Math for Business and Economics: Compendium of Essential Formulas

Preface
Preface to the 2nd, revised and supplemented edition
Preface to the 1st edition
Contents
List of Abbreviations
Chapter 1 Mathematical Signs and Symbols
1.1 Pragmatic Signs
1.2 General Arithmetic Relations and Links
1.3 Sets of Numbers
1.4 Special Numbers and Links
1.5 Limit
1.6 Exponential Functions, Logarithm
1.7 Trigonometric Functions, Hyperbolic Functions
1.8 Vectors, Matrices
1.9 Sets
1.10 Relations
1.11 Functions
1.12 Order Structures
1.13 SI1Multiplying and Dividing Prefixes
1.14 Greek Alphabet
Chapter 2 Logic
2.1 Mathematical Logic
2.2 Propositional Logic
2.2.1 Propositional Variable
2.2.2 Truth Tables
Chapter 3 Arithmetic
3.1 Sets
3.1.1 General
Notation
Bounds, Limits of a Set
3.1.2 Set Relations
Inclusion
Equality
3.1.3 Set Operations
3.1.4 Relations, Laws, Rules of Calculation for Sets
3.1.5 Intervals
3.1.6 Numeral Systems
3.1.6.1 Decimal System (Decadic System)
3.1.6.2 Dual System (Binary System)
3.1.6.3 Roman Numeral System
3.2 Elementary Calculus
3.2.1 Elementary Foundations
3.2.1.1 Axioms
3.2.1.2 Factorisation
3.2.1.3 Relations
3.2.1.4 Absolute Value, Signum
3.2.1.5 Fractions
3.2.1.6 Polynomial Division
3.2.1.7 Horner’s Scheme (Horner’s Method)
3.2.2 Conversions of Terms
3.2.2.1 Binomial Formulas
3.2.2.2 Binomial Theorem
3.2.2.3 General Binomial Theorem for Natural Exponents
3.2.2.4 General Binomial Theorem for Real Exponents
3.2.2.5 Polynomial Terms
3.2.3 Summation and Product Notation
3.2.3.1 Summation Notation
3.2.3.2 Product Notation
3.2.4 Powers, Roots
3.2.5 Logarithms
3.2.6 Factorial
3.2.7 Binomial Coefficient
3.3 Sequences
3.3.1 Definition
Fundamental Terms:
Supremum, Infimum, Limits
3.3.2 Limit of a Sequence
Null Sequence
Improper Limit
3.3.3 Arithmetic and Geometric Sequences
Arithmetic Sequences
Geometric Sequence
3.4 Series
3.4.1 Definition
3.4.2 Arithmetic and Geometric Series
Arithmetic Series
Arithmetic Series of Higher Order
Geometric Series
Infinite Geometric Series
Chapter 4 Algebra
4.1 Fundamental Terms
Equations and Inequations
Universal Equations
Equivalent Transformations of Equations
4.2 Linear Equations
4.2.1 Linear Equations with One Variable
Fractional Equations
Fractional Inequations with One Variable
4.2.2 Linear Inequations with One Variable
4.2.3 Linear Equations with Multiple Variables
4.2.4 Systems of Linear Equations
Equivalent Transformations of Systems of Linear Equations
Solving Systems of Linear Equations
4.2.5 Linear Inequations with Multiple Variables
4.3 Non-linear Equations
4.3.1 Quadratic Equations with One Variable
Completing the Square
4.3.2 Cubic Equations with One Variable
Solving Cubic Equations with One Variable
Solving Cubic Equations with One Variable without Absolute Term
4.3.3 Biquadratic Equations
Solving Biquadratic Equations without Absolute Term
4.3.4 Equations of the nth Degree
4.3.5 Radical Equations
4.4 Transcendental Equations
4.4.1 Exponential Equations
4.4.2 Logarithmic Equations
4.5 Approximation Methods
4.5.1 Regula falsi (Secant Method)
4.5.2 Newton’s Method (Tangent Method)
4.5.3 General Approximation Method (Fixed-point Iteration)
