Modern Algebra & Real Analysis

Modern algebra and real analysis are major branches of mathematics which are typically treated as separate topics presented in separate textbooks. This book presents a unified approach to these branches with a particular focus on the dependencies between them. The work is presented in two parts. The first covers modern algebra, while the second covers real analysis. The first part begins with a review of preliminary definitions and facts of sets, relations and maps, and then leads the reader through group theory, ring theory, linear algebra and field theory. The second part begins with a comparison and contrast of rational and real numbers, and then leads the reader through convergence, continuity, differentiability, integrability and the essential uniqueness of real number constructions. Throughout both developments, the topics are presented with an emphasis on theory over application. A single thread is established across definitions and facts maintaining a single story. The reader is assumed to have no previous knowledge of modern algebra or real analysis. Some experience with logic as used in mathematical proofs such as induction and contraposition will be helpful. The book is not intended as a textbook and is suitable for someone pursuing mathematics as a hobby outside the classroom.