Franz w peren math for business and economics compendium of essential formulas springer (2021)

The use of general descriptive names, registered names, trademarks, service marks, etc.. 1 1.2 General Arithmetic Relations and Links.. 4 1.7 Trigonometric Functions, Hyperbolic Function

Math for Business and Economics

Franz W Peren

Compendium of Essential Formulas

Math for Business and Economics

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Franz W Peren

Bonn-Rhein-Sieg University

Sankt Augustin, Germany

ISBN 978-3-662-63248-2 ISBN 978-3-662-63249-9 (eBook)

https://doi.org/10.1007/978-3-662-63249-9

© Springer-Verlag GmbH Germany, part of Springer Nature 2021

This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software,

or by similar or dissimilar methodology now known or hereafter developed

The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations This Springer imprint is published by the registered company Springer-Verlag GmbH, DE part

of Springer Nature

The registered company address is: Heidelberger Platz 3, 14197 Berlin, Germany

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For my father, Paul.

The following book is based on the author’s expertise in the field of ness mathematics After completing his studies in business administra-tion and mathematics, he started his career working for a global bankand the German government Later he became a professor of businessadministration, specializing in quantitative methods He has been a pro-fessor at the Bonn-Rhein-Sieg University in Sankt Augustin, Germanysince 1995, where he is mainly teaching business mathematics, busi-ness statistics, and operations research He has also previously taughtand conducted research at the University of Victoria in Victoria, BC,Canada and at Columbia University in New York City, New York, USA

busi-To the author’s best knowledge and beliefs, this formulary presents itsmathematical contents in a practical manner, as they are needed formeaningful and relevant application in global business, as well as inuniversities and economic practice

The author would like to thank his academic colleagues who have tributed to this work and to many other projects with creativity, knowl-edge and dedication for more than 25 years In particular, he wouldlike to thank Ms Eva Siebertz and Mr Nawid Schahab, who were in-strumental in managing and creating this formulary Special thanks aregiven to Ms Camilla Demuth, Ms Linh Hoang, and Ms Michelle Jarsen.Should any mistakes remain, such errors shall be exclusively at theexpense of the author The author is thankful in advance to all users ofthis formulary for any constructive comments or suggestions

