Edward Scheinerman, professor of applied mathematics and statistics and vice dean for special projects at the Whiting School of Engineering, has recently published Counting to Continuum (Cambridge University Press), a book that delves into the foundational concepts of real numbers. Known for his contributions to discrete mathematics, Scheinerman applies his expertise to this exploration of the nature of numbers.
In this Q&A, he shares the inspiration behind the book, discusses why understanding the nature of numbers matters, and unpacks the hidden complexities of real numbers.
What inspired you to write this book?
Our Applied Mathematics and Statistics undergraduates are simply amazing. They will take their skills in probability, statistics, optimization, and other areas to solve problems in all walks of life from health care to finance to business logistics. Every day, they use real numbers, but nowhere in our curriculum do they see a mathematically rigorous definition of what real numbers are.
Why is it important that people learn the technical details of what numbers are?
From a practical point of view, it’s not important at all. By analogy, a person can drive a car without any clue about how the engine works. But given how central numbers are to all of mathematics, I think at least some students would want to see exactly what these things we call numbers are.
What are natural versus real numbers? And if real numbers are not what we use to count everyday objects, etc., why are they called real?
The natural numbers are 0, 1, 2, 3, and so on: The numbers we use for counting. The real numbers are those that you can write in decimal notation, including negative numbers and those requiring infinite digits. They form a continuous number line referred to as a continuum.
Neither of these descriptions is a proper mathematical definition. The objective of the book is to give iron-clad precise definitions.
The terms “natural” and “real” are technical—real numbers aren’t “real”; they are abstract concepts like all mathematical objects.
Don’t departments teach this already?
Often the careful development of the real number systems doesn’t occur until graduate school, or for advanced undergraduate majors. I wanted to create a way for students—and not just our AMS majors—to be able to see this material.
What do readers need to know to get the most out of your book?
The book is written at a pre-calculus level. Readers need to be comfortable with high school mathematics topics such as algebra, trigonometry, and the concepts of set and function. To be effective, readers need to do more than read; they also need to engage with the material by doing the exercises in every chapter. The book has plenty of problems to solve and complete solutions.
Is this book for math students only?
Not at all. While I had our AMS students in mind when writing, there are curious people across all fields. Math students want to know about art. Biology students want to know about theater. Not everyone will stop to wonder about what exactly the real numbers are, but many will want to know. This book provides an accessible entry point, whereas most other books on the topic are aimed at advanced mathematics students.
Your last book, The Mathematics Lover’s Companion, came out in 2017?
Yes, I am finishing work on a general-readership book about the mathematics of infinity. That is expected to be available in early 2026.