From Counting to Continuum: What Are Real Numbers, Really? - Hardcover

Edward Scheinerman

Hardcover

ISBN 10: 1009538640 ISBN 13: 9781009538640

Publisher: Cambridge University Press, 2024

Synopsis

Whole numbers, fractions, decimals: we frequently use them, but what are they exactly? Let's explore the core ideas behind them.

About this title

About the Author

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.

Edition details

• Publisher: Cambridge University Press
• Publication date: 2024
• Language: English
• ISBN 10: 1009538640
• ISBN 13: 9781009538640
• Binding: Hardcover
• Number of pages: 228