Eine prägnante, einheitliche Sicht auf die Mathematik und ihre historische Entwicklung. Der Autor richtet sich an Mathematiker, die ein besseres Verständnis für die Zusammenhänge der modernen Mathematik erlangen möchten. Historische Aspekte dienen hier der Motivation und Einheit der Mathematik, ohne dass historische Vorkenntnisse vorausgesetzt werden.
Focusing on the intersection of algebra, analysis, and geometry, this book offers an accessible exploration of mathematics at a calculus level, rooted in the historic conflict between arithmetic and geometry. It emphasizes the unifying role of algebra and presents key concepts through engaging examples like circles and squares. Covering Euclidean ideas alongside modern insights, it caters to high school graduates, graduate students, and researchers alike, featuring exercises that reinforce learning and understanding of fundamental mathematical themes.
Linking set theory with analysis, this text offers a detailed exploration of the real numbers system. It provides a unique introduction to set theory while thoroughly explaining the fundamental concepts of analysis, filling a gap in standard curricula. The book aims to enhance understanding of both mathematical fields through its comprehensive approach.
This book explores the dual nature of algebra as both abstract and applied mathematics, emphasizing its role in solving concrete geometric problems. It traces algebra's evolution from Euclid's era to the 19th century, highlighting its unifying power across various mathematical disciplines and fostering a deeper appreciation for its principles.
Focusing on integer solutions to polynomial equations, this book delves into the interplay between number theory and algebra, particularly through the concepts of rings and ideals. It builds upon the foundational work of Galois, Kummer, and Dedekind, illustrating how these algebraic structures arise from number theory. By exploring the failures of unique prime factorization, it highlights the historical development of "ideal numbers" and their significance in understanding the algebraic properties of integers. This work serves as a comprehensive introduction to key ideas in number theory.
Exploring the evolution of proof, this book highlights its crucial role in shaping mathematical knowledge from ancient to modern times. It traces the journey from Euclid's geometry to the development of algebra and calculus, illustrating how proof methods have transformed. The discussion extends to number theory, non-Euclidean geometry, and logic, revealing the profound implications of proof on arithmetic and the limitations it imposes on theorem validation. Through historical episodes, it provides a fresh perspective on mathematics' foundational principles and its capacity for innovation.
This book explores four distinct approaches to teaching geometry, highlighting the evolution from Euclid's methods to modern concepts like linear algebra and transformation groups. Each approach is presented in two chapters: one concrete and introductory, the other more abstract, illustrating the richness and diversity of geometric understanding.
Exploring the intersection of set theory and mathematical logic, this book delves into their influence on contemporary mathematics, particularly in number theory and combinatorics. It examines how foundational questions about infinity and the nature of proof have shaped mathematical thought. By tracing the evolution of these ideas, the text provides a comprehensive understanding of their relevance in modern mathematical discussions and developments.
Focusing on geometric aspects, this introduction to topology provides a rich historical context and visual interpretation of concepts. The second edition enhances the learning experience with 300 illustrations, a variety of exercises, and challenging open problems. A new chapter dedicated to unsolvable problems adds depth, making this edition a comprehensive resource for both students and enthusiasts of topology.
This book, meant for undergraduate mathematics students and teachers, introduces algebraic number theory through problems from ordinary number theory that can be solved with the help of algebraic numbers, using a suitable generalization of unique prime factorization. The material is motivated by weaving historical information throughout.