Chapter 5 Linear Algebra
5.1 Fundamental Terms
5.1.1 Matrix
5.1.2 Equality/Inequality of Matrices
5.1.3 Transposed Matrix
5.1.4 Vector
5.1.5 Special Matrices and Vectors
5.2 Operations with Matrices
5.2.1 Addition of Matrices
Laws of Addition of Matrices
5.2.2 Multiplication of Matrices
5.2.2.1 Multiplication of a Matrix with a Scalar
Laws of Calculation
5.2.2.2 The Scalar Product of Two Vectors
5.2.2.3 Multiplication of a Matrix by a Column Vector
5.2.2.4 Multiplication of a Row Vector by a Matrix
5.2.2.5 Multiplication of Two Matrices
Rules of Calculation for the Multiplication of Matrices
5.3 The Inverse of a Matrix
5.3.1 Introduction
5.3.2 Determination of the Inverse with the Usage of the Gaussian Elimination Method
Rules of Calculation for Calculating with the Inverse
5.4 The Rank of a Matrix
5.4.1 Definition
5.4.2 Determination of the Rank of a Matrix
5.5 The Determinant of a Matrix
5.5.1 Definition
Minor
5.5.2 Calculation of Determinants
5.5.3 Characteristics of Determinants
5.6 The Adjoint of a Matrix
5.6.1 Definition
5.6.2 Determination of the Inverse with the Usage of the Adjoint
Chapter 6 Combinatorics
6.1 Introduction
6.2 Permutations
Permutation without Repetition
Permutation with Repetition
6.3 Variations
Variation without Repetition
Variation with Repetition
6.4 Combinations
Combination without Repetition
Combination with Repetition
Chapter 7 Financial Mathematics
7.1 Calculation of Interest
7.1.1 Fundamental Terms
7.1.2 Annual Interest
7.1.2.1 Simple Interest Calculation Interest Factor
7.1.2.2 Compound Computation of Interest
7.1.2.3 Composite Interest
7.1.3 Interest During the Period
7.1.3.1 Simple Interest Calculation (linear) Final Capital
7.1.3.2 Simple Interest Using the Nominal Annual Interest Rate Nominal Interest Rate
7.1.3.3 Compound Interest (exponential) Final Capital
7.1.3.4 Interest with Compound Interest Using a Conforming Annual Interest Rate
7.1.3.5 Mixed Interest Final Capital
7.1.3.6 Steady Interest Rate
7.2 Annual Percentage Rate
Effective Annual Percentage Rate
United States
Close-ended Credit
Open-ended Credit
European Union
7.3 Depreciation
7.3.1 Time Depreciation
7.3.1.1 Linear Depreciation
7.3.1.2 Arithmetic-Degressive Depreciation
7.3.1.3 Geometric-Degressive Depreciation
7.3.2 Units of Production Depreciation
7.3.3 Extraordinary Depreciation
7.4 Annuity Calculation
7.4.1 Fundamental Terms
7.4.2 Finite, Regular Annuity
7.4.2.1 Annual Annuity with Annual Interest
7.4.2.2 Annual Annuity with Sub-Annual Interest
7.4.2.3 Sub-Annual Annuity with Annual Interest
7.4.2.4 Sub-Annual Annuity with Sub-Annual Interest
Alternative Calculation Using the ICMA Method
Alternative Calculation Using the ICMA Method
Alternative Calculation Using the ICMA Method
Alternative Calculation Using the ICMA Method
7.4.3 Finite, Variable Annuity
7.4.3.1 Irregular Annuity Amount of Annuity
7.4.3.2 Arithmetic Progressive Annuity
7.4.3.3 Geometric Progressive Annuity
7.4.4 Perpetuity
7.5 Sinking Fund Calculation
7.5.1 Fundamental Terms
7.5.2 Annuity Repayment
7.5.3 Repayment by Instalments
7.5.4 Repayment with Premium
7.5.4.1 Annuity Repayment with Premium
