ARITHMETIC
INTRODUCTION 1 [1] THE AIM OF THIS LECTURE COURSE 1 [1] RELEVANT LITERATURE 3 [4] FIRST PART: ARITHMETIC 7 [6] I CALCULATING WITH NATURAL NUMBERS 9 [6] 1 INTRODUCTION OF NUMBERS IN THE SCHOOLS 9 [6] 2 THE FUNDAMENTAL LAWS OF RECKONING 11 [9] 3 THE LOGICAL FOUNDATIONS OF OPERATIONS WITH INTEGERS 13 [11] 4 PRACTICE IN CALCULATING WITH INTEGERS 19 [18] DESCRIPTION OF THE CALCULATING MACHINE “BRUNSVIGA” 19 [19] II THE FIRST EXTENSION OF THE NOTION OF NUMBER 25 [24] 1 NEGATIVE NUMBERS 25 [24] ON THE HISTORY OF NEGATIVE NUMBERS 27 [27] 2 FRACTIONS 31 [31] 3 IRRATIONAL NUMBERS 33 [34] ON THE NATURE OF SPACE INTUITION (PRECISION AND APPROXIMATION MATHEMATICS) 37 [38] III CONCERNING SPECIAL PROPERTIES OF INTEGERS 39 [40] NUMBER THEORY IN SCHOOLS AND IN UNIVERSITIES 39 [40] SPECIAL DISCUSSIONS OF NUMBER-THEORETIC ISSUES 41 [43] PRIME NUMBERS, DECOMPOSITION INTO PRIME FACTORS 41 [43] TRANSFORMATION OF ORDINARY FRACTIONS INTO DECIMAL NUMBERS 42 [44] CONTINUED FRACTIONS 43 [45] PYTHAGOREAN NUMBERS, FERMAT’S LAST THEOREM 47 [49] THE PROBLEM OF THE DIVISION OF THE CIRCLE 52 [54] IV COMPLEX NUMBERS 59 [61] 1 ORDINARY COMPLEX NUMBERS 59 [61] 2 HIGHER COMPLEX NUMBERS, ESPECIALLY QUATERNIONS 61 [64] REMARKS ON VECTOR CALCULUS 67 [69] 3 QUATERNION MULTIPLICATION – ROTATION-DILATION 69 [71] INTERPRETATION IN THREE-DIMENSIONAL SPACE 72 [74] 4 COMPLEX NUMBERS IN SCHOOL TEACHING 79 [81] CONCERNING THE MODERN DEVELOPMENT AND THE GENERAL STRUCTURE OF MATHEMATICS 81 [82] THE STRUCTURE OF ELEMENTARY ANALYSIS ACCORDING TO TWO PARALLEL PROCESSES OF DEVELOPMENT OF DIFFERING CHARACTER 81 [82] OVERVIEW OF THE HISTORY OF MATHEMATICS 84 [86] SECONDPART: ALGEBRA 89 [93] TEXTBOOKS 89 [93] OUR SPECIFIC GOAL: APPLICATION OF GEOMETRICALLY INTUITIVE METHODS TO THE SOLUTION OF EQUATIONS 90 [93] I REAL EQUATIONS WITH REAL UNKNOWNS 91 [94] 1 EQUATIONS WITH ONE PARAMETER 91 [94] 2 EQUATIONS WITH TWO PARAMETERS 92 [95] CLASSIFICATION ACCORDING TO THE NUMBER OF REAL ROOTS 97 [99] 3 EQUATIONS WITH THREE PARAMETERS ; ; 100 [101] AN APPARATUS FOR THE NUMERICAL SOLUTION OF EQUATIONS 101 [102] THE DISCRIMINANT SURFACE OF THE BIQUADRATIC EQUATION 102 [103] II EQUATIONS IN THE FIELD OF COMPLEX QUANTITIES 109 [109] A THE FUNDAMENTAL THEOREM OF ALGEBRA 109 [109] B EQUATIONS WITH A COMPLEX PARAMETER 112 [112] 1 THE “PURE” EQUATION 118 [119] IRREDUCIBILITY; “IMPOSSIBILITY” OF TRISECTING THE ANGLE 121 [122] 2 THE DIHEDRAL EQUATION 124 [124] 3 THE TETRAHEDRAL, THE OCTAHEDRAL, AND THE ICOSAHEDRAL EQUATIONS 129 [130] 4 CONTINUATION: SETTING UP THE NORMAL EQUATION 133 [134] 5 CONCERNING THE SOLUTION OF THE NORMAL EQUATIONS 139 [140] 6 UNIFORMISATION OF THE NORMAL IRRATIONALITIES BY MEANS OF TRANSCENDENTAL FUNCTIONS 142 [143] THE “CASUS IRREDUCIBILIS” AND THE TRIGONOMETRIC SOLUTION OF THE CUBIC EQUATION 144 [145] 7 SOLVABILITY IN TERMS OF RADICAL SIGNS 147 [148] 8 REDUCTION OF GENERAL EQUATIONS TO NORMAL EQUATIONS 150 [152] THIRDPART: ANALYSIS 153 [155] I LOGARITHMIC AND EXPONENTIAL FUNCTIONS 155 [155] 1 SYSTEMATIC ACCOUNT OF ALGEBRAIC ANALYSIS 155 [155] 2 THE HISTORICAL DEVELOPMENT OF THE THEORY 157 [157] NEPER AND BÜRGI 158 [158] THE 17TH CENTURY: THE AREA OF THE HYPERBOLA 162 [160] EULER AND LAGRANGE: ALGEBRAIC