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Math for Business and Economics: Compendium of Essential Formulas, 4th edition
Author(s): Franz W. Peren－Hochschule Bonn-Rhein-Sieg University of Applied Sciences Sankt Augustin, Germany
Course description
This course extends compendium contains and explains essential mathematical formulas within an economic context. Newly added content introduces non-linear optimization, focusing on maximizing or minimizing real problems using at least one non-linear function and continuous variables. It explains how non-linear programs may include various constraints, are influenced by the properties of the objective functions and the admissible domain, and are characterized by iterative solution processes that ideally converge to feasible, locally optimized solutions.
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, and financial mathematics, including an international comparison between different national methods used in the calculation of interest, optimization of linear models, functions, differential calculus, integral calculus, elasticities, annuity calculation, 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.
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 SI Multiplying 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
3.1.2 Set Relations
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) Decimal powers:
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 with
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
3.3.2 Limit of a Sequence
3.3.3 Arithmetic and Geometric Sequences
3.4 Series
3.4.1 Definition
3.4.2 Arithmetic and Geometric Series
Chapter 4 Algebra
4.1 Fundamental Terms
4.2 Linear Equations
4.2.1 Linear Equations 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
4.2.5 Linear Inequations with Multiple Variables
4.3 Non-linear Equations
4.3.1 Quadratic Equations with One Variable
4.3.2 Cubic Equations with One Variable
4.3.3 Biquadratic Equations
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
5.2.2 Multiplication of Matrices
5.2.2.1 Multiplication of a Matrix with a Scalar
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
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
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
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
6.3 Variations
6.4 Combinations
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
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
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 (a) Annuity repayment with non-included premium
7.5.4.2 Repayment of an Instalment Debt with Premium
7.5.5 Repayment with Discount (Disagio)
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
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
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
8.1.5 Identification of the Type of Optimum
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)
8.2.6 Dual Simplex Algorithm
8.3 Non-linear Optimisation
8.3.1 Introduction
8.3.2 Basic Properties of Non-linear Optimisation
8.3.3 Non-linear Optimisation Methods
8.3.3.1 Search Strategies
8.3.3.2 Deterministic Search Strategies
8.3.3.3 The Nelder-Mead Simplex Search Procedure
8.3.4 Conclusion
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
9.2.2 Non-rational Functions
9.2.2.1 Power Functions
9.2.2.2 Root Function
9.2.2.3 Transcendental Functions
9.2.2.4 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
9.3.2.2 Point Symmetry Point Symmetry to the Point of Origin
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
10.2.3.1 Relative Extrema without Constraint of the Function
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)
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 Theorem5
10.3.4 L’Hospital’s Rule6
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
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
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
12.2 Arc Elasticity
12.3 Point Elasticity
12.4 Price Elasticity of Demand
12.5 Cross Elasticity of Demand
12.6 Income Elasticity of Demand
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
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
13.11.2 Multi-Dimensional Cost Allocation Principles
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
Math for Business and Economics_ Compendium of Essential Formulas Franz W. Peren.pdf
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