Real Numbers

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The Real Numbers

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Author : John Stillwell
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-10-16

The Real Numbers written by John Stillwell and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-10-16 with Mathematics categories.

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.

The Real Number System

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Author : John M. H. Olmsted
language : en
Publisher: Courier Dover Publications
Release Date : 2018-09-12

The Real Number System written by John M. H. Olmsted and has been published by Courier Dover Publications this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-09-12 with Mathematics categories.

Concise but thorough and systematic, this categorical discussion of the real number system presents a series of step-by-step axioms, each illustrated by examples. The highly accessible text is suitable for readers at varying levels of knowledge and experience: advanced high school students and college undergraduates as well as prospective high school and college instructors. The abundance of examples and the wealth of exercises—more than 300, all with answers provided—make this a particularly valuable book for self-study. The first two chapters examine fields and ordered fields, followed by an introduction to natural numbers and mathematical induction. Subsequent chapters explore composite and prime numbers, integers and rational numbers, congruences and finite fields, and polynomials and rational functions. Additional topics include intervals and absolute value, the axiom of completeness, roots and rational exponents, exponents and logarithms, and decimal expansions. A helpful Appendix concludes the text.

Which Numbers Are Real

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Author : Michael Henle
language : en
Publisher: American Mathematical Soc.
Release Date : 2012-12-31

Which Numbers Are Real written by Michael Henle and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-31 with Mathematics categories.

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 And Real Analysis

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Author : Ethan D. Bloch
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-05-27

The Real Numbers And Real Analysis written by Ethan D. Bloch and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-05-27 with Mathematics categories.

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.

Real Numbers

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Author : Joseph T. Sinclair
language : en
Publisher: Real Estate Investment Press
Release Date : 2008-10

Real Numbers written by Joseph T. Sinclair and has been published by Real Estate Investment Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-10 with categories.

Problems In Mathematical Analysis Real Numbers Sequences And Series

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Author : Wiesława J. Kaczor
language : en
Publisher: American Mathematical Soc.
Release Date : 2000

Problems In Mathematical Analysis Real Numbers Sequences And Series written by Wiesława J. Kaczor and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with Mathematics categories.

Solutions for all the problems are provided."--BOOK JACKET.

College Algebra

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Author : OpenStax
language : en
Publisher:
Release Date : 2016-10-11

College Algebra written by OpenStax and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-10-11 with Mathematics categories.

College Algebra provides a comprehensive exploration of algebraic principles and meets scope and sequence requirements for a typical introductory algebra course. The modular approach and richness of content ensure that the book meets the needs of a variety of courses. The text and images in this textbook are grayscale.

Are Numbers Real

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Author : Brian Clegg
language : en
Publisher: Macmillan
Release Date : 2016-12-06

Are Numbers Real written by Brian Clegg and has been published by Macmillan this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-12-06 with Mathematics categories.

Presents 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.

The Emergence Of Number

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Author : John N. Crossley
language : en
Publisher: World Scientific
Release Date : 1987

The Emergence Of Number written by John N. Crossley and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 1987 with Mathematics categories.

This book presents detailed studies of the development of three kinds of number. In the first part the development of the natural numbers from Stone-Age times right up to the present day is examined not only from the point of view of pure history but also taking into account archaeological, anthropological and linguistic evidence. The dramatic change caused by the introduction of logical theories of number in the 19th century is also treated and this part ends with a non-technical account of the very latest developments in the area of G”del's theorem. The second part is concerned with the development of complex numbers and tries to answer the question as to why complex numbers were not introduced before the 16th century and then, by looking at the original materials, shows how they were introduced as a pragmatic device which was only subsequently shown to be theoretically justifiable. The third part concerns the real numbers and examines the distinction that the Greeks made between number and magnitude. It then traces the gradual development of a theory of real numbers up to the precise formulations in the nineteeth century. The importance of the Greek distinction between the number line and the geometric line is brought into sharp focus.This is an new edition of the book which first appeared privately published in 1980 and is now out of print. Substantial revisions have been made throughout the text, incorporating new material which has recently come to light and correcting a few relatively minor errors. The third part on real numbers has been very extensively revised and indeed the last chapter has been almost completely rewritten. Many revisions are the results of comments from earlier readers of the book.

Floating Point Numerics For Games And Simulations

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Author : Nima Badizadegan
language : en
Publisher: CRC Press
Release Date : 2025-05-30

Floating Point Numerics For Games And Simulations written by Nima Badizadegan and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-05-30 with Mathematics categories.

Floating point is ubiquitous in computers, where it is the default way to represent non-integer numbers. However, few people understand it. We all see weird behavior sometimes, and many programmers treat it as a mystical and imprecise system of math that just works until it sometimes doesn’t. We hear that we shouldn’t trust floating point with money, we know that 0.1 + 0.2 does not equal 0.3, and “NaN” shows up in our logs when things break. We rarely hear why any of this is the case, and less about what to do about it. This book pulls back the veil on floating point and shows how this number system we program with every day works. It discusses how to leverage the number system for common calculations, particularly in graphics and simulations, and avoid pitfalls. Further, we will review methods that can give you either better performance or better accuracy on tasks like numerical integration and function approximation, so you can learn to make the right tradeoffs in your programs. This book builds upon a basic knowledge of calculus and linear algebra, working with illustrative examples that demonstrate concepts rather than relying on theoretical proofs. Along the way, we will learn why Minecraft has struggled with boat physics and what the heck John Carmack was thinking with Quake III’s infamous fast reciprocal square root algorithm. By the end of the book, you will be able to understand how to work with floating point in a practical sense, from tracking down and preventing error in small calculations to choosing numerical building blocks for complex 3D simulations. Gives insight into how and why floating-point math works Describes how floating-point error arises and how to avoid it Surveys numerical methods important to graphics and numerical simulations Includes modern techniques to apply to your numerical problems Shows how to hack the floating-point numbers to compute faster and more accurately