Are Numbers Real?

The Uncanny Relationships Between Maths and the Physical World

Author: Brian Clegg

Publisher: Hachette UK

ISBN: 1472139771

Category: Science

Page: 304

View: 3444

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Have you ever wondered what humans did before numbers existed? How they organized their lives, traded goods, or kept track of their treasures? What would your life be like without them? Numbers began as simple representations of everyday things, but mathematics rapidly took on a life of its own, occupying a parallel virtual world. In Are Numbers Real? Brian Clegg explores the way that maths has become more and more detached from reality, yet despite this is driving the development of modern physics. From devising a new counting system based on goats, through the weird and wonderful mathematics of imaginary numbers and infinity to the debate over whether mathematics has too much influence on the direction of science, this fascinating and accessible book opens the reader's eyes to the hidden reality of the strange yet familiar world of numbers.

Eine kleine Geschichte der Unendlichkeit

Author: Brian Clegg

Publisher: Rowohlt Verlag GmbH

ISBN: 364404371X

Category: Fiction

Page: 352

View: 5128

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Philosophen und Mathematiker hat das Nachsinnen über das Wesen des Unendlichen buchstäblich den Verstand geraubt – und dennoch ist es ein Konzept, das immer wieder unser Leben bestimmt. In diesem mit Anekdoten und Geschichten gespickten Buch nimmt uns Brian Clegg mit auf eine Reise durch das Grenzland zwischen dem extrem Großen und dem Ultimativen, von Archimedes, der die Zahl der Sandkörner bestimmte, die das Universum füllen würden, bis zu den neuesten Theorien über die physikalische Realität des Unendlichen.

Precalculus with Calculus Previews

Author: Dennis G. Zill,Jacqueline M. Dewar

Publisher: Jones & Bartlett Publishers

ISBN: 1449645151

Category: Mathematics

Page: 549

View: 9794

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Building off the success of Zill and Dewar's popular Essentials of Precalculus with Calculus Previews, the new Fifth Edition continues to include all of the outstanding features and learning tools found in the original text while incorporating additional topics of coverage that some courses may require. With a continued effort to keep the text complete, yet concise, the authors have included four additional chapters making the text a clear choice for many mainstream courses. Additional chapters include a new chapter on Polar Coordinates, as well as Triangle Trigonometry, Systems of Equations and Inequalities, and Sequences and Series. This student-friendly, full-color text offers numerous exercise sets and examples to aid in students' learning and understanding, and graphs and figures throughout serve to better illuminate key concepts. The exercise sets include engaging problems that focus on algebra, graphing, and function theory, the subtext of so many calculus problems. The authors are careful to use the terminology of calculus in an informal and comprehensible way to facilitate the student's successful transition into future calculus courses. New to the Fifth Edition: Includes a new Chapter 8, Polar Coordinates. A new appendix on Complex Numbers has been added. Available with a new graphing calculator manual, Exploring Mathematics: Solving Problems with the TI-84 Plus Graphing Calculator. Available with WebAssign Key Features: Vibrant full-color design illuminates key concepts and improves students' comprehension of graphs and figures. Translating Words into Functions section illustrates how to translate a verbal description into a symbolic representation of a function and demonstrates these translations with actual calculus problems. Chapter Review Exercises include problems that focus on algebra, graphing and function theory, the sub-text of so many calculus problems. Review questions include conceptual fill-in-the-blank and true-false questions, as well as numerous thought-provoking exercises. The Calculus Preview found at the end of each chapter offers students a glimpse of a single calculus concept along with the algebraic, logarithmic, and trigonometric manipulations that are necessary for the successful completion of typical problems related to that concept.

