All in One Mathematics Cheat Sheet, Cheat Sheet of Mathematics

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Page 1 of 330 ALL IN ONE MATHEMATICS CHEAT SHEET V2.10 eiπ + 1 = 0 CONTAINING FORMULAE FOR ELEMENTARY, HIGH SCHOOL AND UNIVERSITY MATHEMATICS COMPILED FROM MANY SOURCES BY ALEX SPARTALIS 2009-2013 4/9/2013 9:44:00 PM Euler’s Identity: Page 2 of 330 REVISION HISTORY 2.1. 08/06/2012 UPDATED: Format NEW: Multivariable Calculus UPDATED: Convergence tests UPDATED: Composite Functions 2.2. 10/07/2012 NEW: Three Phase – Delta & Y NEW: Electrical Power 2.3. 14/08/2012 NEW: Factorial NEW: Electromagnetics NEW: Linear Algebra NEW: Mathematical Symbols NEW: Algebraic Identities NEW: Graph Theory UPDATED: Linear Algebra UPDATED: Linear Transformations 2.4. 31/08/2012 NEW: Graphical Functions NEW: Prime numbers NEW: Power Series Expansion NEW: Inner Products UPDATED: Pi Formulas UPDATED: General Trigonometric Functions Expansion UPDATED: Linear Algebra UPDATED: Matrix Inverse 2.5. 10/09/2012 NEW: Machin-Like Formulae NEW: Infinite Summations To Pi NEW: Classical Mechanics NEW: Relativistic Formulae NEW: Statistical Distributions NEW: Logarithm Power Series NEW: Spherical Triangle Identities NEW: Bernoulli Expansion UPDATED: Pi Formulas UPDATED: Logarithm Identities UPDATED: Riemann Zeta Function UPDATED: Eigenvalues and Eigenvectors 2.6. 3/10/2012 NEW: QR Factorisation NEW: Jordan Forms NEW: Macroeconomics NEW: Golden Ratio & Fibonacci Sequence NEW: Complex Vectors and Matrices NEW: Numerical Computations for Matrices UPDATED: Prime Numbers UPDATED: Errors within Matrix Formula 2.7. 25/10/2012 NEW: USV Decomposition NEW: Ordinary Differential Equations Using Matrices NEW: Exponential Identities UPDATED: Matrix Inverse CORRECTION: Left and Right Matrix Inverse 2.8. 31/12/2012 NEW: Applications of Functions NEW: Higher Order Integration NEW: Root Expansions Page 5 of 330 PRISIM: 40 PYRAMID: 40 TETRAHEDRON: 40 OCTAHEDRON: 40 DODECAHEDRON: 40 ICOSAHEDRON: 40 3.3 SURFACE AREA: 40 CUBE: 40 CUBOIDS: 40 TETRAHEDRON: 41 OCTAHEDRON: 41 DODECAHEDRON: 41 ICOSAHEDRON: 41 CYLINDER: 41 3.4 MISCELLANEOUS: 41 DIAGONAL OF A RECTANGLE 41 DIAGONAL OF A CUBOID 41 LONGEST DIAGONAL (EVEN SIDES) 41 LONGEST DIAGONAL (ODD SIDES) 41 TOTAL LENGTH OF EDGES (CUBE): 41 TOTAL LENGTH OF EDGES (CUBOID): 41 CIRCUMFERENCE 41 PERIMETER OF RECTANGLE 41 SEMI PERIMETER 41 EULER’S FORMULA 41 3.5 ABBREVIATIONS (3.1, 3.2, 3.3, 3.4) 41 PART 4: ALGEBRA & ARITHMETIC 43 4.1 POLYNOMIAL FORMULA: 43 QUDARATIC: 