Understanding the natural numbers, which we use to count things, comes naturally. Meanwhile, the real numbers, which include a wide range of numbers from whole numbers to fractions to exotic ones like p, are, frankly, really difficult to describe rigorously. Instead of waiting to take a theorem-proof graduate course to appreciate the real numbers, readers new to university-level mathematics can explore the core ideas behind the construction of the real numbers in this friendly introduction. Beginning with the intuitive notion of counting, the book progresses step-by-step to the real numbers. Each sort of number is defined in terms of a simpler kind by developing an equivalence relation on a previous idea. We find the finite sets' equivalence classes are the natural numbers. Integers are equivalence classes of pairs of natural numbers. Modular numbers are equivalence classes of integers. And so forth. Exercises and their solutions are included.
Whole numbers, fractions, decimals: we frequently use them, but what are they exactly? How about exotic numbers like the square root of 2 or p? You don't need to be a graduate student to explore the core ideas behind the construction of the real numbers: just read this book.
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'This book takes us on a fascinating journey through the world of turning intuition into rigor. Deep and elegant ideas are presented at just the right level of detail to keep the reader interested and engaged. A perfect introduction for anyone who is open to seeing the beauty of mathematics.' Maria Chudnovsky, Princeton University 'Providing a careful, rigorous construction of the field of real numbers is among the greatest intellectual achievements in human history. This outstanding book will take you on an engaging and exciting guided tour of the real numbers that explains the mysteries and conveys the magic of this beautiful conceptual foundation for mathematical analysis.' James R. Schatz, Johns Hopkins University 'Scheinerman is a master expositor who here presents a patient and thorough entry into the world of real numbers, providing sufficient precision and detail to make the definitions mathematically correct but in a way that is accessible to a wide audience.' David Bressoud, Macalester College
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Whole numbers, fractions, decimals: we frequently use them, but what are they exactly? Let's explore the core ideas behind them.
Preface;
0. Prelude;
1. Fundamentals;
2. N: natural numbers;
3. Z:
integers;
4. Zm: modular arithmetic;
5. Q: rational numbers;
6. R: real
numbers I, Dedekind cuts;
7. R: real numbers II, Cauchy sequences;
8. R: real
numbers III, complete ordered fields;
9. C: complex numbers;
10. Further
extensions; Answers to exercises; Bibliography; Index.
Edward Scheinerman is Professor of Applied Mathematics and Statistics at Johns Hopkins University. He is the author of various books including textbooks, a research monograph, and a volume for general readership, The Mathematics Lover's Companion (2017). He has twice been awarded the Mathematical Association of America's Lester R. Ford Award for outstanding mathematical exposition and has received numerous teaching awards at Johns Hopkins. His research publications are in discrete mathematics.
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