Foundations of Mathematics: A Preparatory Course

Introduction and Foreword
Preface to the Second Edition
Preface to the Third Edition
Preface to the Fourth Edition
Contents
1: Elementary Calculation Methods
1.1 Basic Arithmetic Operations
Commutative Law and Associative Law
Addition and Multiplication Are Commutative and Associative
Distributive Law
Products of Negative Numbers
1.2 Fractions and Rational Numbers
Extending Fractions
Shortening Fractions
Sum of Two Fractions with the Same Denominator
Sum of Two Fractions
Product of Two Fractions
Product of a Fraction and an Integer
Dividing Fractions
Rule of Three
1.3 Bracket Calculation
Bracket Rule
Nested Brackets
1.4 Powers and Roots
Exponentiation with Natural Numbers
Power Laws
Powers with Negative Exponents
Powers with Exponent 1/2 - Square Roots
Powers with a Root Break in the Exponent - nth Roots
Exponentiation with Any Rational Number
Identities for ap/q
Real Numbers
Arithmetic Rules for Real Numbers
1.5 Special Expressions and Notations
Multiplying Out Brackets
Binomial Formulas
Summarizing Terms with the Same Name
Factor Out
Empty Sums
Factorial
2: Basic Information About Functions
2.1 Definition Range, Value Set and Image Set
Function, Domain and Set of Values
The Value Pool Does Not Have to Be Exhausted
Interval
2.2 Concatenation of Functions; Monotonicity and Invertibility
Concatenation of Functions
Reverse Function
Monotone Function
Strictly Monotonic Function
Strict Monotonicity and Reversibility
2.3 Power and Root Functions
Power Functions
n-th Root Function
2.4 Polynomials and Rational Functions
Polynomials
Zero
Zeros of Polynomials
Rational Function
Polarity of a Rational Function
2.5 Exponential and Logarithmic Functions
Exponential Function to General Base
Behavior of the Exponential Function to Base a
Calculation Rules for the Exponential Function
Logarithm to Base a
Monotonicity Behavior of the Logarithm Function to the Base a
Calculation Rules for the Logarithm Function
Conversion of Logarithms to Different Bases
3: Equations and Inequalities
Equation
Transformation of Equations
3.1 Linear Equations
Solution of Linear Equation
3.2 Quadratic Equations
Quadratic Equation
Solution of Quadratic Equations
Decomposition of Second Degree Polynomials into Linear Factors
3.3 Higher Order Polynomial Equations
Polynomial Equations Without Absolute Element
Biquadratic Equation
Solution of Biquadratic Equations
3.4 Root and Exponential Equations
Root Equation
Solution of Root Equations
Exponential Equation
3.5 Inequalities
Inequality
Transformation of Inequalities
Multiplication of an Inequality by a Negative Number
4: Geometry
4.1 Triangles and Trigonometric Functions
Triangle Inequality
Sum of the Angles on the Triangle
Intersection of the Central Perpendicular and the Circumcircle
Intersection of the Angle Bisector and the Incircle
Intersection of the Bisectors
Area of the Triangle
Heron´s Formula
Pythagorean Theorem
Euclid´s Theorem
Height Theorem
Thales´ Theorem
Sine, Cosine and Tangent
Sum of the Squares of the Sine and Cosine
Radians
Trigonometric Functions
Addition Theorems of Sine and Cosine
Cosine Theorem
Sine Theorem
Tangent Theorem
4.2 Plane Geometric Figures
Sum of the Interior Angles of a Quadrilateral
Length of the Diagonals in the Quadrilateral
Area of Parallelograms
Area of the Trapezoid
Sum of the Interior Angles of an N-corner
Area of the Regular N-corner
Definition of Circles
Circular Area
Circumference of the Circle
5: Introduction to Linear Algebra
5.1 Vectors
Three Dimensional Vector
Vector
Addition and Subtraction of Vectors
Multiplication of a Vector by a Number
Scalar Product
Calculation Rules for the Scalar Product
Linear Combination
Linear Dependence and Independence
Linear Dependence and Independence
Linear Dependence
5.2 Matrices
Matrix
Matrix
Addition and Subtraction of Matrices
Multiplication of a Matrix by a Number
Rules of Calculation for Matrices
Matrix Multiplication
Condition for Matrix Multiplication
Calculation Rules for the Matrix Product
Unit Matrix
Inverse Matrix
5.3 Linear Systems of Equations
Linear System of Equations
Permitted Manipulations of Systems of Equations
Gauss Algorithm
Unsolvable Linear Systems of Equations
Gauss Algorithm
Inversion of Matrices
5.4 Analytical Geometry
Two Dimensional Vectors
Three Dimensional Vectors
Negative Vector
Multiplication of a Vector by a Number
Addition of Vectors
Vertical Vectors
Vector Product
Calculation of the Parameter-Free Plane Equation
6: Differential and Integral Calculus
6.1 First Derivative of Functions and Derivative Rules
First Derivative of Constant Functions
First Derivative of Linear Functions
First Derivative of the Quadratic Power Function f (x) = x2
First Derivative of the Power Function pi (x) = xi