43 CUBIC: 43 4.2 FUNDAMENTALS OF ARITHMETIC: 45 RATIONAL NUMBERS: 45 IRRATIONAL NUMBERS: 45 4.3 ALGEBRAIC EXPANSION: 45 BABYLONIAN IDENTITY: 45 COMMON PRODUCTS AND FACTORS: 45 BINOMIAL THEOREM: 45 BINOMIAL EXPANSION: 45 DIFFERENCE OF TWO SQUARES: 46 BRAHMAGUPTA–FIBONACCI IDENTITY: 46 DEGEN'S EIGHT-SQUARE IDENTITY: 46 4.4 ROOT EXPANSIONS: 47 4.5 LIMIT MANIPULATIONS: 47 L’HOPITAL’S RULE: 47 4.6 SUMMATION MANIPULATIONS: 48 4.7 COMMON FUNCTIONS: 48 CONSTANT FUNCTION: 48 LINE/LINEAR FUNCTION: 48 PARABOLA/QUADRATIC FUNCTION: 49 CIRCLE: 49 ELLIPSE: 49 HYPERBOLA: 49 4.8 LINEAR ALGEBRA: 50 Page 6 of 330 VECTOR SPACE AXIOMS: 50 SUBSPACE: 50 COMMON SPACES: 50 ROWSPACE OF A SPANNING SET IN RN 51 COLUMNSPACE OF A SPANNING SET IN RN 51 NULLSPACE: 51 NULLITY: 51 LINEAR DEPENDENCE: 51 BASIS: 51 STANDARD BASIS: 52 ORTHOGONAL COMPLEMENT: 52 ORTHONORMAL BASIS: 52 GRAM-SCHMIDT PROCESS: 52 COORDINATE VECTOR: 53 DIMENSION: 53 4.9 COMPLEX VECTOR SPACES: 53 FORM: 53 DOT PRODUCT: 53 INNER PRODUCT: 54 4.10 LINEAR TRANSITIONS & TRANSFORMATIONS: 54 TRANSITION MATRIX: 54 CHANGE OF BASIS TRANSITION MATRIX: 54 TRANSFORMATION MATRIX: 54 4.11 INNER PRODUCTS: 54 DEFINITION: 54 AXIOMS: 54 UNIT VECTOR: 55 CAVCHY-SCHUARZ INEQUALITY: 55 INNER PRODUCT SPACE: 55 ANGLE BETWEEN TWO VECTORS: 55 DISTANCE BETWEEN TWO VECTORS: 55 GENERALISED PYTHAGORAS FOR ORTHOGONAL VECTORS: 55 4.12 PRIME NUMBERS: 55 DETERMINATE: 55 LIST OF PRIME NUMBERS: 55 FUNDAMENTAL THEORY OF ARITHMETIC: 56 LAPRANGE’S THEOREM: 56 ADDITIVE PRIMES: 56 ANNIHILATING PRIMES: 56 BELL NUMBER PRIMES: 56 CAROL PRIMES: 56 CENTERED DECAGONAL PRIMES: 56 CENTERED HEPTAGONAL PRIMES: 56 CENTERED SQUARE PRIMES: 57 CENTERED TRIANGULAR PRIMES: 57 CHEN PRIMES: 57 CIRCULAR PRIMES: 57 COUSIN PRIMES: 57 CUBAN PRIMES: 57 CULLEN PRIMES: 57 DIHEDRAL PRIMES: 57 DOUBLE FACTORIAL PRIMES: 57 DOUBLE MERSENNE PRIMES: 58 EISENSTEIN PRIMES WITHOUT IMAGINARY PART: 58 EMIRPS: 58 EUCLID PRIMES: 58 Page 7 of 330 EVEN PRIME: 58 FACTORIAL PRIMES: 58 FERMAT PRIMES: 58 FIBONACCI PRIMES: 58 FORTUNATE PRIMES: 58 GAUSSIAN PRIMES: 58 GENERALIZED FERMAT PRIMES BASE 10: 58 GENOCCHI NUMBER PRIMES: 59 GILDA'S PRIMES: 59 GOOD PRIMES: 59 HAPPY PRIMES: 59 HARMONIC PRIMES: 59 HIGGS PRIMES FOR SQUARES: 59 HIGHLY COTOTIENT NUMBER PRIMES: 59 IRREGULAR PRIMES: 59 (P, P−5) IRREGULAR PRIMES: 59 (P, P−9) IRREGULAR PRIMES: 59 ISOLATED PRIMES: 59 KYNEA PRIMES: 59 LEFT-TRUNCATABLE PRIMES: 60 LEYLAND PRIMES: 60 LONG PRIMES: 60 LUCAS PRIMES: 60 LUCKY PRIMES: 60 MARKOV PRIMES: 60 MERSENNE PRIMES: 60 