A Bridge to Advanced Mathematics. From Natural to Complex Numbers

Cover
Title page
Copyright
Contents
Preface
	1. The Content of the Book
	2. How to Use This Book?
Chapter 1. Natural Numbers N
	1.1. Basic Properties
	1.2. The Principle of Induction
	1.3. Arithmetic and Geometric Progressions
	1.4. The Least Element Principle
	1.5. There are 10 Kinds of People in the World
	1.6. Divisibility
	1.7. Counting and Binomial Formula
	1.8. More Exercises
Chapter 2. Integer Numbers Z
	2.1. Basic Properties
	2.2. Integer Division
	2.3. Euclidean Algorithm Revisited
	2.4. Congruences and Modular Arithmetic
	2.5. Modular Equations
	2.6. The Chinese Remainder Theorem
	2.7. Fermat and Euler Theorems
	2.8. More Exercises
Chapter 3. Rational Numbers Q
	3.1. Basic Properties
	3.2. Not Everything Is Rational
	3.3. Fractions and Decimal Representations
	3.4. Finite Continued Fractions
	3.5. Farey Sequences and Pick’s Formula
	3.6. Ford Circles and Stern–Brocot Trees
	3.7. Egyptian Fractions
	3.8. More Exercises
Chapter 4. Real Numbers R
	4.1. Basic Properties
	4.2. The Real Numbers Form a Field
	4.3. Order and Absolute Value
	4.4. Completeness
	4.5. Supremum and Infimum of a Set
	4.6. Roots and Powers
	4.7. Expansion of Real Numbers
	4.8. More Exercises
Chapter 5. Sequences of Real Numbers
	5.1. Basic Properties
	5.2. Convergent and Divergent Sequences
	5.3. The Monotone Convergence Theorem and Its Applications
	5.4. Subsequences
	5.5. Cauchy Sequences
	5.6. Infinite Series
	5.7. Infinite Continued Fractions
	5.8. More Exercises
Chapter 6. Complex Numbers C
	6.1. Basic Properties
	6.2. The Conjugate and the Absolute Value
	6.3. Polar Representation of Complex Numbers
	6.4. Roots of Complex Numbers
	6.5. Geometric Applications
	6.6. Sequences of Complex Numbers
	6.7. Infinite Series of Complex Numbers
	6.8. More Exercises
Epilogue
Appendix. Sets, Functions, and Relations
	A.1. Logic
	A.2. Sets
	A.3. Functions
	A.4. Cardinality of Sets
	A.5. Relations
	A.6. Proofs
	A.7. Peano’s Axioms and the Construction of Integers
	A.8. More Exercises
Bibliography
Index
Back Cover