From Counting to Continuum

What are real numbers, really?

Forthcoming from Cambridge University Press

When it comes to numbers, the natural numbers are, well, perfectly natural. They’re the numbers we use to count things. On the other hand, the real numbers are, well, really difficult to describe rigorously.

The objective is to present a description of the real numbers that starts with the intuitive notion of counting and progresses step-by-step to the real numbers. It is meant for mathematics majors (pure and applied) to get an early glimpse at just what these things we call numbers are. We don’t have to wait until senior-year (or later!) analysis.

A recurrent theme is to define each sort of number in terms of a simpler kind. A typical step forward involves developing an equivalence relation on a previous idea and then the new numbers are the equivalence classes. Thus we start with finite sets; their equivalence classes are the natural numbers. Integers are equivalence classes of pairs of natural numbers. Modular numbers are equivalence classes of integers. Rational numbers are equivalence classes of pairs of integers. And so forth.

Learn more:

• Table of Contents
• Preface