Essentials of Precalculus with Calculus Previews

Author: Dennis G. Zill,Jacqueline M. Dewar

Publisher: Jones & Bartlett Publishers

ISBN: 1449614973

Category: Mathematics

Page: 440

View: 6297

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Perfect for the one-term course, Essentials of Precalculus with Calculus Previews, Fifth Edition provides a complete, yet concise, introduction to precalculus concepts, focusing on important topics that will be of direct and immediate use in most calculus courses. Consistent with Professor Zill's eloquent writing style , this full-color text offers numerous exercise sets and examples to aid in student comprehension, while graphs and figures throughout serve to illuminate key concepts. The exercise sets include engaging problems that focus on algebra, graphing, and function theory, the sub-text of many calculus problems. The authors are careful to use calculus terminology in an informal and accessible way to facilitate the students successful transition into future calculus courses. With an outstanding collection of student and instructor resources, Essentials of Precalculus with Calculus Previews offers a complete teaching and learning package. Key Features: • Available with WebAssign Online Homework and Grading System • Vibrant four-color design illuminates key concepts and improves students' comprehension of graphs and figures. • Translating Words into Functions section illustrates how to translate a verbal description into a symbolic representation of a function and demonstrates these translations with actual calculus problems. • Chapter Review Exercises include problems that focus on the algebra, graphing, and function theory, the sub-text of so many calculus problems. Review questions include conceptual fill--in-the-blank and true/false, as well as numerous thought-provoking exercises. • The Calculus Preview found at the end of each chapter offers students a glimpse of a single calculus concept along with the algebraic, logarithmic, and trigonometric manipulations that are necessary for the successful completion on typical problems related to that concept. • Provides a complete teaching and learning program with numerous student and instructor resources, including the Student Resource Manual, WebAssign Access, Complete eLearning Center, and • Complete Instructor Solutions Manual. • Includes a new section on simple harmonic motion in Chapter 4. • A new section of parametric equations, as well as a new calculus preview of 3-space, has been added to Chapter 6. • Rotation of polar graphs is now discussed in Section 6.6 • The discussion of the hyperbolic functions in Section 5.4 has been expanded. • Numerous new problems have been added throughout the text. • The final exam at the end of the text has been expanded.

Number and Numbers

Author: Alain Badiou

Publisher: Polity

ISBN: 0745638791

Category: Mathematics

Page: 240

View: 2114

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Badiou goes through the most important innovations in the 19th-century theory of numbers, covering Frege, Dedekind, Peano and Cantor, and presents his own theory of numbers, one that has broader implications for social and political philosophy.

Which Numbers Are Real?

Author: Michael Henle

Publisher: MAA

ISBN: 0883857774

Category: Mathematics

Page: 219

View: 6610

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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.

The Real Numbers

An Introduction to Set Theory and Analysis

Author: John Stillwell

Publisher: Springer Science & Business Media

ISBN: 331901577X

Category: Mathematics

Page: 244

View: 1890

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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.

Real World Numbers

The Number System

Author: Matthew Hill

Publisher: AuthorHouse

ISBN: 1467026670

Category: Education

Page: 36

View: 3970

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This book provides support in keeping with the major goals of National Council of Teachers of Mathematics curriculum. It provides an important mathematical topic, the number system, which will be learned through K-8th grade, and used through high school and college. The instructional emphasis is designed to communicate knowledge and skills in mathematics across different grade levels, while offering the opportunity for children to learn about the number system in a fun and easy way. The book focuses on key areas of important emphasis, necessary for building math fluency in pre-algebra and algebra.

High Performance MySQL

Optimization, Backups, and Replication

Author: Baron Schwartz,Peter Zaitsev,Vadim Tkachenko

Publisher: "O'Reilly Media, Inc."

ISBN: 1449332498

Category: Computers

Page: 826

View: 6574

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How can you bring out MySQL’s full power? With High Performance MySQL, you’ll learn advanced techniques for everything from designing schemas, indexes, and queries to tuning your MySQL server, operating system, and hardware to their fullest potential. This guide also teaches you safe and practical ways to scale applications through replication, load balancing, high availability, and failover. Updated to reflect recent advances in MySQL and InnoDB performance, features, and tools, this third edition not only offers specific examples of how MySQL works, it also teaches you why this system works as it does, with illustrative stories and case studies that demonstrate MySQL’s principles in action. With this book, you’ll learn how to think in MySQL. Learn the effects of new features in MySQL 5.5, including stored procedures, partitioned databases, triggers, and views Implement improvements in replication, high availability, and clustering Achieve high performance when running MySQL in the cloud Optimize advanced querying features, such as full-text searches Take advantage of modern multi-core CPUs and solid-state disks Explore backup and recovery strategies—including new tools for hot online backups

