## Description

This book grew from an article I wrote in 2008 for the centenary of Felix Klein’s Elementary Mathematics from an Advanced Standpoint. The article reflected on Klein’s view of elementary mathematics, which I found to be surprisingly modern, and made some comments on how his view might change in the light of today’s mathematics. With further reflection I realized that a discussion of elementary mathematics today should include not only some topics that are elementary from the twenty-first-century viewpoint, but also a more precise explanation of the term “elementary” than was possible in Klein’s day.

So, the first goal of the book is to give a bird’s eye view of elementary mathematics and its treasures. This view will sometimes be “from an advanced standpoint,” but nevertheless as elementary as possible. Readers with a good high school training in mathematics should be able to understand most of the book, though no doubt everyone will experience some difficulties, due to the wide range of topics. Bear in mind what G. H. Hardy (1942) said in a review of the excellent book What is Mathematics? by Courant and Robbins (1941): “a book on mathematics without difficulties would be worthless.”

The second goal of the book is to explain what “elementary” means, or at least to explain why certain pieces of mathematics seem to be “more elementary” than others. It might be thought that the concept of “elementary” changes continually as mathematics advances. Indeed, some topics now considered part of elementary mathematics are there because some great advance made them elementary. One such advance was the use of algebra in geometry, due to Fermat and Descartes. On the other hand, some concepts have remained persistently difficult. One is the concept of real number, which has caused headaches since the time of Euclid. Advances in logic in the twentieth century help to explain why the real numbers remain an “advanced” concept, and this idea will be gradually elaborated in the second half of the book. We will see how elementary mathematics collides with the real number concept from various directions, and how logic identifies the advanced nature of the real numbers—and, more generally, the nature of infinity—in various ways.

Those are the goals of the book. Here is how they are implemented. Chapter 1 briefly introduces eight topics that are important at the elementary level—arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, and logic—with some illustrative examples. The next eight chapters develop these topics in more detail, laying down their basic principles, solving some interesting problems, and making connections between them. Algebra is used in geometry, geometry in arithmetic, combinatorics in probability, logic in computation, and so on. Ideas are densely interlocked, even at the elementary level! The mathematical details are supplemented by historical and philosophical remarks at the end of each chapter, intended to give an overview of where the ideas came from and how they shape the concept of elementary mathematics.

Since we are exploring the scope and limits of elementary mathematics we cannot help crossing the boundary into advanced mathematics on occasion. We warn the reader of these incursions with a star (∗) in the titles of sections and subsections that touch upon advanced concepts. In chapter 10 we finally cross the line in earnest, with examples of non-elementary mathematics in each of the eight topics above. The purpose of these examples is to answer some questions that arose in the elementary chapters, showing that, with just small steps into the infinite, it is possible to solve interesting problems beyond the reach of elementary methods.