrecht Beutelspacher ― founder of the renowned interactive mathematics museum, Mathematikum ― characterizes the wealth of experiences that numbers have to offer. In addition, he considers the many things that can be described by numbers and discusses which numbers possess special fascinations and pose lasting mysteries. Starting with natural numbers, the book examines representations of numbers, rational and irrational numbers, transcendental numbers, and imaginary and complex numbers. Readers will explore the history of numbers from Pythagoras to Fermat and discover such practical applications as cryptography and barcodes. A thoughtful and enlightening introduction to the past, present, and future of numbers, this volume will captivate mathematicians and nonmathematicians alike.
Category: Mathematics

Elements Of Real Anyalsis

Author : M.D.Raisinghania
ISBN : 9788121903066
Genre : Mathematics
File Size : 31.52 MB
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This book is an attempt to make presentation of Elements of Real Analysis more lucid. The book contains examples and exercises meant to help a proper understanding of the text. For B.A., B.Sc. and Honours (Mathematics and Physics), M.A. and M.Sc. (Mathematics) students of various Universities/ Institutions.As per UGC Model Curriculum and for I.A.S. and Various other competitive exams.
Category: Mathematics

By Parallel Reasoning

Author : Paul Bartha
ISBN : 9780199717057
Genre : Science
File Size : 83.7 MB
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In By Parallel Reasoning Paul Bartha proposes a normative theory of analogical arguments and raises questions and proposes answers regarding (i.) criteria for evaluating analogical arguments, (ii.) the philosophical justification for analogical reasoning, and (iii.) the place of scientific analogies in the context of theoretical confirmation.
Category: Science

Undergraduate Topology

Author : Aisling McCluskey
ISBN : 9780191006739
Genre : Mathematics
File Size : 40.38 MB
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This textbook offers an accessible, modern introduction at undergraduate level to an area known variously as general topology, point-set topology or analytic topology with a particular focus on helping students to build theory for themselves. It is the result of several years of the authors' combined university teaching experience stimulated by sustained interest in advanced mathematical thinking and learning, alongside established research careers in analytic topology. Point-set topology is a discipline that needs relatively little background knowledge, but sufficient determination to grasp ideas precisely and to argue with straight and careful logic. Research and long experience in undergraduate mathematics education suggests that an optimal way to learn such a subject is to teach it to yourself, pro-actively, by guided reading of brief skeleton notes and by doing your own spadework to fill in the details and to flesh out the examples. This text will facilitate such an approach for those learners who opt to do it this way and for those instructors who would like to encourage this so-called 'Moore approach', even for a modest segment of the teaching term or for part of the class. In reality, most students simply do not have the combination of time, background and motivation needed to implement such a plan fully. The accessibility, flexibility and completeness of this text enable it to be used equally effectively for more conventional instructor-led courses. Critically, it furnishes a rich variety of exercises and examples, many of which have specimen solutions, through which to gain in confidence and competence.
Category: Mathematics


Author : Robert Geroch
ISBN : 9781927763179
Genre : Topology
File Size : 27.61 MB
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This book is about the branch of mathematics called topology. But its larger purpose is to illustrate how mathematics works: The interplay between intuition on the one hand and a pure mathematical formulation on the other. Thus, we develop the axioms for a topological space, formulate definitions within the context of those axioms and actually prove theorems from the axioms. But underlying all this is our intuition about topology. It is this intuition that guides and gives "meaning" to the definitions we make and to the theorems we prove. No prior knowledge of mathematics is assumed. In fact, these were originally the notes for a course for freshman non-scientists. This book, including over 100 figures and problem sets with solutions, should be of interest to those who would like to understand what mathematics is all about, as well as those who would like to learn about the this important branch of mathematics.
Category: Topology