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Essentials of Mathematics

Margie Hale


This beautifully written book is intended to be used by first or second year undergraduate university students to bridge the gap from school mathematics to "what comes next."...This is a wonderful book.... It is a thought-provoking resource for anyone involved in curriculum development and teaching of mathematics at the university entrance level...It should be essential reading for every enthusiastic undergraduate. — Cheryl E. Praeger, Gazette, Australian Mathematical Society

The presentations style is conversational, employing frequent thought questions and directions for readers to try for themselves. A unique and refreshing features of this book is the introduction of the interesting and fun problems in mathematics that students do not usually encounter in a core course.... The book breaks the mold of traditional texts by portraying mathematics in a different light. Through this broader view of mathematics, this text may attract more undergraduate students into the beautiful and delightful world of mathematics. — The American Statistician

This book is well written, and will, I'm sure be useful. I would love to teach a course from it. — Marion Cohen, MAA Online

Every mathematician must make the transition from the calculations of high school to the structural and theoretical approaches of graduate school. Essentials of Mathematics provides the knowledge needed to move onto advanced mathematical work, and a glimpse of what being a mathematician might be like. No other book takes this particular holistic approach to the task.

The content is of two types. There is material for a "Transitions" course at the sophomore level; introductions to logic and set theory, discussions of proof writing and proof discovery, and introductions to the number systems (natural, rational, real, and complex). The material is presented in a fashion suitable for a Moore Method course, although such an approach is not necessary. An accompanying Instructor's Manual provides support for all flavors of teaching styles. In addition to presenting the important results for student proof, each area provides warm-up and follow-up exercises to help students internalize the material.

The second type of content is an introduction to the professional culture of mathematics. There are many things that mathematicians know but weren't exactly taught. To give college students a sense of the mathematical universe, the book includes narratives on this kind of information. There are sections on pure and applied mathematics, the philosophy of mathematics, ethics in mathematical work, professional (including student) organizations, famous theorems, famous unsolved problems, famous mathematicians, discussions of the nature of mathematics research, and more. The prerequisites for a course based on this book include the content of high school mathematics and a certain level of mathematical maturity. The student must be willing to think on an abstract level. Two semesters of calculus indicates a readiness for this material.

Print on demand (POD) books are not returnable because they are printed at your request. Damaged books will, of course, be replaced (customer support information is on your receipt). All print on demand books are paperback.

* As a textbook, Essentials of Mathematics does have DRM. Our DRM protected PDFs can be downloaded to three computers. iOS tablets can open secure PDFs using the AWReader app (available in the App Store). The iOS app uses the native iPad PDF reader so it is a very basic reader, no frills. Linux is not supported at this time for our secure PDFs.

Electronic ISBN: 9780883859827

Print ISBN: 9780883857298

Chapter 0: Mathematics


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