MERSENNE PRIME EXPONENTS: 60 MILLS PRIMES: 60 MINIMAL PRIMES: 60 MOTZKIN PRIMES: 60 NEWMAN–SHANKS–WILLIAMS PRIMES: 61 NON-GENEROUS PRIMES: 61 ODD PRIMES: 61 PADOVAN PRIMES: 61 PALINDROMIC PRIMES: 61 PALINDROMIC WING PRIMES: 61 PARTITION PRIMES: 61 PELL PRIMES: 61 PERMUTABLE PRIMES: 61 PERRIN PRIMES: 61 PIERPONT PRIMES: 61 PILLAI PRIMES: 62 PRIMEVAL PRIMES: 62 PRIMORIAL PRIMES: 62 PROTH PRIMES: 62 PYTHAGOREAN PRIMES: 62 PRIME QUADRUPLETS: 62 PRIMES OF BINARY QUADRATIC FORM: 62 QUARTAN PRIMES: 62 RAMANUJAN PRIMES: 62 REGULAR PRIMES: 62 REPUNIT PRIMES: 62 PRIMES IN RESIDUE CLASSES: 62 RIGHT-TRUNCATABLE PRIMES: 63 SAFE PRIMES: 63 SELF PRIMES IN BASE 10: 63 Page 10 of 330 5.12 DISCRETE RANDOM VARIABLES: 77 STANDARD DEVIATION: 77 EXPECTED VALUE: 77 VARIANCE: 78 PROBABILITY MASS FUNCTION: 78 CUMULATIVE DISTRIBUTION FUNCTION: 78 5.13 COMMON DRVS: 78 BERNOULLI TRIAL: 78 BINOMIAL TRIAL: 78 POISSON DISTRIBUTION: 78 GEOMETRIC BINOMIAL TRIAL: 79 NEGATIVE BINOMIAL TRIAL: 79 HYPERGEOMETRIC TRIAL: 79 5.14 CONTINUOUS RANDOM VARIABLES: 79 PROBABILITY DENSITY FUNCTION: 79 CUMULATIVE DISTRIBUTION FUNCTION: 79 INTERVAL PROBABILITY: 79 EXPECTED VALUE: 80 VARIANCE: 80 5.15 COMMON CRVS: 80 UNIFORM DISTRIBUTION: 80 EXPONENTIAL DISTRIBUTION: 80 NORMAL DISTRIBUTION: 81 5.16 BIVARIABLE DISCRETE: 81 PROBABILITY: 81 MARGINAL DISTRIBUTION: 82 EXPECTED VALUE: 82 INDEPENDENCE: 82 COVARIANCE: 82 5.17 BIVARIABLE CONTINUOUS: 82 CONDITIONS: 82 PROBABILITY: 82 MARGINAL DISTRIBUTION: 82 MEASURE: 83 EXPECTED VALUE: 83 INDEPENDENCE: 83 CONDITIONAL: 83 COVARIANCE: 83 CORRELATION COEFFICIENT: 83 BIVARIATE UNIFROM DISTRIBUTION: 83 MULTIVARIATE UNIFORM DISTRIBUTION: 83 BIVARIATE NORMAL DISTRIBUTION: 83 5.18 FUNCTIONS OF RANDOM VARIABLES: 84 SUMS (DISCRETE): 84 SUMS (CONTINUOUS): 84 QUOTIENTS (DISCRETE): 84 QUOTIENTS (CONTINUOUS): 84 MAXIMUM: 85 MINIMUM: 85 ORDER STATISTICS: 85 5.19 TRANSFORMATION OF THE JOINT DENSITY: 86 BIVARIATE FUNCTIONS: 86 MULTIVARIATE FUNCTIONS: 86 JACOBIAN: 86 JOINT DENSITY: 86 ABBREVIATIONS 86 Page 11 of 330 PART 6: STATISTICAL ANALYSIS 88 6.1 GENERAL PRINCIPLES: 88 MEAN SQUARE VALUE OF X: 88 F-STATISTIC OF X: 88 F-STATISTIC OF THE NULL HYPOTHESIS: 88 P-VALUE: 88 RELATIVE EFFICIENCY: 88 6.2 CONTINUOUS REPLICATE DESIGN (CRD): 88 TREATMENTS: 88 FACTORS: 88 REPLICATIONS PER TREATMENT: 88 TOTAL TREATMENTS: 88 MATHEMATICAL MODEL: 88 TEST FOR TREATMENT EFFECT: 89 ANOVA: 89 6.3 RANDOMISED BLOCK DESIGN (RBD): 89 TREATMENTS: 89 FACTORS: 89 REPLICATIONS PER TREATMENT: 89 TOTAL TREATMENTS: 89 MATHEMATICAL