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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, a 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.
The Tree of Mathematics
The field of mathematics is composed of many subfields, some of which you have studied, and others of which you may not even be aware. These areas are logically related to each other in various ways. One way to look at mathematics is as a tree:
The tree starts at the bottom, the "trunk," shows dependencies and connections among the fields. You can learn about some areas in any order, but to obtain a sound theory we begin at ground level. This tree is somewhat abbreviated for simplicity. A list of their "official" fields of mathematics appears at the end of this chapter.
You can see that the three main branches of mathematics-algebra, geometry, and analysis-grow out of three basic areas. Logic comprises the rules by which mathematicians operate, the "grammar" of the language. Set theory provides the vocabulary. And the Number Systems comprise the most basic content from which the various branches grow. The course material in this book will acquaint you with the segments on the trunk so that you can make the climb into the canopy.
It is hoped that readers of this book will gain the following:
- facility in interpreting and using mathematical notation;
- a background in elementary logic and practice in reasoning;
- experience with sets and set notation
- practice in constructing proofs and evaluating the proofs of others;
- an introduction to the subject matter and activities of mathematics, including the analysis of examples, formulation of conjectures and reading and writing proofs;
- an introduction to the professional culture inhabited by mathematicians;
- and, and eagerness to do more mathematics.
The best progress toward these goals will come from a combination of reading the book and talking with your professor and classmates.
0.1 The Tree of Mathematics
0.2 What is Mathematics
0.3 Pure vs. Applied
0.4 What Kind of People are Mathematicians?
0.5 Mathematics Subject Classification
1.1 Mathematical Systems
1.2 Warm-up Exercises
1.3 Essentials of Logic
1.4 Further Exercises
1.5 The Axiomatic Method
1.6 The Results of Gödel
2. Set Theory
2.2 Warm-Up Exercises
2.3 Essentials of Set Theory
2.4 Further Exercises
2.6 Axiomatic Set Theory
3. The Natural Numbers
3.1 Mathematical Symbols and Notation
3.2 Warm-up Exercises
3.3 Essentials of Natural Number System
3.4 Further Exercises
3.5 Cantor's Infinite Arithmetic
4. The Positive Rational Numbers
4.1 Philosophy of Mathematics
4.2 Warm-up Exercises
4.3 Essentials of the Positive of Rational Numbers
4.4 Further Exercises
5. The Real Numbers
5.1 Famous Mathematical Objects
5.2 Warm-up Exercises
5.3 Essentials of the Positive Real Numbers
5.4 Essentials of the Real Number System
5.5 Further Exercises
5.6 Important Properties of the Real Number Line
6. The Complex Numbers
6.1 Famous Mathematicians
6.2 Warm-up Exercises
6.3 Essentials of the Complex Number System
6.4 Further Exercises
6.5 Important Properties of the Complex Numbers
7. And Beyond...
7.1 What Is Mathematical Research?
7.2 Famous Theorems
7.3 Famous Unsolved Problems
7.4 Professional Organizations
7.6 Extracurricular Activities