7.5.4.2 Repayment of an Instalment Debt with Premium
7.5.5 Repayment with Discount (Disagio)
Annuity Repayment with Discount
7.5.5.1 Annuity Repayment with Discount when Immediately Booked as Interest Expense
7.5.5.2 Annuity Repayment with Discount when a Disagio is Included in Prepaid Expenses
7.5.5.3 Instalment Repayment with Discount when Immediately Booked as Interest Expense
7.5.5.4 Instalment Repayment with Discount when a Disagio is Included in Prepaid Expenses
7.5.6 Grace Periods
(1) Grace Periods for Annuity Repayment
k Residual Amount at the Beginning of the Year Interest Amount Repayment Instalment Annuity
(2) Grace Periods for Repayment by Instalments
k Residual Amount at the Beginning of the Year Interest Amount Repayment Instalment Annuity
7.5.7 Rounded Annuities
7.5.7.1 Percentage Annuity
7.5.7.2 Repayment of Bonds
7.5.8 Repayment During the Year
7.5.8.1 Annuity Repayment During the Year
7.5.8.2 Repayment by Instalments During the Year
7.6 Investment Calculation
7.6.1 Fundamental Terms
7.6.2 Fundamentals of Financial Mathematics
7.6.3 Methods of Static Investment Calculation
Cost Comparison Method
Profit Comparison Method
Amortisation Calculation (Pay-back Method, Pay-off Method or Pay-out Method)
Profitability Calculation
7.6.4 Methods of Dynamic Investment Calculation
7.6.4.1 Net Present Value Method (Net Present Value, Amount of Capital, Final Asset Value)
7.6.4.2 Annuity Method
7.6.4.3 Internal Rate of Return Method
Chapter 8 Optimisation of Linear Models
8.1 Lagrange Method
8.1.1 Introduction
8.1.2 Formation of the Lagrange Function
8.1.3 Determination of the Solution
8.1.4 Interpretation of λ
8.2 Linear Optimisation
8.2.1 Introduction
8.2.2 The Linear Programming Approach
8.2.3 Graphical Solution
8.2.4 Primal Simplex Algorithm
8.2.5 Simplex Tableau (Basic Structure)
Primal Simplex Algorithm | Linear Programming Approach
8.2.6 Dual Simplex Algorithm
Chapter 9 Functions
9.1 Introduction
9.1.1 Composition of Functions
9.1.2 Inverse Function
9.2 Classification of Functions
9.2.1 Rational Functions
9.2.1.1 Polynomial Functions
9.2.1.2 Broken Rational Functions
Proper Broken Rational Functions
Improper Broken Rational Functions
Characteristics
Constraints in the domain
Discontinuities
9.2.2 Non-rational Functions
9.2.2.1 Power Functions
9.2.2.2 Root Function
9.2.2.3 Transcendental Functions
9.2.2.3.1 Exponential Functions
9.2.2.3.2 Logarithmic Functions
9.2.2.4 Trigonometric Functions (Angle Functions/Circular Functions)
9.3 Characteristics of Real Functions
9.3.1 Boundedness
9.3.2 Symmetry
9.3.2.1 Axial Symmetry Axial Symmetry to the y-Axis
9.3.2.2 Point Symmetry Point Symmetry to the Point of Origin
Point Symmetry to the Point of Origin
Point Symmetry to any Arbitrary Point
9.3.3 Transformations
9.3.3.1 Vertex Form
9.3.4 Continuity
9.3.5 Infinite Discontinuities
9.3.6 Removable Discontinuities
9.3.7 Jump Discontinuities
9.3.8 Homogeneity
9.3.9 Periodicity
9.3.10 Zeros
9.3.11 Local Extremes
9.3.12 Monotonicity
9.3.13 Concavity and Convexity | Inflection Points
9.3.14 Asymptotes
9.3.14.1 Horizontal Asymptotes
9.3.14.2 Vertical Asymptote