Example 6.1
Secant of f Through x and x + h
Differential Quotient and First Derivative of f in x
Tangent of f in x
Example 6.2
First Derivative of the Exponential Function exp(x) = ex
First Derivative of the Sine Function sin(x) and Cosine Function cos(x)
First Derivative of the Root Function , x [a, b] with a > 0
Example 6.3
Example 6.4
Example 6.5
First Derivative of the Factor Product λ f - Factor Rule
First Derivative of the Sum of Functions f + g - Sum Rule
Example 6.6
Example 6.7
First Derivative of Polynomial Functions p
Example 6.8
First Derivative of the Product of Functions f g - Product Rule
Example 6.8 (continued)
Example 6.9
Example 6.10
First Derivative of Chained Functions fg - Chain Rule
Example 6.11
Example 6.12
Example 6.13
First Derivative of the Quotient of Functions fg - Quotient Rule
Example 6.14
Example 6.15
Example 6.16
6.2 Applications of Derivatives and Curve Sketching
Monotonicity and Sign of the First Derivative
Example 6.17
Example 6.18
Strict Monotonicity and Sign of the First Derivative
Example 6.19
First Derivative of the Inverse Function
Example 6.20
Example 6.21
Example 6.22
Global Extrema
Global Extrema of Monotone Functions f : [a,b]
Local Extrema
Necessary Criterion for Local Extremes
Example 6.23
Example 6.24
Higher Derivatives
The i-th Derivative p(i) of Polynomial Functions p
Example 6.25
Sufficient Criterion for Local Extremes
Example 6.26
Example 6.27
Example 6.28
Convexity and Sign of the Second Derivative
Inflection Points
Example 6.26 (continued)
Necessary and Sufficient Criterion for Inflection Points
Example 6.29
Example 6.30
6.3 Integration of Functions
Integration of the Power Function pi(x) = xi, Where i {0}
Integrability and Definite Integral
Example 6.31
Properties of the Definite Integral
Integration Rules for the Factor Product λ f and the Sum f + g
Example 6.32
Example 6.33
Integration of Polynomials
Differentiability of the Integral Function of Continuous Functions
Example 6.34
Example 6.35
Primitive Functions Differ only by Additive Constants
Main Theorem of Differential and Integral Calculus
Example 6.36
Example 6.37
Product Integration: Partial Integration
Example 6.38
Example 6.39
Example 6.40
Substitution Rule
Example 6.41
Example 6.42
Example 6.43
7: Fundamentals of Probability Theory
7.1 Combinatorics
Experiment with Finitely Many Equally Probable Outcomes
Urn Models
Selection with Backspacing with Attention to the Sequence
Binomial Coefficient
Selection with Backspacing without Regard to the Sequence
Selection Without Backspacing with Consideration of the Sequence
Arrays
Selection Without Backspacing Without Consideration of the Sequence
7.2 Relative Frequency and Classical Definition of Probability
Random Experiment and Random Event
Absolute and Relative Frequency
Classical Definition of Probability
Probability of Counter Event
Incompatible Events
Sum of Events
Sum of Incompatible Events
7.3 Axiomatic Definition of Probability
Axiomatic Definition of Probability
Rules for Calculating Probabilities
Product of Events
Sum of any Events
7.4 Conditional Probabilities
Conditional Probability
8: Descriptive Statistics
8.1 Introduction
Descriptive Statistics
8.2 Representation Methods
Statistical Scales
Absolute and Relative Frequencies
Cumulative and Relative Cumulative Frequencies
Bar and Column Diagrams
8.3 Position and Spreading Dimensions
Modal Value
Arithmetic Mean
Arithmetic Mean for Class Division
Median
Range
Mean Absolute Deviation
Dispersion and Standard Deviation
9: Complex Numbers
9.1 The Imaginary Unit i and the Set of Complex Numbers
Imaginary Unit
Complex Numbers
9.2 Basic Arithmetic Operations for Complex Numbers
Reciprocal of a Complex Number
Division of Complex Numbers
9.3 The Gaussian Number Plane and the Trigonometric Form of Complex Numbers
Absolute Value
Angle of a Complex Number
Trigonometric Form of a Complex Number
9.4 Powers and Roots of Complex Numbers
Multiplication and Division of Complex Numbers in Trigonometric Form
Exponentiation of Complex Numbers
Roots of Complex Numbers
9.5 Complete Solution of Quadratic and Biquadratic Equations
Complete Solution of Quadratic Equations
Complete Solution of Biquadratic Equations
10: Formulary
10.1 Chapter 1: Basics
10.2 Chapter 2: Functions
10.3 Chapter 3: Equations and Inequalities
10.4 Chapter 4: Geometry
10.5 Chapter 5: Linear Algebra
10.6 Chapter 6: Differential and Integral Calculus
10.7 Chapter 7: Probability Calculation
10.8 Chapter 8: Descriptive Statistics
10.9 Chapter 9: Complex Numbers
Appendix: Solutions of the Exercises
Exercises Chap. 1
Exercises Chap. 2
Exercises Chap. 3
Exercises Chap. 4
Exercises Chap. 5
Exercises Chap. 6
Exercises Chap. 7
Exercises Chap. 8
Exercises Chap. 9
Bibliography
Index