MODEL: 89 TEST FOR TREATMENT EFFECT: 90 TEST FOR BLOCK EFFECT: 90 RELATIVE EFFICIENCY: 90 ANOVA: 90 6.4 LATIN SQUARE DESIGN (LSD): 90 TREATMENTS: 90 FACTORS: 90 REPLICATIONS PER TREATMENT: 90 TOTAL TREATMENTS: 90 MATHEMATICAL MODEL: 90 TEST FOR TREATMENT EFFECT: 91 RELATIVE EFFICIENCY: 91 ANOVA: 91 6.5 ANALYSIS OF COVARIANCE: 91 MATHEMATICAL MODEL: 91 ASSUMPTIONS: 91 6.6 RESPONSE SURFACE METHODOLOGY: 92 DEFINITION: 92 1ST ORDER: 92 2ND ORDER: 92 COMMON DESIGNS 92 CRITERION FOR DETERMINING THE OPTIMATILITY OF A DESIGN: 92 6.7 FACTORIAL OF THE FORM 2N: 92 GENERAL DEFINITION: 92 CONTRASTS FOR A 22 DESIGN: 92 SUM OF SQUARES FOR A 22 DESIGN: 92 HYPOTHESIS FOR A CRD 22 DESIGN: 92 HYPOTHESIS FOR A RBD 22 DESIGN: 93 6.8 GENERAL FACTORIAL: 93 GENERAL DEFINITION: 93 ORDER: 93 DEGREES OF FREEDOM FOR MAIN EFFECTS: 93 DEGREES OF FREEDOM FOR HIGHER ORDER EFFECTS: 93 Page 12 of 330 6.9 ANOVA ASSUMPTIONS: 93 ASSUMPTIONS: 93 LEVENE’S TEST: 94 6.10 CONTRASTS: 94 LINEAR CONTRAST: 94 ESTIMATED MEAN OF CONTRAST: 94 ESTIMATED VARIANCE OF CONTRAST: 94 F OF CONTRAST: 94 ORTHOGONAL CONTRASTS: 94 6.11 POST ANOVA MULTIPLE COMPARISONS: 94 BONDERRONI METHOD: 94 FISHER’S LEAST SIGNIFICANT DIFFERENCE: 94 TUKEY’S W PROCEDURE: 94 SCHEFFE’S METHOD: 94 PART 7: PI 96 7.1 AREA: 96 CIRCLE: 96 CYCLIC QUADRILATERAL: 96 AREA OF A SECTOR (DEGREES) 96 AREA OF A SECTOR (RADIANS) 96 AREA OF A SEGMENT (DEGREES) 96 AREA OF AN ANNULUS: 96 ELLIPSE: 96 7.2 VOLUME: 96 SPHERE: 96 CAP OF A SPHERE: 96 CONE: 96 ICE-CREAM & CONE: 96 DOUGHNUT: 96 SAUSAGE: 96 ELLIPSOID: 96 7.3 SURFACE AREA: 96 SPHERE: 96 HEMISPHERE: 96 DOUGHNUT: 96 SAUSAGE: 96 CONE: 96 7.4 MISELANIOUS: 97 LENGTH OF ARC (DEGREES) 97 LENGTH OF CHORD (DEGREES) 97 PERIMETER OF AN ELLIPSE 97 7.6 PI: 97 ARCHIMEDES’ BOUNDS: 97 JOHN WALLIS: 97 ISAAC NEWTON: 97 JAMES GREGORY: 97 SCHULZ VON STRASSNITZKY: 97 JOHN MACHIN: 97 LEONARD EULER: 97 JOZEF HOENE-WRONSKI: 97 FRANCISCUS VIETA: 97 INTEGRALS: 98 INFINITE SERIES: 98 Page 15 of 330 10.2 EXPONENTIAL IDENTITIES: 121 10.3 LOG IDENTITIES: 121 10.4 LAWS FOR LOG TABLES: 122 10.5 COMPLEX NUMBERS: 122 10.6 LIMITS INVOLVING LOGARITHMIC TERMS 122 PART 11: COMPLEX NUMBERS 123 11.1 GENERAL: 123 FUNDAMENTAL: 123 STANDARD FORM: 123 POLAR FORM: 123 ARGUMENT: 123 MODULUS: 123 CONJUGATE: 123 EXPONENTIAL: 123 DE MOIVRE’S FORMULA: 123 EULER’S IDENTITY: 123 11.2 OPERATIONS: 123 ADDITION: 123 SUBTRACTION: 123 MULTIPLICATION: 123 DIVISION: 123 SUM OF SQUARES: 123 11.3 IDENTITIES: 123 EXPONENTIAL: 123 LOGARITHMIC: 123 TRIGONOMETRIC: 