VII

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1.1 Pragmatic Signs 1

1.2 General Arithmetic Relations and Links 1

1.3 Sets of Numbers 2

1.4 Special Numbers and Links 3

1.5 Limit 4

1.6 Exponential Functions, Logarithm 4

1.7 Trigonometric Functions, Hyperbolic Functions 4

1.8 Vectors, Matrices 5

1.9 Sets 6

1.10 Relations 7

1.11 Functions 7

1.12 Order Structures 7

1.13 SI Multiplying and Dividing Prefixes 8

1.14 Greek Alphabet 9

X Contents

2.1 Mathematical Logic 11

2.2 Propositional Logic 11

2.2.1 Propositional Variable 11

2.2.2 Truth Tables 12

3 Arithmetic 15 3.1 Sets 15

3.1.1 General 15

3.1.2 Set Relations 16

3.1.3 Set Operations 17

3.1.4 Relations, Laws, Rules of Calculation for Sets 19 3.1.5 Intervals 21

3.1.6 Numeral Systems 22

3.1.6.1 Decimal System (Decadic System) 23 3.1.6.2 Dual System (Binary System) 23

3.1.6.3 Roman Numeral System 24

3.2 Elementary Calculus 24

3.2.1 Elementary Foundations 24

3.2.1.1 Axioms 25

3.2.1.2 Factorisation 25

3.2.1.3 Relations 26

3.2.1.4 Absolute Value, Signum 26

3.2.1.5 Fractions 27

3.2.1.6 Polynomial Division 27

3.2.2 Conversions of Terms 29

3.2.2.1 Binomial Formulas 29

3.2.2.2 Binomial Theorem 29

3.2.2.3 General Binomial Theorem for Nat-ural Exponents 30

3.2.2.4 General Binomial Theorem for Real Exponents 30

3.2.2.5 Polynomial Terms 30

3.2.3 Summation and Product Notation 30

3.2.3.1 Summation Notation 31

3.2.3.2 Product Notation 32

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Contents XI

3.2.4 Powers, Roots 33

3.2.5 Logarithms 36

3.2.6 Factorial 38

3.2.7 Binomial Coefficient 39

3.3 Sequences 41

3.3.1 Definition 41

3.3.2 Limit of a Sequence 44

3.3.3 Arithmetic and Geometric Sequences 46

3.4 Series 46

3.4.1 Definition 46

3.4.2 Arithmetic and Geometric Series 47

4 Algebra 51 4.1 Fundamental Terms 51

4.2 Linear Equations 53

4.2.1 Linear Equations with One Variable 53

4.2.2 Linear Inequalities with One Variable 56

4.2.3 Linear Equations with Multiple Variables 56

4.2.4 Systems of Linear Equations 57

4.2.5 Linear Inequalities with Multiple Variables 61

4.3 Non-linear Equations 62

4.3.1 Quadratic Equations with One Variable 62

4.3.2 Cubic Equations with One Variable 65

4.3.3 Biquadratic Equations 67

4.3.4 Equations of the nthDegree 68

4.3.5 Radical Equations 69

4.4 Transcendental Equations 71

4.4.1 Exponential Equations 71

4.4.2 Logarithmic Equations 73

4.5 Approximation Methods 75

4.5.1 Regula falsi (Secant Method) 75

4.5.2 Newton’s Method (Tangent Method) 77

XII Contents

4.5.3 General Approximation Method (Fixed-point

Iteration) 80

5 Linear Algebra 87 5.1 Fundamental Terms 87

5.1.1 Matrix 87

5.1.2 Equality/Inequality of Matrices 88

5.1.3 Transposed Matrix 89

5.1.4 Vector 89

5.1.5 Special Matrices and Vectors 92

5.2 Operations with Matrices 94

5.2.1 Addition of Matrices 94

5.2.2 Multiplication of Matrices 96

5.2.2.1 Multiplication of a Matrix with a Scalar 96

5.2.2.2 The Scalar Product of Two Vec-tors 98

5.2.2.3 Multiplication of a Matrix by a Col-umn Vector 100

5.2.2.4 Multiplication of a Row Vector by a Matrix 102

5.2.2.5 Multiplication of Two Matrices 103

5.3 The Inverse of a Matrix 107

5.3.1 Introduction 107

5.3.2 Determination of the Inverse with the Usage of the Gaussian Elimination Method 109

5.4 The Rank of a Matrix 113

5.4.1 Definition 113

5.4.2 Determination of the Rank of a Matrix 113

5.5 The Determinant of a Matrix 117

5.5.1 Definition 117

5.5.2 Calculation of Determinants 118

5.5.3 Characteristics of Determinants 124

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Contents XIII

5.6 The Adjoint of a Matrix 125

5.6.1 Definition 125

5.6.2 Determination of the Inverse with the Usage of the Adjoint 127

6 Combinatorics 129 6.1 Introduction 129

6.2 Permutations 133

6.3 Variations 135

6.4 Combinations 136

7 Financial Mathematics 141 7.1 Calculation of Interest 141

7.1.1 Fundamental Terms 141

7.1.2 Annual Interest 142

7.1.2.1 Simple Interest Calculation 142

7.1.2.2 Compound Computation of Inter-est 144

7.1.2.3 Composite Interest 146

7.1.3 Interest During the Period 155

7.1.3.1 Simple Interest Calculation (linear) 155 7.1.3.2 Simple Interest Using the Nomi-nal Annual Interest Rate 156

7.1.3.3 Compound Interest (exponential) 157 7.1.3.4 Interest with Compound Interest Using a Conforming Annual Inter-est Rate 157

