Math for Business and Economics: Compendium of Essential Formulas, 2nd Edition
- Length: 655 pages
- Edition: 2
- Language: English
- Publisher: Springer
- Publication Date: 2023-03-14
- ISBN-10: 3662669749
- ISBN-13: 9783662669747
- Sales Rank: #0 (See Top 100 Books)
This 2nd edition, revised and extended compendium contains and explains essential mathematical formulas within an economic context. Newly added content focuses on financial mathematics, now including an international comparison between different national methods used in the calculation of interest. Further, the annuity calculation now contains unique content.
A broad range of aids and supportive examples will help readers to understand the formulas and their practical applications. This mathematical formulary is presented in a practice-oriented, clear, and understandable manner, as it is needed for meaningful and relevant application in global business, as well as in the academic setting and economic practice. The topics presented include, but are not limited to: mathematical signs and symbols, logic, arithmetic, algebra, linear algebra, combinatorics, financial mathematics, optimisation of linear models, functions, differential calculus, integral calculus, elasticities, economic functions, and the Peren Theorem.
Given its scope, the book offers an indispensable reference guide and is a must-read for undergraduate and graduate students, as well as managers, scholars, and lecturers in business, politics, and economics.
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
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