Unlike some other reproductions of classic texts (1) We have not used OCR(Optical Character Recognition), as this leads to bad quality books with introduced typos. (2) In books where there are images such as portraits, maps, sketches etc We have endeavoured to keep the quality of these images, so they represent accurately the original artefact. Although occasionally there may be certain imperfections with these old texts, we feel they deserve to be made available for future generations to enjoy.

This lively introductory text exposes the student to the rewards of a rigorous study of functions of a real variable. In each chapter, informal discussions of questions that give analysis its inherent fascination are followed by precise, but not overly formal, developments of the techniques needed to make sense of them. By focusing on the unifying themes of approximation and the resolution of paradoxes that arise in the transition from the finite to the infinite, the text turns what could be a daunting cascade of definitions and theorems into a coherent and engaging progression of ideas. Acutely aware of the need for rigor, the student is much better prepared to understand what constitutes a proper mathematical proof and how to write one. Fifteen years of classroom experience with the first edition of Understanding Analysis have solidified and refined the central narrative of the second edition. Roughly 150 new exercises join a selection of the best exercises from the first edition, and three more project-style sections have been added. Investigations of Euler's computation of (2), the Weierstrass Approximation Theorem, and the gamma function are now among the book's cohort of seminal results serving as motivation and payoff for the beginning student to master the methods of analysis.

How playwrights from Alfred Jarry and Samuel Beckett to Tom Stoppard and Simon McBurney brought the power of abstract mathematics to the human stage The discovery of alternate geometries, paradoxes of the infinite, incompleteness, and chaos theory revealed that, despite its reputation for certainty, mathematical truth is not immutable, perfect, or even perfectible. Beginning in the last century, a handful of adventurous playwrights took inspiration from the fractures of modern mathematics to expand their own artistic boundaries. Originating in the early avant-garde, mathematics-infused theater reached a popular apex in Tom Stoppard’s 1993 play Arcadia. In The Proof Stage, mathematician Stephen Abbott explores this unlikely collaboration of theater and mathematics. He probes the impact of mathematics on such influential writers as Alfred Jarry, Samuel Beckett, Bertolt Brecht, and Stoppard, and delves into the life and mathematics of Alan Turing as they are rendered onstage. The result is an unexpected story about the mutually illuminating relationship between proofs and plays—from Euclid and Euripides to Gödel and Godot. Theater is uniquely poised to discover the soulful, human truths embedded in the austere theorems of mathematics, but this is a difficult feat. It took Stoppard twenty-five years of experimenting with the creative possibilities of mathematics before he succeeded in making fractal geometry and chaos theory integral to Arcadia’s emotional arc. In addition to charting Stoppard’s journey, Abbott examines the post-Arcadia wave of ambitious works by Michael Frayn, David Auburn, Simon McBurney, Snoo Wilson, John Mighton, and others. Collectively, these gifted playwrights transform the great philosophical upheavals of mathematics into profound and sometimes poignant revelations about the human journey.

This lively introductory text exposes the student to the rewards of a rigorous study of functions of a real variable. In each chapter, informal discussions of questions that give analysis its inherent fascination are followed by precise, but not overly formal, developments of the techniques needed to make sense of them. By focusing on the unifying themes of approximation and the resolution of paradoxes that arise in the transition from the finite to the infinite, the text turns what could be a daunting cascade of definitions and theorems into a coherent and engaging progression of ideas. Acutely aware of the need for rigor, the student is much better prepared to understand what constitutes a proper mathematical proof and how to write one. Fifteen years of classroom experience with the first edition of Understanding Analysis have solidified and refined the central narrative of the second edition. Roughly 150 new exercises join a selection of the best exercises from the first edition, and three more project-style sections have been added. Investigations of Euler's computation of (2), the Weierstrass Approximation Theorem, and the gamma function are now among the book's cohort of seminal results serving as motivation and payoff for the beginning student to master the methods of analysis.

Getting certified to teach high school mathematics typically requires completing a course in real analysis. Yet most teachers point out real analysis content bears little resemblance to secondary mathematics and report it does not influence their teaching in any significant way. This textbook is our attempt to change the narrative. It is our belief that analysis can be a meaningful part of a teacher's mathematical education and preparation for teaching. This book is a companion text. It is intended to be a supplemental resource, used in conjunction with a more traditional real analysis book. The textbook is based on our efforts to identify ways that studying real analysis can provide future teachers with genuine opportunities to think about teaching secondary mathematics. It focuses on how mathematical ideas are connected to the practice of teaching secondary mathematics-and not just the content of secondary mathematics itself. Discussions around pedagogy are premised on the belief that the way mathematicians do mathematics can be useful for how we think about teaching mathematics. The book uses particular situations in teaching to make explicit ways that the content of real analysis might be important for teaching secondary mathematics, and how mathematical practices prevalent in the study of real analysis can be incorporated as practices for teaching.This textbook will be of particular interest to mathematics instructors-and mathematics teacher educators-thinking about how the mathematics of real analysis might be applicable to secondary teaching, as well as to any prospective (or current) teacher who has wondered about what the purpose of taking such courses could be.

This popular science title covers adhesion science in an easily accessible entertaining manner. As well as outlining types of adhesion and their importance in everyday life, the book covers interesting future applications of adhesion and inspiration taken from nature. Ideal for students and the scientifically minded reader this book provides a fascinating introduction to the science of what makes things stick.