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Exploring Mathematics
An Engaging Introduction to Proof



Part of Cambridge Mathematical Textbooks

  • Date Published: September 2017
  • availability: In stock
  • format: Hardback
  • isbn: 9781107128989

£ 40.99

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About the Authors
  • Exploring Mathematics gives students experience with doing mathematics - interrogating mathematical claims, exploring definitions, forming conjectures, attempting proofs, and presenting results - and engages them with examples, exercises, and projects that pique their interest. Written with a minimal number of pre-requisites, this text can be used by college students in their first and second years of study, and by independent readers who want an accessible introduction to theoretical mathematics. Core topics include proof techniques, sets, functions, relations, and cardinality, with selected additional topics that provide many possibilities for further exploration. With a problem-based approach to investigating the material, students develop interesting examples and theorems through numerous exercises and projects. In-text exercises, with complete solutions or robust hints included in an appendix, help students explore and master the topics being presented. The end-of-chapter exercises and projects provide students with opportunities to confirm their understanding of core material, learn new concepts, and develop mathematical creativity.

    • Mixes the creativity and rigor that is essential to conducting mathematical enquiry and includes model proofs as well as a range of accessible topics, such as Fibonacci numbers and games, to encourage students to explore and to build intuition and background
    • Engages the reader thanks to in-text exercises with complete solutions and robust hints included in an appendix and helps students develop the skill of frequently interrogating the definitions, examples, and arguments presented
    • Offers students the opportunity to practice their mathematical writing skills in response to sometimes challenging questions, and with less emphasis on formal logic and rote proof-writing exercises
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    Reviews & endorsements

    'Wonderful list of topics, entertaining presentation, well-chosen problems - this is the way I want my grandchildren to engage with the beauty of mathematics. Too many students enter college with tools from calculus but no concept of how mathematicians think, and hardly any exposure to sets, logic, numbers, groups, or probability. How can we expect them to decide about continuing with mathematics, without a glimpse of the wonders ahead? A marvelous world lies at your students' doorstep, and Exploring Mathematics makes them players, not just observers, in that world. You'll want all your math majors, and perhaps your computer science majors as well, to have this experience.' Peter Winkler, Dartmouth College, New Hampshire

    'Meier and Smith have written a wonderful introduction to higher mathematics, showing both the thrill of abstraction and the beauty in discovering proofs.' Tom Garrity, Williams College, Massachusetts

    'I think that this is an excellent book. It is playful and serious at the same time. The book takes the unusual step of explaining not just mathematical topics but also what math is about and how to do it. The book is full of interesting topics, exercises, and examples. I think that this book could really fire up the imagination of a student wanting to get a start in real mathematics.' Richard Schwartz, Brown University, Rhode Island

    'The biggest step in studying mathematics is learning to write proofs. After calculus, students discover that truth is not a matter of a calculation, but a careful argument, juggling concepts within formal logic. Exploring Mathematics is a guide to this new level. Rich with exercises and projects, the book provides a well-grounded introduction to proof. 'Then the fun begins'. The reader participates by doing what mathematicians do, experimenting, formulating conjectures, exploring foundations for the basis of a proof. This book engages the reader fully to reach the goal of learning to fashion real mathematics. A valuable addition to every undergraduate library.' John McCleary, Vassar College, New York

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    Product details

    • Date Published: September 2017
    • format: Hardback
    • isbn: 9781107128989
    • length: 338 pages
    • dimensions: 260 x 183 x 20 mm
    • weight: 0.88kg
    • contains: 60 b/w illus.
    • availability: In stock
  • Table of Contents

    1. Let's play!
    2. Discovering and presenting mathematics
    3. Sets
    4. The integers and the fundamental theorem of arithmetic
    5. Functions
    6. Relations
    7. Cardinality
    8. The real numbers
    9. Probability and randomness
    10. Algebra and symmetry
    11. Projects
    Appendix A. Solutions, answers, or hints to in-text exercises

  • Resources for

    Exploring Mathematics

    John Meier, Derek Smith

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  • Authors

    John Meier, Lafayette College, Pennsylvania
    John Meier is Professor of Mathematics at Lafayette College, Pennsylvania, where he also served as Dean of the Curriculum. His research focuses on geometric group theory and involves algorithmic, combinatorial, geometric and topological issues that arise in the study of infinite groups. In addition to teaching awards from Cornell University, New York, and Lafayette College, Professor Meier is the proud recipient of the James Crawford Teaching Prize from the Eastern Pennsylvania and Delaware section of the Mathematical Association of America.

    Derek Smith, Lafayette College, Pennsylvania
    Derek Smith is Associate Professor of Mathematics at Lafayette College, Pennsylvania. His research focuses on algebra, combinatorics, and geometry. He has taught a wide variety of undergraduate courses in mathematics and other subjects in both the United States and Europe. He is the recipient of multiple teaching awards at Lafayette, and his work has been supported by the Mathematical Association of America and the National Science Foundation. Professor Smith is a former editor of the problem section of Math Horizons.

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