ANALYSIS 163 [164] THE 19TH CENTURY: FUNCTIONS OF COMPLEX VARIABLES 165 [166] 3 REMARKS ABOUT TEACHING IN SCHOOLS 166 [167] 4 THE STANDPOINT OF FUNCTION THEORY 168 [169] THE PASSAGE TO THE LIMIT FROM THE POWER TO THE EXPONENTIAL FUNCTION 172 [173] II THE GONIOMETRIC FUNCTIONS 175 [175] 1 THEORY OF THE GONIOMETRIC FUNCTIONS 175 [175] EXACT COMPARISON WITH THE THEORY OF LOGARITHMS 177 [176] 2 TRIGONOMETRIC TABLES 184 [183] A PURELY TRIGONOMETRIC TABLES 184 [183] B LOGARITHMIC-TRIGONOMETRIC TABLES 186 [185] 3 APPLICATIONS OF GONIOMETRIC FUNCTIONS 189 [188] A TRIGONOMETRY, IN PARTICULAR, SPHERICAL TRIGONOMETRY 189 [189] FORMULAS OF SECOND KIND: TRIANGLES OF FIRST AND SECOND KIND 194 [194] THE AREA OF SPHERICAL TRIANGLES 197 [197] B THEORY OF SMALL OSCILLATIONS, ESPECIALLY THOSE OF THE PENDULUM 201 [201] TEACHING IN SCHOOLS (HIDDEN INFINITESIMAL CALCULUS) 202 [202] C REPRESENTATION OF PERIODIC FUNCTIONS BY MEANS OF SERIES OF GONIOMETRIC FUNCTIONS (TRIGONOMETRIC SERIES) 205 [205] APPROXIMATION BY SERIES WITH A FINITE NUMBER OF TERMS 206 [206] ESTIMATION OF ERRORS; CONVERGENCE OF INFINITE SERIES 211 [211] THE GIBBS PHENOMENON 214 [214] EXCURSIONS CONCERNING THE GENERAL NOTION OF FUNCTION 216 [215] HISTORICAL IMPORTANCE OF TRIGONOMETRIC SERIES; THE ROLE OF FOURIER 221 [221] III CONCERNING INFINITESIMAL CALCULUS PROPER 225 [223] 1 GENERAL CONSIDERATIONS IN INFINITESIMAL CALCULUS 225 [223] EMERGENCE OF THE INFINITESIMAL CALCULUS BY THE SPECIFICITY OF OUR SENSE INTUITION 226 [224] THE LOGICAL FOUNDATION OF DIFFERENTIAL AND INTEGRAL CALCULUS BY MEANS OF THE LIMIT CONCEPT (NEWTON AND HIS SUCCESSORS UP TO CAUCHY) 229 [228] CONSTRUCTION OF THE INFINITESIMAL CALCULUS BASED ON “DIFFERENTIALS” (LEIBNIZ AND HIS FOLLOWERS) 232 [231] THE ACTUAL INFINITELY SMALL QUANTITIES IN THE AXIOMATICS OF GEOMETRY 236 [234] THE REACTION: THE DERIVATIVE CALCULUS OF LAGRANGE 238 [237] FORM AND IMPORTANCE OF THE INFINITESIMAL CALCULUS IN THE PRESENT STATE OF TEACHING 239 [239] 2 TAYLOR’S THEOREM 241 [241] THE FIRST PARABOLAS OF OSCULATION 241 [241] INCREASING THE ORDER: QUESTIONS OF CONVERGENCE 245 [244] GENERALISING TAYLOR’S THEOREM TO A THEOREM OF FINITE DIFFERENCES 248 [246] CAUCHY’S ESTIMATE OF THE ERROR 251 [249] HISTORICAL REMARKS ABOUT TAYLOR AND MACLAURIN 253 [251] 3 HISTORICAL AND PEDAGOGICAL CONSIDERATIONS 254 [253] REMARKS ABOUT TEXTBOOKS FOR THE INFINITESIMAL CALCULUS 254 [253] CHARACTERISING OUR PROPER PRESENTATION 256 [254] IV SUPPLEMENT 259 [256] IVA TRANSCENDENCE OF THE NUMBERS E AND 259 [256] HISTORICAL ASPECTS 259 [256] PROOF OF THE TRANSCENDENCE OF E, 260 [257] PROOF OF THE TRANSCENDENCE OF 266 [263] MORE ON TRANSCENDENT AND ALGEBRAIC NUMBERS 272 [269] IVB SET THEORY 274 [271] 1 THE POTENCY OF SETS 274 [271] DENUMERABILITY OF RATIONAL AND ALGEBRAIC NUMBERS 275 [273] THE CONTINUUM NOT BEING DENUMERABLE 278 [276] POTENCY OF CONTINUA OF MORE DIMENSIONS 280 [278] SETS WITH HIGHER POTENCIES 283 [281] 2 ARRANGEMENT OF THE ELEMENTS OF A SET 286 [283] TYPES OF ARRANGEMENT OF DENUMERABLE SETS 286 [284] THE CONTINUITY OF THE CONTINUUM 287 [285] INVARIANCE OF DIMENSION FOR CONTINUOUS ONE-TO-ONE REPRESENTATIONS 288 [286] CLOSING REMARKS 290 [287] IMPORTANCE AND GOALS OF SET THEORY 290 [289]