This book contains my answer to that question. The purpose of the book is to tell the beginning student of advanced mathematics the basic set- theoretic facts of life, and to do so with the minimum of philosophical discourse and logical formalism. The point of view throughout is that of a prospective mathematician anxious to study groups, or integrals, or manifolds. From this point of view the concepts and methods of this book are merely some of the standard mathematical tools; the expert specialist will find nothing new here. Scholarly bibliographical credits and references are out of place in a purely expository book such as this one. The student who gets interested in set theory for its own sake should know, however, that there is much more to the subject than there is in this book. One of the most beautiful sources of set-theoretic wisdom is still Hausdorff's Set theory. A recent and highly readable addition to the literature, with an extensive and up-to-date bibliography, is Axiomatic set theory by Suppes.

    Topics covered include matrix multiplication, row reduction, matrix inverse, orthogonality and computation. The self-teaching book is loaded with examples and graphics and provides a wide array of probing problems, accompanying solutions, and a glossary. Chapter 1: Introduction to Vectors; Chapter 2: Solving Linear Equations; Chapter 3: Vector Spaces and Subspaces; Chapter 4: Orthogonality; Chapter 5: Determinants; Chapter 6: Eigenvalues and Eigenvectors; Chapter 7: Linear Transformations; Chapter 8: Applications; Chapter 9: Numerical Linear Algebra; Chapter 10: Complex Vectors and Matrices; Solutions to Selected Exercises; Final Exam. Matrix Factorizations. Conceptual Questions for Review. Glossary: A Dictionary for Linear Algebra Index Teaching Codes Linear Algebra in a Nutshell.

    Linear algebra is tightly integrated into the text.

    Even when dealing with standard material, Stillwell manages to dramatize it and to make it worth rethinking. In short, his book is a splendid addition to the genre of works that build royal roads to mathematical culture for the many." (Mathematical Intelligencer)This second edition includes new chapters on Chinese and Indian number theory, on hypercomplex numbers, and on algebraic number theory. Many more exercises have been added, as well as commentary to the exercises explaining how they relate to the preceding section, and how they foreshadow later topics.

    The novel approach taken here banishes determinants to the end of the book and focuses on the central goal of linear algebra: understanding the structure of linear operators on vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. For example, the book presents - without having defined determinants - a clean proof that every linear operator on a finite-dimensional complex vector space (or an odd-dimensional real vector space) has an eigenvalue. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. This second edition includes a new section on orthogonal projections and minimization problems. The sections on self-adjoint operators, normal operators, and the spectral theorem have been rewritten. New examples and new exercises have been added, several proofs have been simplified, and hundreds of minor improvements have been made throughout the text.

    A comprehensive introduction to the subject, this book shows in detail how such problems can be solved numerically with great efficiency. The focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. The text contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance, and economics.

    It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity. Topics include sets, logic, counting, methods of conditional and non-conditional proof, disproof, induction, relations, functions and infinite cardinality.

    The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. To help students construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. Previous Edition Hb (1994) 0-521-44116-1 Previous Edition Pb (1994) 0-521-44663-5

    These themes include mathematical reasoning, combinatorial analysis, discrete structures, algorithmic thinking, and enhanced problem-solving skills through modeling. Its intent is to demonstrate the relevance and practicality of discrete mathematics to all students. The Fifth Edition includes a more thorough and linear presentation of logic, proof types and proof writing, and mathematical reasoning. This enhanced coverage will provide students with a solid understanding of the material as it relates to their immediate field of study and other relevant subjects. The inclusion of applications and examples to key topics has been significantly addressed to add clarity to every subject. True to the Fourth Edition, the text-specific web site supplements the subject matter in meaningful ways, offering additional material for students and instructors. Discrete math is an active subject with new discoveries made every year. The continual growth and updates to the web site reflect the active nature of the topics being discussed. The book is appropriate for a one- or two-term introductory discrete mathematics course to be taken by students in a wide variety of majors, including computer science, mathematics, and engineering. College Algebra is the only explicit prerequisite.