Warum wir arbeiten

TED Books

Author: Barry Schwartz

Publisher: S. Fischer Verlag

ISBN: 3104036233

Category: Literary Collections

Page: 112

View: 7356

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Der Psychologieprofessor Barry Schwartz findet in seinem TED-Book ›Warum wir arbeiten‹ eine ganz überraschende, komplexe und eindringliche Antwort auf diese auf den ersten Blick so simple Frage. Er zerstört den Mythos, dass es bei der Arbeit nur ums Geldverdienen gehe und fordert – unterstützt durch zahlreiche Studien und Anekdoten –, dass wir neue Wege finden müssen, um unser Verhältnis zur Arbeit neu zu definieren. Schwartz nimmt uns mit in Krankenhäuser und Friseursalons, Fabriken und Vorstandsetagen und zeigt, welche unterschiedlichen Formen die Arbeit annehmen kann, welche Funktion sie für unsere Kultur innehat und wie jeder von uns seinen eigenen Weg zum Glück am Arbeitsplatz findet.

Prime Obsession

Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

Author: John Derbyshire

Publisher: Joseph Henry Press

ISBN: 0309141257

Category: Science

Page: 446

View: 8050

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In August 1859 Bernhard Riemann, a little-known 32-year old mathematician, presented a paper to the Berlin Academy titled: "On the Number of Prime Numbers Less Than a Given Quantity." In the middle of that paper, Riemann made an incidental remark â€" a guess, a hypothesis. What he tossed out to the assembled mathematicians that day has proven to be almost cruelly compelling to countless scholars in the ensuing years. Today, after 150 years of careful research and exhaustive study, the question remains. Is the hypothesis true or false? Riemann's basic inquiry, the primary topic of his paper, concerned a straightforward but nevertheless important matter of arithmetic â€" defining a precise formula to track and identify the occurrence of prime numbers. But it is that incidental remark â€" the Riemann Hypothesis â€" that is the truly astonishing legacy of his 1859 paper. Because Riemann was able to see beyond the pattern of the primes to discern traces of something mysterious and mathematically elegant shrouded in the shadows â€" subtle variations in the distribution of those prime numbers. Brilliant for its clarity, astounding for its potential consequences, the Hypothesis took on enormous importance in mathematics. Indeed, the successful solution to this puzzle would herald a revolution in prime number theory. Proving or disproving it became the greatest challenge of the age. It has become clear that the Riemann Hypothesis, whose resolution seems to hang tantalizingly just beyond our grasp, holds the key to a variety of scientific and mathematical investigations. The making and breaking of modern codes, which depend on the properties of the prime numbers, have roots in the Hypothesis. In a series of extraordinary developments during the 1970s, it emerged that even the physics of the atomic nucleus is connected in ways not yet fully understood to this strange conundrum. Hunting down the solution to the Riemann Hypothesis has become an obsession for many â€" the veritable "great white whale" of mathematical research. Yet despite determined efforts by generations of mathematicians, the Riemann Hypothesis defies resolution. Alternating passages of extraordinarily lucid mathematical exposition with chapters of elegantly composed biography and history, Prime Obsession is a fascinating and fluent account of an epic mathematical mystery that continues to challenge and excite the world. Posited a century and a half ago, the Riemann Hypothesis is an intellectual feast for the cognoscenti and the curious alike. Not just a story of numbers and calculations, Prime Obsession is the engrossing tale of a relentless hunt for an elusive proof â€" and those who have been consumed by it.

Die Wirklichkeit, die nicht so ist, ...

