[Book Cover]

Mathematical Thinking: Problem-Solving and Proofs, 2/e

John P. D'Angelo, University of Illinois
Douglas B. West, University of Illinois

Coming December, 1999 by Prentice Hall Engineering/Science/Mathematics

Copyright 2000, 480 pp.
Cloth
ISBN 0-13-014412-6


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Summary

For one/two-term courses in Transition to Advanced Mathematics or Introduction to Proofs. Also suitable for courses in Analysis or Discrete Math. This text is designed to prepare students thoroughly in the logical thinking skills necessary to understand and communicate fundamental ideas and proofs in mathematics—skills vital for success throughout the upperclass mathematics curriculum. The text offers both discrete and continuous mathematics, allowing instructors to emphasize one or to present the fundamentals of both. It begins by discussing mathematical language and proof techniques (including induction), applies them to easily-understood questions in elementary number theory and counting, and then develops additional techniques of proof via important topics in discrete and continuous mathematics. The stimulating exercises are acclaimed for their exceptional quality.

Features


NEW—A clearly outlined transition course—Rearranges material to facilitate a clearly defined and more accessible transition course using Chs. 1-5, initial parts of Chs. 6,8 and Chs. 13-14.

  • By narrowing the focus, makes it easy to present a course with rich content to beginning students in a transition course without overwhelming them.
NEW—“Approaches to Problems”—In selected chapters. Summarizes key points and presents problem-solving strategies relevant to exercises.
  • In the transition course, helps students organize their understanding of the chapter, avoid typical pitfalls, and learn ways to approach problems of moderate difficulty.
NEW—A clearly outlined analysis course—Now contains an excellent course in analysis using Part I as background, touching briefly on Ch. 8, and covering Part IV in depth.
  • Provides review reading on proof methods in Part I while being as thorough and accessible as introductory texts in Part IV.
NEW—Expanded and improved selection of exercises—New, easier exercises check mastery of concepts; some difficult exercises are clarified.
  • Enlarged selection of easier exercises provides greater encouragement for beginning students; clarifications make other exercises more accessible. NEW—Reorganization of material—Provides smoother development and clearer focus on essential material.
  • Makes it easier for students to follow the mathematical development and how to know what assumption can be used when working problems.
NEW—Definitions in bold—Terms being defined are in bold type with almost all definitions in numbered terms.
  • Makes definitions easier for students to find.
NEW—More accessible presentation—Some terse discussions expanded, examples added, and more computations placed in displays.
  • Makes material easier for students to comprehend and conveys a greater sense of progress by making pages less dense.
Engaging examples—Interesting applications introduce and motivate the underlying mathematics.
  • Engage student interest and commitment from the beginning and keep the material lively.
Logical Organization—Introduces concepts as needed with each item carefully selected. Distinguishes between Lemmas/Theorems (the mathematical development) and Examples/Solutions (the illustration or application of mathematical results).
  • Enables students to find fundamental results easily and learn more efficiently. Helps students understand the difference between mathematical tools and their application in problem-solving.
Flexibility in course design—Mathematical background in Part I can be treated quickly with strong students or in detail for beginners. Rich variety of subsequent topics permits a broad introduction to mathematics or a focus on discrete mathematics (Part III) or Analysis (Part IV).
  • Enables instructors to design courses appropriate to students' abilities; a two-semester treatment offers a thorough introduction to mathematics.
Emphasis on understanding rather than manipulation—Stresses full comprehension rather than rote symbolic manipulation for mastery of proof techniques and mathematical ideas.
  • Helps students develop critical thinking skills and appreciation of coherent arguments, preparing them both for later courses in mathematics and for problem-solving situations outside school.
Emphasis on clear communication—Discusses the use of language and requires written arguments in many exercises.
  • Helps students develop or remediate their written communication skills, both in forming English sentences and in presenting coherent arguments. Supports the efforts of instructors to apply such skills in a mathematical context.
Richness of Topics—After the elementary material, provides a wealth of “intellectual highs” from diophantine equations to Fermat's Little Theorem to Pythagorean triples, Bertrand's Ballot Problem, the pigeonhole principle, the Euler totient, Hall's Marriage Condition, the theory of calculus, interchange of limiting operations, series of functions, the existence of continuous nowhere differentiable functions, complex numbers, and the Fundamental Theorem of Algebra.
  • Permits the inclusion of topics to enrich the mathematical experience. Depending on the talents of students, these can be presented in class or left for outside reading. Hints for selected exercises—Provides immediate hints for some exercises and hints for others in an appendix.
  • Gives students the flexibility to learn at their own pace; weaker students have more opportunity to be successful, and stronger students have more opportunity to be stimulated.
Superior exercise sets—Offers over 850 exercises ranging from relatively straightforward applications of ideas in the text to subtle problems requiring some ingenuity.
  • Helps students at all levels to understand the ideas of the course and to broaden their mathematical interests.
Gradation of exercises—Distinguishes easier exercises by (–), harder by (+), and particularly valuable or instructive exercises by (!).
  • Aids instructor in selecting appropriate exercises and students in practicing for tests.
Instructor's Manual—Contains solutions to exercises and pedagogical suggestions. Only available directly through editor.


Table of Contents
I. ELEMENTARY CONCEPTS.
    1. Numbers, Sets and Functions.
    2. Language and Proofs.
    3. Properties of Functions.
    4. Induction.

II. PROPERTIES OF NUMBERS.
    5. Cardinality and Counting.
    6. Divisibility.
    7. Modular Arithmetic.
    8. The Rational Numbers.
III. DISCRETE MATHEMATICS.
    9. Combinatorial Reasoning.
    10. Two Principles of Counting.
    11. Graph Theory.
    12. Recurrence Relations.
IV. CONTINUOUS MATHEMATICS.
    13. The Real Numbers.
    14. Sequences and Series.
    15. Continuity.
    16. Differentiation.
    17. Integration.
    18. The Complex Numbers.


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