7.1.3.5 Mixed Interest 159

7.1.3.6 Steady Interest Rate 160

7.2 Annual Percentage Rate 164

7.3 Depreciation 169

7.3.1 Time Depreciation 169

XIV Contents

7.3.1.2 Arithmetic-Degressive Depreciation 170 7.3.1.3 Geometric-Degressive

Deprecia-tion 172

7.3.2 Units of Production Depreciation 174

7.3.3 Extraordinary Depreciation 175

7.4 Annuity Calculation 176

7.4.1 Fundamental Terms 176

7.4.2 Finite, Regular Annuity 179

7.4.2.1 Annual Annuity with Annual Interest 179 7.4.2.2 Annual Annuity with Sub-Annual Interest 183

7.4.2.3 Sub-Annual Annuity with Annual Interest 186

7.4.2.4 Sub-Annual Annuity with Sub-Annual Interest 190

7.4.3 Finite, Variable Annuity 192

7.4.3.1 Irregular Annuity 192

7.4.3.2 Arithmetic Progressive Annuity 193

7.4.3.3 Geometric Progressive Annuity 195

7.4.4 Perpetuity 198

7.5 Sinking Fund Calculation 199

7.5.1 Fundamental Terms 200

7.5.2 Annuity Repayment 202

7.5.3 Repayment by Instalments 204

7.5.4 Repayment with Premium 206

7.5.4.1 Annuity Repayment with Premium 206 7.5.4.2 Repayment of an Instalment Debt with Premium 210

7.5.5 Grace Periods 212

7.5.6 Rounded Annuities 214

7.5.6.1 Percentage Annuity 214

7.5.6.2 Repayment of Bonds 217

7.5.7 Repayment During the Year 222

7.5.7.1 Annuity Repayment During the Year 222 7.5.7.2 Repayment by Instalments During the Year 229

7.6 Investment Calculation 234

7.6.1 Fundamental Terms 235

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Contents XV

7.6.2 Fundamentals of Financial Mathematics 238

7.6.3 Methods of Static Investment Calculation 240

7.6.4 Methods of Dynamic Investment Calcula-tion 241

7.6.4.1 Net Present Value Method (Net Present Value, Amount of Capital, Final Asset Value) 241

7.6.4.2 Annuity Method 244

7.6.4.3 Internal Rate of Return Method 246

8 Optimisation of Linear Models 249 8.1 Lagrange Method 249

8.1.1 Introduction 249

8.1.2 Formation of the Lagrange Function 249

8.1.3 Determination of the Solution 250

8.1.4 Interpretation ofl 251

8.2 Linear Optimisation 254

8.2.1 Introduction 254

8.2.2 The Linear Programming Approach 254

8.2.3 Graphical Solution 255

8.2.4 Simplex Algorithm 258

8.2.5 Simplex Tableau (Basic Structure) 259

9 Functions 265 9.1 Introduction 265

9.1.1 Composition of Functions 269

9.1.2 Inverse Function 271

9.2 Classification of Functions 273

9.2.1 Rational Functions 274

9.2.1.1 Polynomial Functions 274

9.2.1.2 Broken Rational Functions 274

9.2.2 Non-rational Functions 278

9.2.2.1 Power Functions 278

9.2.2.2 Root Function 281

XVI Contents

9.2.2.3 Transcendental Functions 282

9.2.2.3.1 Exponential Functions 282 9.2.2.3.2 Logarithmic Functions 288

9.2.2.4 Trigonometric Functions (Angle Func-tions/Circular Functions) 294

9.3 Characteristics of Real Functions 322

9.3.1 Boundedness 322

9.3.2 Symmetry 324

9.3.2.1 Axial Symmetry 324

9.3.2.2 Point Symmetry 326

9.3.3 Transformations 329

9.3.3.1 Vertex Form 331

9.3.4 Continuity 334

9.3.5 Infinite Discontinuities 334

9.3.6 Removable Discontinuities 336

9.3.7 Jump Discontinuities 337

9.3.8 Homogeneity 338

9.3.9 Periodicity 339

9.3.10 Zeros 339

9.3.11 Local Extremes 340

9.3.12 Monotonicity 341

9.3.13 Concavity and Convexity | Inflection Points 342 9.3.14 Asymptotes 344

9.3.14.1 Horizontal Asymptotes 345

9.3.14.2 Vertical Asymptote 347

9.3.14.3 Oblique Asymptote 348

9.3.14.4 Asymptotic Curve 349

9.3.15 Tangent Lines to a Curve 350

9.3.16 Normal Lines to a Curve 351

9.4 Exercises 352

10 Differential Calculus 357 10.1 Differentiation of Functions with One Independent Variable 357

10.1.1 General 357

10.1.2 First Derivative of Elementary Functions 360

10.1.3 Derivation Rules 362

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Contents XVII

10.1.4 Higher Derivations 36410.1.5 Differentiation of Functions with Parame-

ters 36510.1.6 Curve Sketching 36510.2 Differentiation of Functions with More Than One In-

dependent Variable 37510.2.1 Partial Derivatives (1stOrder) 37510.2.2 Partial Derivatives (2nd Order) 37810.2.3 Local Extrema of the Function f = f (x, y) 380

