Real Numbers And Computers

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A Dictionary of Real Numbers

Author : Jonathan Borwein
Publisher : Springer Science & Business Media
ISBN 13 : 1461585104
Total Pages : 433 pages
Book Rating : 4.4/5 (615 download)

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Book Synopsis A Dictionary of Real Numbers by : Jonathan Borwein

Download or read book A Dictionary of Real Numbers written by Jonathan Borwein and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 433 pages. Available in PDF, EPUB and Kindle. Book excerpt: How 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.

Computability and Complexity in Analysis

Author : Jens Blanck
Publisher : Springer Science & Business Media
ISBN 13 : 3540421971
Total Pages : 403 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis Computability and Complexity in Analysis by : Jens Blanck

Download or read book Computability and Complexity in Analysis written by Jens Blanck and published by Springer Science & Business Media. This book was released on 2001-05-23 with total page 403 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the thoroughly refereed post-proceedings of the 4th International Workshop on Computability and Complexity in Analysis, CCA 2000, held in Swansea, UK, in September 2000. The 23 revised full papers presented together with a report on a competition of exact real arithmetic systems were carefully selected from 28 papers accepted for presentation during the workshop. The book presents a cross-section on current research on computability and complexity in analysis.

Real Computing Made Real

Author : Forman S. Acton
Publisher : Courier Corporation
ISBN 13 : 0486152936
Total Pages : 288 pages
Book Rating : 4.4/5 (861 download)

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Book Synopsis Real Computing Made Real by : Forman S. Acton

Download or read book Real Computing Made Real written by Forman S. Acton and published by Courier Corporation. This book was released on 2013-01-18 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt: This concise guide to trouble-shooting offers practical advice on detecting and removing the bugs, preserving significant figures, avoiding extraneous solutions, and finding efficient iterative processes for solving nonlinear equations. 1996 edition.

Handbook of Computability and Complexity in Analysis

Author : Vasco Brattka
Publisher : Springer Nature
ISBN 13 : 3030592340
Total Pages : 427 pages
Book Rating : 4.0/5 (35 download)

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Book Synopsis Handbook of Computability and Complexity in Analysis by : Vasco Brattka

Download or read book Handbook of Computability and Complexity in Analysis written by Vasco Brattka and published by Springer Nature. This book was released on 2021-06-04 with total page 427 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computable analysis is the modern theory of computability and complexity in analysis that arose out of Turing's seminal work in the 1930s. This was motivated by questions such as: which real numbers and real number functions are computable, and which mathematical tasks in analysis can be solved by algorithmic means? Nowadays this theory has many different facets that embrace topics from computability theory, algorithmic randomness, computational complexity, dynamical systems, fractals, and analog computers, up to logic, descriptive set theory, constructivism, and reverse mathematics. In recent decades computable analysis has invaded many branches of analysis, and researchers have studied computability and complexity questions arising from real and complex analysis, functional analysis, and the theory of differential equations, up to (geometric) measure theory and topology. This handbook represents the first coherent cross-section through most active research topics on the more theoretical side of the field. It contains 11 chapters grouped into parts on computability in analysis; complexity, dynamics, and randomness; and constructivity, logic, and descriptive complexity. All chapters are written by leading experts working at the cutting edge of the respective topic. Researchers and graduate students in the areas of theoretical computer science and mathematical logic will find systematic introductions into many branches of computable analysis, and a wealth of information and references that will help them to navigate the modern research literature in this field.

Numbers

Author : John Tabak
Publisher : Infobase Publishing
ISBN 13 : 0816068747
Total Pages : 239 pages
Book Rating : 4.8/5 (16 download)

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Book Synopsis Numbers by : John Tabak

Download or read book Numbers written by John Tabak and published by Infobase Publishing. This book was released on 2014-05-14 with total page 239 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents a survey of the history and evolution of the use of numbers and numerical quantities by different civilizations around the world.

Complexity and Real Computation

Author : Lenore Blum
Publisher : Springer Science & Business Media
ISBN 13 : 1461207010
Total Pages : 456 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Complexity and Real Computation by : Lenore Blum

Download or read book Complexity and Real Computation written by Lenore Blum and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt: The classical theory of computation has its origins in the work of Goedel, Turing, Church, and Kleene and has been an extraordinarily successful framework for theoretical computer science. The thesis of this book, however, is that it provides an inadequate foundation for modern scientific computation where most of the algorithms are real number algorithms. The goal of this book is to develop a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing. Along the way, the authors consider such fundamental problems as: * Is the Mandelbrot set decidable? * For simple quadratic maps, is the Julia set a halting set? * What is the real complexity of Newton's method? * Is there an algorithm for deciding the knapsack problem in a ploynomial number of steps? * Is the Hilbert Nullstellensatz intractable? * Is the problem of locating a real zero of a degree four polynomial intractable? * Is linear programming tractable over the reals? The book is divided into three parts: The first part provides an extensive introduction and then proves the fundamental NP-completeness theorems of Cook-Karp and their extensions to more general number fields as the real and complex numbers. The later parts of the book develop a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing.

