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CONTENTS
Authors ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. xxv
Preface . ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. xxvii
Main Notation .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. xxix
Part I. Definitions, Formulas, Methods, and Theorems 1
1. Arithmetic and Elementary Algebra .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 3
1.1. Real Numbers . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 3
1.1.1. Integer Numbers .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 3
1.1.2. Real, Rational, and Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2. Equalities and Inequalities. Arithmetic Operations. Absolute Value . . . . . . . . . . . . . . . . 5
1.2.1. Equalities and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2. Addition and Multiplication of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3. Ratios and Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.4. Percentage .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 7
1.2.5. Absolute Value of a Number (Modulus of a Number) . . . . . . . . . . . . . . . . . . . . . . 8
1.3. Powers and Logarithms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 8
1.3.1. Powers and Roots . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 8
1.3.2. Logarithms .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 9
1.4. Binomial Theorem and Related Formulas .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 10
1.4.1. Factorials. Binomial Coefficients. Binomial Theorem .. ... .. .. .. .. .. .. .. .. 10
1.4.2. Related Formulas . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 10
1.5. Arithmetic and Geometric Progressions. Finite Sums and Products . . . . . . . . . . . . . . . . . 11
1.5.1. Arithmetic and Geometric Progressions . . . . . . . . . . . . . ... .. .. .. .. .. .. .. .. 11
1.5.2. Finite Series and Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6. Mean Values and Inequalities of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6.1. Arithmetic Mean, Geometric Mean, and Other Mean Values. Inequalities for
Mean Values .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 13
1.6.2. Inequalities of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.7. Some Mathematical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.7.1. Proof by Contradiction . .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 15
1.7.2. Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7.3. Proof by Counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7.4. Method of Undetermined Coefficients . . . . . . . . . . . . . . . ... .. .. .. .. .. .. .. .. 17
References for Chapter 1 . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 18
2. Elementary Functions .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 19
2.1. Power, Exponential, and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1. Power Function: y = x α .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 19
2.1.2. Exponential Function: y = a x .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 21
2.1.3. Logarithmic Function: y =log x .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 22
a
2.2. Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.1. Trigonometric Circle. Definition of Trigonometric Functions . . . . . . . . . . . . . . . 24
2.2.2. Graphs of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.3. Properties of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
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