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How to Think Like a Mathematician
A Companion to Undergraduate Mathematics

$41.99 (P)

  • Date Published: February 2009
  • availability: In stock
  • format: Paperback
  • isbn: 9780521719780

$ 41.99 (P)

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About the Authors
  • Looking for a head start in your undergraduate degree in mathematics? Maybe you've already started your degree and feel bewildered by the subject you previously loved? Don't panic! This friendly companion will ease your transition to real mathematical thinking. Working through the book you will develop an arsenal of techniques to help you unlock the meaning of definitions, theorems and proofs, solve problems, and write mathematics effectively. All the major methods of proof - direct method, cases, induction, contradiction and contrapositive - are featured. Concrete examples are used throughout, and you'll get plenty of practice on topics common to many courses such as divisors, Euclidean algorithms, modular arithmetic, equivalence relations, and injectivity and surjectivity of functions. The material has been tested by real students over many years so all the essentials are covered. With over 300 exercises to help you test your progress, you'll soon learn how to think like a mathematician.

    • Encourages a questioning and active nature rather than a passive one, leading the reader to develop a deeper understanding of mathematics
    • Emphasises the use of examples and counterexamples to illuminate theorems
    • Essential for any starting undergraduates in mathematics; also of benefit to engineers and physicists who need high-level mathematics, and students that require logic such as computer scientists, philosophers, and linguists
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    Reviews & endorsements

    "In this book, Houston has created a primer on the fundamental abstract ideas of mathematics; the primary emphasis is on demonstrating the many principles and tactics used in proofs. The material is explained in ways that are comprehensible, which will be a great help for people who seem to hit the wall regarding what to do when confronted with the creation of a proof... In this book, Houston takes a systematic and gentle approach to explaining the ideas of mathematics and how tactics of reasoning can be combined with those ideas to generate what would be considered a convincing proof."
    Charles Ashbacher, Journal of Recreational Mathematics

    "The author provides concise, crisp explanations, including definitions, examples, tips, remarks, warnings, and idea-reinforcing questions. Houston expresses thoughts clearly and concisely, and includes succinct remarks to make points, clarify arguments, and reveal subleties."
    W.R. Lee, Choice Magazine

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    Product details

    • Date Published: February 2009
    • format: Paperback
    • isbn: 9780521719780
    • length: 274 pages
    • dimensions: 254 x 195 x 19 mm
    • weight: 0.54kg
    • contains: 1 b/w illus. 10 tables 335 exercises
    • availability: In stock
  • Table of Contents

    Part I. Study Skills For Mathematicians:
    1. Sets and functions
    2. Reading mathematics
    3. Writing mathematics I
    4. Writing mathematics II
    5. How to solve problems
    Part II. How To Think Logically:
    6. Making a statement
    7. Implications
    8. Finer points concerning implications
    9. Converse and equivalence
    10. Quantifiers – For all and There exists
    11. Complexity and negation of quantifiers
    12. Examples and counterexamples
    13. Summary of logic
    Part III. Definitions, Theorems and Proofs:
    14. Definitions, theorems and proofs
    15. How to read a definition
    16. How to read a theorem
    17. Proof
    18. How to read a proof
    19. A study of Pythagoras' Theorem
    Part IV. Techniques of Proof:
    20. Techniques of proof I: direct method
    21. Some common mistakes
    22. Techniques of proof II: proof by cases
    23. Techniques of proof III: Contradiction
    24. Techniques of proof IV: Induction
    25. More sophisticated induction techniques
    26. Techniques of proof V: contrapositive method
    Part V. Mathematics That All Good Mathematicians Need:
    27. Divisors
    28. The Euclidean Algorithm
    29. Modular arithmetic
    30. Injective, surjective, bijective – and a bit about infinity
    31. Equivalence relations
    Part VI. Closing Remarks:
    32. Putting it all together
    33. Generalization and specialization
    34. True understanding
    35. The biggest secret
    Appendices: A. Greek alphabet
    B. Commonly used symbols and notation
    C. How to prove that …

  • Resources for

    How to Think Like a Mathematician

    Kevin Houston

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  • Instructors have used or reviewed this title for the following courses

    • Discrete Mathematics ll
    • Discrete Structures
    • Introduction To Proofs
    • Introduction to Abstract Mathematics
    • Introduction to Advanced Mathematics
    • Introduction to Higher Mathematics
    • Mathematical Foundations
    • Problem Solving & Proof
    • Proofs
    • Transitions to Advanced Mathematics
  • Author

    Kevin Houston, University of Leeds
    Kevin Houston is Senior Lecturer in Mathematics at the University of Leeds.

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