Elements of Mathematics

Elements of Mathematics takes readers on a fascinating tour that begins in elementary mathematics―but, as John Stillwell shows, this subject is not as elementary or straightforward as one might think. Not all topics that are part of today’s elementary mathematics were always considered as such, and great mathematical advances and discoveries had to occur in order for certain subjects to become “elementary.” Stillwell examines elementary mathematics from a distinctive twenty-first-century viewpoint and describes not only the beauty and scope of the discipline, but also its limits.

From Gaussian integers to propositional logic, Stillwell delves into arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, and logic. He discusses how each area ties into more advanced topics to build mathematics as a whole. Through a rich collection of basic principles, vivid examples, and interesting problems, Stillwell demonstrates that elementary mathematics becomes advanced with the intervention of infinity. Infinity has been observed throughout mathematical history, but the recent development of “reverse mathematics” confirms that infinity is essential for proving well-known theorems, and helps to determine the nature, contours, and borders of elementary mathematics.

Elements of Mathematics gives readers, from high school students to professional mathematicians, the highlights of elementary mathematics and glimpses of the parts of math beyond its boundaries.

Table of Contents

1 Elementary Topics

2 Arithmetic

3 Computation

4 Algebra

5 Geometry

6 Calculus

7 Combinatorics

8 Probability

9 Logic

10 Some Advanced Mathematics

Review

“The coherence of mathematics is on display in this masterful, beautifully written synthesis. With probing questions, lucid explanations, and fascinating examples and stories, John Stillwell reveals where the seemingly separate branches of math came from, how they intertwine, and why the hidden unifier is the deepest idea of all: infinity.”―Steven Strogatz, Cornell University and author of The Joy of x

“This is a beautifully written overview and excursion through elementary mathematics, written to appeal to nonspecialists. The book is distinguished by its breadth of scope and precise details. It will be a classic in the field.”―David Bressoud, Macalester College

“[A] sophisticated treatment of topics usually described as elementary.”―John Allen Paulos

About the Author

John Stillwell is professor of mathematics at the University of San Francisco. He is the author of Reverse Mathematics: Proofs from the Inside Out (Princeton).