9.3.14.3 Oblique Asymptote
9.3.14.4 Asymptotic Curve
9.3.15 Tangent Lines to a Curve
9.3.16 Normal Lines to a Curve
9.4 Exercises
Chapter 10 Differential Calculus
10.1 Differentiation of Functions with One Independent Variable
10.1.1 General
10.1.2 First Derivative of Elementary Functions
10.1.3 Derivation Rules
10.1.4 Higher Derivations
10.1.5 Differentiation of Functions with Parameters
10.1.6 Curve Sketching
10.2 Differentiation of Functions with More Than One Independent Variable
10.2.1 Partial Derivatives (1st Order)
10.2.2 Partial Derivatives (2nd Order)
10.2.3 Local Extrema of the Function f = f (x, y)
10.2.3.1 Relative Extrema without Constraint of the Function f = f (x, y)
necessary conditions
sufficient conditions
10.2.3.2 Relative Extrema with m Constraints of the Function f = f (x1, . . . , xn) with m < n
10.2.4 Differentials of the Function f = f (x1, ..., xn)
Partial Differential (1st Order)
Total Differential (1st Order)
10.3 Theorems of Differentiable Functions
10.3.1 Mean Value Theorem for Differential Calculus
10.3.2 Generalized Mean Value Theorem for Differential Calculus
10.3.3 Rolle’s Theorem
10.3.4 L’Hospital’s Rule
10.3.5 Bounds Theorem for Differential Calculus
Chapter 11 Integral Calculus
11.1 Introduction
11.2 The Indefinite Integral
11.2.1 Definition/Determining the Antiderivative
Antiderivative
Indefinite Integral
11.2.2 Elementary Calculation Rules for the Indefinite Integral
11.3 The Definite Integral
11.3.1 Introduction
11.3.2 Relationship between the Definite and the Indefinite Integral
Variation of the Upper Limit
Addition of the Absolute Values
11.3.3 Special Techniques of Integration
11.3.3.1 Partial Integration
11.3.3.2 Integration by Substitution
11.4 Multiple Integrals
11.5 Integral Calculus and Economic Problems
11.5.1 Cost Functions
11.5.2 Revenue Function (= Sales Function)
11.5.3 Profit Functions
Chapter 12 Elasticities
12.1 Definition of Elasticity
Absolute Changes
Relative Changes
12.2 Arc Elasticity
12.3 Point Elasticity
12.4 Price Elasticity of Demand εxp
12.5 Cross Elasticity of Demand εxApB
12.6 Income Elasticity of Demand εxy
Chapter 13 Economic Functions
13.1 Supply Function
13.2 Demand Function / Inverse Demand Function
13.3 Market Equilibrium
13.4 Buyer’s Market and Seller’s Market
13.5 Supply Gap
13.6 Demand Gap
13.7 Revenue Function
1. The price p is constant
2. The price p = p(x) is variable
13.8 Cost Functions
13.9 Neoclassical Cost Function
13.10 Cost Function According to the Law of Diminishing Returns
13.11 Direct Costs versus Indirect Costs
13.11.1 One-Dimensional Cost Allocation Principles
Principle of Causation
Principle of Utilisation
Principle of Averages
Principle of Plausibility
Principle of Financial Viability
13.11.2 Multi-Dimensional Cost Allocation Principles
Principle of Decision
Principle of Identity
13.12 Profit Function
Chapter 14 The Peren Theorem: The Mathematical Frame in Which We Live
Synopsis
The Current Human Lifestyle Cannot be Continued
The Peren Theorem
Options for Securing Human Livelihood
Individual Prosperity Effects
Appendix A Financial Mathematical Factors
Appendix B Bibliography
Index