Exploratory Examples for Real Analysis

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.