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From Counting to Continuum - Edward Scheinerman

From Counting to Continuum

What Are Real Numbers, Really?
Buch | Hardcover
228 Seiten
2024
Cambridge University Press (Verlag)
978-1-009-53864-0 (ISBN)
CHF 87,25 inkl. MwSt
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Whole numbers, fractions, decimals: we frequently use them, but what are they exactly? How about exotic numbers like the square root of 2 or π? 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.
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 π, 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.

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.

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.

Erscheint lt. Verlag 31.12.2024
Zusatzinfo Worked examples or Exercises
Verlagsort Cambridge
Sprache englisch
Themenwelt Mathematik / Informatik Mathematik Analysis
Mathematik / Informatik Mathematik Arithmetik / Zahlentheorie
ISBN-10 1-009-53864-0 / 1009538640
ISBN-13 978-1-009-53864-0 / 9781009538640
Zustand Neuware
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