Numbers and Computers

Author : Ronald T. Kneusel
Publisher : Springer
ISBN 13 : 3319505084
Total Pages : 346 pages
Book Rating : 4.3/5 (195 download)

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Book Synopsis Numbers and Computers by : Ronald T. Kneusel

Download or read book Numbers and Computers written by Ronald T. Kneusel and published by Springer. This book was released on 2017-02-09 with total page 346 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a book about numbers and how those numbers are represented in and operated on by computers. It is crucial that developers understand this area because the numerical operations allowed by computers, and the limitations of those operations, especially in the area of floating point math, affect virtually everything people try to do with computers. This book aims to fill this gap by exploring, in sufficient but not overwhelming detail, just what it is that computers do with numbers. Divided into two parts, the first deals with standard representations of integers and floating point numbers, while the second examines several other number representations. Details are explained thoroughly, with clarity and specificity. Each chapter ends with a summary, recommendations, carefully selected references, and exercises to review the key points. Topics covered include interval arithmetic, fixed-point numbers, big integers and rational arithmetic. This new edition has three new chapters: Pitfalls of Floating-Point Numbers (and How to Avoid Them), Arbitrary Precision Floating Point, and Other Number Systems. This book is for anyone who develops software including software engineers, scientists, computer science students, engineering students and anyone who programs for fun.

A Number for your Thoughts

Author : Lines M E
Publisher : CRC Press
ISBN 13 : 9780852744956
Total Pages : 228 pages
Book Rating : 4.7/5 (449 download)

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Book Synopsis A Number for your Thoughts by : Lines M E

Download or read book A Number for your Thoughts written by Lines M E and published by CRC Press. This book was released on 1986-01-01 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: Why do we count the way we do? What is a prime number or a friendly, perfect, or weird one? How many are there and who has found the largest yet known? What is the Baffling Law of Benford and can you really believe it? Do most numbers you meet in every day life really begin with a 1, 2, or 3? What is so special about 6174? Can cubes, as well as squares, be magic? What secrets lie hidden in decimals? How do we count the infinite, and is one infinity really larger than another? These and many other fascinating questions about the familiar 1, 2, and 3 are collected in this adventure into the world of numbers. Both entertaining and informative, A Number for Your Thoughts: Facts and Speculations about Numbers from Euclid to the Latest Computers contains a collection of the most interesting facts and speculations about numbers from the time of Euclid to the most recent computer research. Requiring little or no prior knowledge of mathematics, the book takes the reader from the origins of counting to number problems that have baffled the world's greatest experts for centuries, and from the simplest notions of elementary number properties all the way to counting the infinite.

The Real Number System

Author : John M. H. Olmsted
Publisher : Courier Dover Publications
ISBN 13 : 048682764X
Total Pages : 241 pages
Book Rating : 4.4/5 (868 download)

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Book Synopsis The Real Number System by : John M. H. Olmsted

Download or read book The Real Number System written by John M. H. Olmsted and published by Courier Dover Publications. This book was released on 2018-09-12 with total page 241 pages. Available in PDF, EPUB and Kindle. Book excerpt: Concise but thorough and systematic, this categorical discussion presents a series of step-by-step axioms. The highly accessible text includes numerous examples and more than 300 exercises, all with answers. 1962 edition.

Real Numbers and Computers

Author :
Publisher :
ISBN 13 :
Total Pages : 170 pages
Book Rating : 4.:/5 (397 download)

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Book Synopsis Real Numbers and Computers by :

Download or read book Real Numbers and Computers written by and published by . This book was released on 1996 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Theorem Proving with the Real Numbers

Author : John Harrison
Publisher : Springer Science & Business Media
ISBN 13 : 1447115910
Total Pages : 193 pages
Book Rating : 4.4/5 (471 download)

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Book Synopsis Theorem Proving with the Real Numbers by : John Harrison

Download or read book Theorem Proving with the Real Numbers written by John Harrison and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 193 pages. Available in PDF, EPUB and Kindle. Book excerpt: This 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.

The Real Numbers

Author : John Stillwell
Publisher : Springer Science & Business Media
ISBN 13 : 331901577X
Total Pages : 253 pages
Book Rating : 4.3/5 (19 download)

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Book Synopsis The Real Numbers by : John Stillwell

Download or read book The Real Numbers written by John Stillwell and published by Springer Science & Business Media. This book was released on 2013-10-16 with total page 253 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Connecting Discrete Mathematics and Computer Science

Author : David Liben-Nowell
Publisher : Cambridge University Press
ISBN 13 : 1009174746
Total Pages : 694 pages
Book Rating : 4.0/5 (91 download)

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Book Synopsis Connecting Discrete Mathematics and Computer Science by : David Liben-Nowell