Eine Reise in die Welt der Quantengravitation

Author: Carlo Rovelli

Publisher: Rowohlt Verlag GmbH

ISBN: 3644052514

Category: Science

Page: 320

View: 6145

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Was ist Wirklichkeit? Existieren Raum und Zeit tatsächlich, wenn wir uns anschicken, die elementarsten Grundlagen unserer Existenz zu erforschen? Wie viel davon können wir überhaupt verstehen? Carlo Rovelli beschäftigt sich seit vielen Jahren damit, die Grenzen unseres Verstehens zu erweitern. In diesem Buch nimmt er uns mit auf eine Reise, die von dem Realitätsverständnis der griechischen Klassik bis zur Schleifenquantengravitation führt. Ein großer Physiker unserer Zeit macht sich auf, uns ein neues Welt-Bild zu zeichnen: mit einem physikalischen Universum ohne Zeit, einer Raumzeit, die aus Schleifen und Körnchen besteht und in der Unendlichkeit nicht existiert. Einer Kosmologie, die ohne Urknall und Paralleluniversen auskommt und hier zum ersten Mal von einem ihrer «Erfinder» für ein breites Publikum einfach und ausführlich erklärt wird. Ein Buch über «die großen Herausforderungen der gegenwärtigen Naturwissenschaften, die all unser Wissen über die Natur in Frage stellen» (Rovelli).

Algebra in Ancient and Modern Times

Author: V. S. Varadarajan

Publisher: American Mathematical Soc.

ISBN: 9780821809891

Category: Mathematics

Page: 142

View: 2620

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From the reviews: This is a fine book on two counts. First ... there is the singularly excellent treatment of the solution of biquadratic equations. Second, it paints a strong picture of mathematics as a very long sequence of accomplishments, each building on the ones before, in a way that beginning mathematicians can understand and appreciate it. It paints the picture in a concise and economical style, the style that mathematicians find elegant. I would particularly recommend Algebra in Ancient and Modern Times to strong high school students, to high school algebra teachers, to people who want a history of mathematics with a lot of mathematics in the history, and to anyone who needs to know how to find an analytic solution to a nasty fourth degree polynomial. -- MAA Online Varadarajan spins a captivating tale, and the mathematics is first-rate. The book belongs on the shelf of any teacher of algebra ... The great treasure of this book is the discussion of the work of the great Hindu mathematicians Aryabhata (c.476-550), Brahmagupta (c.598-665), and Bhaskara (c.1114-1185). Teachers of mathematics history will be especially interested in Varadarajan's exposition of the remarkable cakravala, an algorithm for solving $X^2 - NY^2= \pm 1$. The book contains many exercises that enhance and supplement the text and that also include historical information. Many of the exercises ask readers to apply the historical techniques. Some of the exercises are quite difficult and will challenge any student. --Mathematics Teacher This text offers a special account of Indian work in diophantine equations during the 6th through 12th centuries and Italian work on solutions of cubic and biquadratic equations from the 11th through 16th centuries. The volume traces the historical development of algebra and the theory of equations from ancient times to the beginning of modern algebra, outlining some modern themes such as the fundamental theorem of algebra, Clifford algebras, and quaternions. It is geared toward undergraduates who have no background in calculus. V. S. Varadarajan is a professor of mathematics at the University of California, Los Angeles.

Numbers, Sets and Axioms

The Apparatus of Mathematics

Author: A. G. Hamilton

Publisher: Cambridge University Press

ISBN: 9780521287616

Category: Mathematics

Page: 255

View: 6033

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Following the success of Logic for Mathematicians, Dr Hamilton has written a text for mathematicians and students of mathematics that contains a description and discussion of the fundamental conceptual and formal apparatus upon which modern pure mathematics relies. The author's intention is to remove some of the mystery that surrounds the foundations of mathematics. He emphasises the intuitive basis of mathematics; the basic notions are numbers and sets and they are considered both informally and formally. The role of axiom systems is part of the discussion but their limitations are pointed out. Formal set theory has its place in the book but Dr Hamilton recognises that this is a part of mathematics and not the basis on which it rests. Throughout, the abstract ideas are liberally illustrated by examples so this account should be well-suited, both specifically as a course text and, more broadly, as background reading. The reader is presumed to have some mathematical experience but no knowledge of mathematical logic is required.