Real Numbers

John Stillwell is a professor of mathematics at the University of San Francisco. He is also an accomplished author, having published several books with Springer, including Mathematics and Its History; The Four Pillars of Geometry; Elements of Algebra; Numbers and Geometry; and many more.

From the book reviews: "This book is an interesting introduction to set theory and real analysis embedded in properties of the real numbers. ... The 300-plus problems are frequently challenging and will interest both upper-level undergraduate students and readers with a strong mathematical background. ... A list of approximately 100 references at the end of the book will help students to further explore the topic. ... Summing Up: Recommended. Lower-division undergraduates." (D. P. Turner, Choice, Vol. 51 (11), August, 2014) "This is an informal look at the nature of the real numbers ... . There are extensive historical notes about the evolution of real analysis and our understanding of real numbers. ... Stillwell has deliberately set out to provide a different sort of construction where you understand what the foundation is supporting and why it is important. I think this is very successful, and his book ... is much more informative and enjoyable." (Allen Stenger, MAA Reviews, February, 2014) "This book will be fully appreciated by either professional mathematicians or those students, who already have passed a course in analysis or set theory. ... The book contains a quantity of motivation examples, worked examples and exercises, what makes it suitable also for self-study." (Vladimir Janis, zbMATH, 2014) "The book offers a rigorous foundation of the real number system. It is intended for senior undergraduates who have already studied calculus, but a wide range of readers will find something interesting, new, or instructive in it. ... This is an extremely reader-friendly book. It is full of interesting examples, very clear explanations, historical background, applications. Each new idea comes after proper motivation." (Laszlo Imre Szabo, Acta Scientiarum Mathematicarum (Szeged), Vol. 80 (1-2), 2014)

• The Fundamental Questions.- From Discrete to Continuous.- Infinite Sets.- Functions and Limits.- Open Sets and Continuity.- Ordinals.- The Axiom of Choice.- Borel Sets.- Measure Theory.- Reflections.- Bibliography.- Index.