    Focused on groups, rings and fields, this text gives students a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures. KEY TOPICS: Sets and Relations; GROUPS AND SUBGROUPS; Introduction and Examples; Binary Operations; Isomorphic Binary Structures; Groups; Subgroups; Cyclic Groups; Generators and Cayley Digraphs; PERMUTATIONS, COSETS, AND DIRECT PRODUCTS; Groups of Permutations; Orbits, Cycles, and the Alternating Groups; Cosets and the Theorem of Lagrange; Direct Products and Finitely Generated Abelian Groups; Plane Isometries; HOMOMORPHISMS AND FACTOR GROUPS; Homomorphisms; Factor Groups; Factor-Group Computations and Simple Groups; Group Action on a Set; Applications of G-Sets to Counting; RINGS AND FIELDS; Rings and Fields; Integral Domains; Fermat's and Euler's Theorems; The Field of Quotients of an Integral Domain; Rings of Polynomials; Factorization of Polynomials over a Field; Noncommutative Examples; Ordered Rings and Fields; IDEALS AND FACTOR RINGS; Homomorphisms and Factor Rings; Prime and Maximal Ideas; Gr�bner Bases for Ideals; EXTENSION FIELDS; Introduction to Extension Fields; Vector Spaces; Algebraic Extensions; Geometric Constructions; Finite Fields; ADVANCED GROUP THEORY; Isomorphism Theorems; Series of Groups; Sylow Theorems; Applications of the Sylow Theory; Free Abelian Groups; Free Groups; Group Presentations; GROUPS IN TOPOLOGY; Simplicial Complexes and Homology Groups; Computations of Homology Groups; More Homology Computations and Applications; Homological Algebra; Factorization; Unique Factorization Domains; Euclidean Domains; Gaussian Integers and Multiplicative Norms; AUTOMORPHISMS AND GALOIS THEORY; Automorphisms of Fields; The Isomorphism Extension Theorem; Splitting Fields; Separable Extensions; Totally Inseparable Extensions; Galois Theory; Illustrations of Galois Theory; Cyclotomic Extensions; Insolvability of the Quintic; Matrix Algebra MARKET: For all readers interested in abstract algebra.

    The Calculus Lifesaver provides students with the essential tools they need not only to learn calculus, but to excel at it.All of the material in this user-friendly study guide has been proven to get results. The book arose from Adrian Banner's popular calculus review course at Princeton University, which he developed especially for students who are motivated to earn A's but get only average grades on exams. The complete course will be available for free on the Web in a series of videotaped lectures. This study guide works as a supplement to any single-variable calculus course or textbook. Coupled with a selection of exercises, the book can also be used as a textbook in its own right. The style is informal, non-intimidating, and even entertaining, without sacrificing comprehensiveness. The author elaborates standard course material with scores of detailed examples that treat the reader to an inner monologue--the train of thought students should be following in order to solve the problem--providing the necessary reasoning as well as the solution. The book's emphasis is on building problem-solving skills. Examples range from easy to difficult and illustrate the in-depth presentation of theory.The Calculus Lifesaver combines ease of use and readability with the depth of content and mathematical rigor of the best calculus textbooks. It is an indispensable volume for any student seeking to master calculus.Serves as a companion to any single-variable calculus textbookInformal, entertaining, and not intimidatingInformative videos that follow the book--a full forty-eight hours of Banner's Princeton calculus-review course--is available at Adrian Banner lecturesMore than 475 examples (ranging from easy to hard) provide step-by-step reasoningTheorems and methods justified and connections made to actual practiceDifficult topics such as improper integrals and infinite series covered in detailTried and tested by students taking freshman calculus

    This book will offer students a broad outline of essential mathematics and will help to fill in the gaps in their knowledge. The author explains the basic points and a few key results of all the most important undergraduate topics in mathematics, emphasizing the intuitions behind the subject. The topics include linear algebra, vector calculus, differential and analytical geometry, real analysis, point-set topology, probability, complex analysis, set theory, algorithms, and more. An annotated bibliography offers a guide to further reading and to more rigorous foundations.

    The basic techniques and theorems of analysis are presented in such a way that the intimate connections between its various branches are strongly emphasized. The traditionally separate subjects of 'real analysis' and 'complex analysis' are thus united in one volume. Some of the basic ideas from functional analysis are also included. This is the only book to take this unique approach. The third edition includes a new chapter on differentiation. Proofs of theorems presented in the book are concise and complete and many challenging exercises appear at the end of each chapter. The book is arranged so that each chapter builds upon the other, giving students a gradual understanding of the subject.This text is part of the Walter Rudin Student Series in Advanced Mathematics.

    With a new introduction, three new chapters, modernized language and methods throughout, and an appendix of challenging and enjoyable practice problems, Calculus Made Easy has been thoroughly updated for the modern reader.

    It reveals the attraction that has drawn leading mathematicians and amateurs alike to number theory over the course of history.