Download or read book Connecting Discrete Mathematics and Computer Science written by David Liben-Nowell and published by Cambridge University Press. This book was released on 2022-08-04 with total page 694 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computer science majors taking a non-programming-based course like discrete mathematics might ask 'Why do I need to learn this?' Written with these students in mind, this text introduces the mathematical foundations of computer science by providing a comprehensive treatment of standard technical topics while simultaneously illustrating some of the broad-ranging applications of that material throughout the field. Chapters on core topics from discrete structures – like logic, proofs, number theory, counting, probability, graphs – are augmented with around 60 'computer science connections' pages introducing their applications: for example, game trees (logic), triangulation of scenes in computer graphics (induction), the Enigma machine (counting), algorithmic bias (relations), differential privacy (probability), and paired kidney transplants (graphs). Pedagogical features include 'Why You Might Care' sections, quick-reference chapter guides and key terms and results summaries, problem-solving and writing tips, 'Taking it Further' asides with more technical details, and around 1700 exercises, 435 worked examples, and 480 figures.

Introduction to Numerical Computations

Author : James S. Vandergraft
Publisher : Academic Press
ISBN 13 : 1483267091
Total Pages : 388 pages
Book Rating : 4.4/5 (832 download)

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Book Synopsis Introduction to Numerical Computations by : James S. Vandergraft

Download or read book Introduction to Numerical Computations written by James S. Vandergraft and published by Academic Press. This book was released on 2014-05-10 with total page 388 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computer Science and Applied Mathematics: Introduction to Numerical Computations, Second Edition introduces numerical algorithms as they are used in practice. This edition covers the usual topics contained in introductory numerical analysis textbooks that include all of the well-known and most frequently used algorithms for interpolation and approximation, numerical differentiation and integration, solution of linear systems and nonlinear equations, and solving ordinary differential equations. A complete discussion of computer arithmetic, problems that arise in the computer evaluation of functions, and cubic spline interpolation are also provided. This text likewise discusses the Newton formulas for interpolation and adaptive methods for integration. The level of this book is suitable for advanced undergraduate students and readers with elementary mathematical background.

Real Numbers

Author : Jean E. Cunningham
Publisher : Jcc Press
ISBN 13 : 9780999380109
Total Pages : 200 pages
Book Rating : 4.3/5 (81 download)

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Book Synopsis Real Numbers by : Jean E. Cunningham

Download or read book Real Numbers written by Jean E. Cunningham and published by Jcc Press. This book was released on 2017-09-30 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: How management accounting evolved with Lean principles.

Computer Arithmetic and Validity

Author : Ulrich Kulisch
Publisher : Walter de Gruyter
ISBN 13 : 3110301792
Total Pages : 456 pages
Book Rating : 4.1/5 (13 download)

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Book Synopsis Computer Arithmetic and Validity by : Ulrich Kulisch

Download or read book Computer Arithmetic and Validity written by Ulrich Kulisch and published by Walter de Gruyter. This book was released on 2013-04-30 with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the revised and extended second edition of the successful basic book on computer arithmetic. It is consistent with the newest recent standard developments in the field. The book shows how the arithmetic and mathematical capability of the digital computer can be enhanced in a quite natural way. The work is motivated by the desire and the need to improve the accuracy of numerical computing and to control the quality of the computed results (validity). The accuracy requirements for the elementary floating-point operations are extended to the customary product spaces of computations including interval spaces. The mathematical properties of these models are extracted into an axiomatic approach which leads to a general theory of computer arithmetic. Detailed methods and circuits for the implementation of this advanced computer arithmetic on digital computers are developed in part two of the book. Part three then illustrates by a number of sample applications how this extended computer arithmetic can be used to compute highly accurate and mathematically verified results. The book can be used as a high-level undergraduate textbook but also as reference work for research in computer arithmetic and applied mathematics.

Computer Science Illuminated

Author : Nell Dale
Publisher : Jones & Bartlett Learning
ISBN 13 : 0763776467
Total Pages : 672 pages
Book Rating : 4.7/5 (637 download)

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Book Synopsis Computer Science Illuminated by : Nell Dale

Download or read book Computer Science Illuminated written by Nell Dale and published by Jones & Bartlett Learning. This book was released on 2010-03-10 with total page 672 pages. Available in PDF, EPUB and Kindle. Book excerpt: Revised and updated with the latest information in the field, the Fourth Edition of Computer Science Illuminated continues to engage and enlighten students on the fundamental concepts and diverse capabilities of computing. Written by two of today’s most respected computer science educators, Nell Dale and John Lewis, the text provides a broad overview of the many aspects of the discipline from a generic view point. Separate program language chapters are available as bundle items for those instructors who would like to explore a particular programming language with their students. The many layers of computing are thoroughly explained beginning with the information layer, working through the hardware, programming, operating systems, application, and communication layers, and ending with a discussion on the limitations of computing. Perfect for introductory computing and computer science courses, the fourth edition's thorough presentation of computing systems provides computer science majors with a solid foundation for further study, and offers non-majors a comprehensive and complete introduction to computing.