10.2.3.1 Relative Extrema without Constraint

of the Function f = f (x, y) 38010.2.3.2 Relative Extrema with m Constraints

of the Function f = f (x1, , xn)with m < n 38410.2.4 Differentials of the Function f = f (x1, ,xn) 38810.3 Theorems of Differentiable Functions 39010.3.1 Mean Value Theorem for Differential Cal-

culus 39010.3.2 Generalized Mean Value Theorem for Dif-

ferential Calculus 39110.3.3 Rolle’s Theorem 39110.3.4 L’Hospital’s Rule 39210.3.5 Bounds Theorem for Differential Calculus 393

11.1 Introduction 39511.2 The Indefinite Integral 39611.2.1 Definition/Determining the Antiderivative 39611.2.2 Elementary Calculation Rules for the Indefi-

nite Integral 39911.3 The Definite Integral 40011.3.1 Introduction 40011.3.2 Relationship between the Definite and the

Indefinite Integral 404

XVIII Contents

11.3.3 Special Techniques of Integration 409

11.3.3.1 Partial Integration 409

11.3.3.2 Integration by Substitution 411

11.4 Multiple Integrals 412

11.5 Integral Calculus and Economic Problems 413

11.5.1 Cost Functions 413

11.5.2 Revenue Function (= Sales Function) 415

11.5.3 Profit Functions 416

12 Elasticities 421 12.1 Definition of Elasticity 421

12.2 Arc Elasticity 422

12.3 Point Elasticity 427

12.4 Price Elasticity of Demandexp 430

12.5 Cross Elasticity of DemandexApB 435

12.6 Income Elasticity of Demandexy 436

13 Economic Functions 439 13.1 Supply Function 439

13.2 Demand Function / Inverse Demand Function 441

13.3 Market Equilibrium 443

13.4 Buyer’s Market and Seller’s Market 444

13.5 Supply Gap 445

13.6 Demand Gap 445

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Contents XIX

13.7 Revenue Function 44613.8 Cost Functions 45213.9 Neoclassical Cost Function 46013.10 Cost Function According to the Law of Diminishing

Returns 46713.11 Direct Costs versus Indirect Costs 47813.11.1 One-Dimensional Cost Allocation Principles 48113.11.2 Multi-Dimensional Cost Allocation Principles 48313.12 Profit Function 486

14 The Peren Theorem

The Mathematical Frame in Which We Live 495

List of Abbreviations

XXII List of Abbreviations

XXIV List of Abbreviations

Chapter 1

Mathematical Signs and Symbols

applica-tions For definitions see dedicated passage

1.2 General Arithmetic Relations and Links

(a, b are figures, elements, objects)

a < b aless than b, fundamental term, e.g 6 < 2

a > b agreater than b, e.g 3 > 8

1

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2 1 Mathematical Signs and Symbols

a  b aless than or (at most) equal to b, a  8 is equivalent to

a b agreater than or (at least) equal to b, is equivalent to b  a

b| a,b 2 Z, b 6= 0}

Q+

R+

1.4 Special Numbers and Links 3]a,•[ open, unbounded interval starting at a, {x | a < x}

[a,b] closed interval from a to b, {x | a  x  b}

[a,•[ closed, unbounded interval starting at a,{x | a  x}

[a,b[ left-closed, right-open interval from a to b, {x | a  x < b}

1.4 Special Numbers and Links

a[i] ain the ithposition; e.g 5; 6; 7; a[2]=6

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4 1 Mathematical Signs and Symbols

