# Essentials of Mathematics: Introduction to Theory, Proof, and the Professional Culture

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*Margie Hale*

MAA Press: An Imprint of the American Mathematical Society

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. Essentials of Mathematics is designed as both a textbook and outside reading for college students who want to prepare themselves for mathematics courses beyond the first-year level and take courses in which proofs play a major role. There are also narratives on the nature of mathematics and the mathematics profession.

#### Reviews & Endorsements

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 feature 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

# Table of Contents

## Essentials of Mathematics: Introduction to Theory, Proof, and the Professional Culture

- Cover cover11
- copyright page ii3
- Title page iii4
- Contents vii8
- Preface xi12
- Acknowledgments xv16
- Notation xvii18
- 0 Mathematics 120
- 1 Logic 1130
- 2 Set Theory 2746
- 3 The Natural Numbers 4968
- 4 The Positive Rational Numbers 6180
- 5 The Real Numbers 7998
- 6 The Complex Numbers 103122
- 7 And Beyond . . . 129148
- References 165184
- Index 171190