Mathematical Thinking

Description

New and updated features of this title

• 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.
• “Approaches to Problems”: In selected chapters. Summarizes key points and presents problem-solving strategies relevant to exercises.
• 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.
• Expanded and improved selection of exercises: New, easier exercises check mastery of concepts; some difficult exercises are clarified.
• Reorganization of material: Provides smoother development and clearer focus on essential material.
• Definitions in bold: Terms being defined are in bold type with almost all definitions in numbered terms.

Mathematical Thinking, 2nd Edition is designed to prepare students with the logical thinking skills necessary to understand and communicate fundamental ideas and proofs in mathematics (skills vital for success throughout the upperclass mathematics curriculum). It 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, 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 exercises are acclaimed for their exceptional quality.

This title is part of the Pearson Modern Classics series. Pearson Modern Classics are acclaimed titles at a value price.

• I. ELEMENTARY CONCEPTS.
• 1. Numbers, Sets and Functions.
• 2. Language and Proofs.
• 3. Induction.
• 4. Bijections and Cardinality.
• II. PROPERTIES OF NUMBERS.
• 5. Combinatorial Reasoning.
• 6. Divisibility.
• 7. Modular Arithmetic.
• 8. The Rational Numbers.
• III. DISCRETE MATHEMATICS.
• 9. Probability.
• 10. Two Principles of Counting.
• 11. Graph Theory.
• 12. Recurrence Relations.
• IV. CONTINUOUS MATHEMATICS.
• 13. The Real Numbers.
• 14. Sequences and Series.
• 15. Continuous Functions.
• 16. Differentiation.
• 17. Integration.
• 18. The Complex Numbers.

Hallmark features of this title

• Engaging examples: Interesting applications introduce and motivate the underlying mathematics.
• 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).
• 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).
• Emphasis on understanding rather than manipulation: Stresses full comprehension rather than rote symbolic manipulation for mastery of proof techniques and mathematical ideas.
• Emphasis on clear communication: Discusses the use of language and requires written arguments in many exercises.
• Hints for selected exercises: Provides immediate hints for some exercises and hints for others in an appendix.

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 emphasise 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.

Additional information