1.5 Limit

lim

x!0 f (x) = a ais the limit of the function f (x) for x towards 0,

i.e xx!0gradually approaches the value 0,the function’s value f (x) converges (limits)towards a

lim

x!• f (x) = b bis the limit of the function f (x) for x towards•

lim

x!5 f (x) = c cis the limit of the function f (x) for x towards 5

1.6 Exponential Functions, Logarithm

1.8 Vectors, Matrices 5

1.8 Vectors, Matrices

a,b,x,y, signs for vectors, also !a ,!b ,!x ,!y

vectors

A = (ai j)m,n-matrix A

element ai j(ithrow, jthcolumn)

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6 1 Mathematical Signs and Symbols

C identity (unit) matrix; diagonal matrix, whoseelements of the main diagonals are all 1and whose remaining elements are all 0

proper inclusion relation “included and unequal”

which are not included in B

1.12 Order Structures 7

1.10 Relations

set of all (ordered) pairs from A and B

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8 1 Mathematical Signs and Symbols

1.13 SI1Multiplying and Dividing Prefixes

8 x universal quantifier for all x (applies)

9 x existential quantifier there is (at least) one x for

which it applies

2.2 Propositional Logic

2.2.1 Propositional Variable

a, b, are letters or other symbols which can be used as

place-holders for statements or truths

A Ais a statement that can be true (t) or false ( f ).

truth values t (true); f (false)

Examples: The statement “7 is a prime number” is true,

the statement “8 3 = 4” is false,

“7x + 4 = 25” is only valid when “x = 3”

“3” is called solution

v(A) v(A)is referred to as the truth value of the statement A.v(A) = 1means that A is true and v(A) = 0 means that A isfalse

¬ A The negation ¬A (or ¯A) of the statement A is true when A isfalse and false when A is true

A ^ B The conjunction A ^ B is true when both statements are true

and false when there is at least one false statement

A _ B The disjunction A _ B is true when there is at least one true

statement, and false when both statements are false

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2.2 Propositional Logic 13

A ) B The implication A ) B means: When A is true, B is also true

Ais considered as condition (premise) and B as quence (conclusion) A ) B is only false when a false

conse-conclusion is drawn from a true premise

A , B The equivalence A , B means: When A is true, B is also true

and vice versa A , B is only false when one of the ments is true and the other one is false

state-9 “There is” (e.g.: 9x 2Q : x2=4means: there is a rational ber x with x2=4)

num-8 “For all” (e.g.: 8x 2 Q : x2 0means: for all rational numbers xwith x2 0)

{a1, ,an} set with the elements a1, ,an

(no elements included)

a, b 2 A , a 2 A ^ b 2 A

i.e set equality)

A ✓ B ^ A 6= B, proper inclusion relation “includedand unequal”

¯A = G\A (G is the universal set)

15

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16 3 Arithmetic

A ⇥ B = {(a,b) | a 2 A ^ b 2 B}

P(A)is the set of all subsets T of A

Bounds, Limits of a Set

A universal set, SU, is bounded upwards (or downwards) if it has at leastone upper (or lower) bound B If both conditions apply, SUis bounded:

Intersection of two sets A \ B;

conjunction: “A and B”

Aand B are conjunct for: A \ B = {x | x 2 A ^ x 2 B}

Aand B are disjoint for: A \ B = /0

Relative complement of two sets A \ B, “A without B”

A \ B = {x | x 2 A ^ x /2 B}

Symmetric difference of A and B

A4B = (A [ B) \ (A \ B)

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18 3 Arithmetic

Complement of the set B

Set of all elements,

which are not included in B

¯B = {x | x 2 A^x /2 B}

Power set of B

Set of all subsets of a set B

P(B) = {x | x ✓ B} always valid: /0 2 P(B) and B 2 P(B)

Product (cartesian) of two sets A ⇥ B, “ A cross B ”

A ⇥ B (product of two sets) is the set of all ordered pairs of elements

(a, b)with a 2 A and b 2 B

A ⇥ B = {(a, b) | a 2 A; b 2 B} A ⇥ B 6= B ⇥ A

The product set A1⇥ A2⇥ ⇥ An, n 1, is the set of all ordered k-tuples(x1, ,xn) of the elements x1of A1, x2of A2, , xnof An

20 3 ArithmeticProduct relations

• infinite (half-open) intervals•; • are “improper

numbers” in R with • < a; a < • for all a 2 R