123 HYPERBOLIC: 124 PART 12: DIFFERENTIATION 125 12.1 GENERAL RULES: 125 PLUS OR MINUS: 125 PRODUCT RULE: 125 QUOTIENT RULE: 125 POWER RULE: 125 CHAIN RULE: 125 BLOB RULE: 125 BASE A LOG: 125 NATURAL LOG: 125 EXPONENTIAL (X): 125 FIRST PRINCIPLES: 125 ANGLE OF INTERSECTION BETWEEN TWO CURVES: 126 12.2 EXPONETIAL FUNCTIONS: 126 12.3 LOGARITHMIC FUNCTIONS: 126 12.4 TRIGONOMETRIC FUNCTIONS: 126 12.5 HYPERBOLIC FUNCTIONS: 127 12.5 PARTIAL DIFFERENTIATION: 127 FIRST PRINCIPLES: 127 GRADIENT: 128 TOTAL DIFFERENTIAL: 128 CHAIN RULE: 128 IMPLICIT DIFFERENTIATION: 129 Page 16 of 330 HIGHER ORDER DERIVATIVES: 129 PART 13: INTEGRATION 130 13.1 GENERAL RULES: 130 POWER RULE: 130 BY PARTS: 130 CONSTANTS: 130 13.2 RATIONAL FUNCTIONS: 130 13.3 TRIGONOMETRIC FUNCTIONS (SINE): 131 13.4 TRIGONOMETRIC FUNCTIONS (COSINE): 132 13.5 TRIGONOMETRIC FUNCTIONS (TANGENT): 133 13.6 TRIGONOMETRIC FUNCTIONS (SECANT): 133 13.7 TRIGONOMETRIC FUNCTIONS (COTANGENT): 134 13.8 TRIGONOMETRIC FUNCTIONS (SINE & COSINE): 134 13.9 TRIGONOMETRIC FUNCTIONS (SINE & TANGENT): 136 13.10 TRIGONOMETRIC FUNCTIONS (COSINE & TANGENT): 136 13.11 TRIGONOMETRIC FUNCTIONS (SINE & COTANGENT): 136 13.12 TRIGONOMETRIC FUNCTIONS (COSINE & COTANGENT): 136 13.13 TRIGONOMETRIC FUNCTIONS (ARCSINE): 136 13.14 TRIGONOMETRIC FUNCTIONS (ARCCOSINE): 137 13.15 TRIGONOMETRIC FUNCTIONS (ARCTANGENT): 137 13.16 TRIGONOMETRIC FUNCTIONS (ARCCOSECANT): 137 13.17 TRIGONOMETRIC FUNCTIONS (ARCSECANT): 138 13.18 TRIGONOMETRIC FUNCTIONS (ARCCOTANGENT): 138 13.19 EXPONETIAL FUNCTIONS 138 13.20 LOGARITHMIC FUNCTIONS 140 13.21 HYPERBOLIC FUNCTIONS 141 13.22 INVERSE HYPERBOLIC FUNCTIONS 143 13.23 ABSOLUTE VALUE FUNCTIONS 144 13.24 SUMMARY TABLE 144 13.25 SQUARE ROOT PROOFS 145 13.26 CARTESIAN APPLICATIONS 148 AREA UNDER THE CURVE: 148 VOLUME: 148 VOLUME ABOUT X AXIS: 148 VOLUME ABOUT Y AXIS: 149 SURFACE AREA ABOUT X AXIS: 149 LENGTH WRT X-ORDINATES: 149 LENGTH WRT Y-ORDINATES: 149 LENGTH PARAMETRICALLY: 149 LINE INTEGRAL OF A SCALAR FIELD: 149 LINE INTEGRAL OF A VECTOR FIELD: 149 AREA OF A SURFACE: 149 13.27 HIGHER ORDER INTEGRATION 149 PROPERTIES OF DOUBLE INTEGRALS: 150 VOLUME USING DOUBLE INTEGRALS: 150 VOLUME USING TRIPLE INTEGRALS: 151 CENTRE OF MASS: 153 13.28 WORKING IN DIFFERENT COORDINATE SYSTEMS: 153 CARTESIAN: 153 POLAR: 153 CYLINDRICAL: 153 SPHERICAL: 154 CARTESIAN TO POLAR: 154 Page 17 of 330 POLAR TO CARTESIAN: 154 CARTESIAN TO CYLINDRICAL: 154 CYLINDRICAL TO CARTESIAN: 154 SPHERICAL TO CARTESIAN: 154 PART 14: FUNCTIONS 155 14.1 ODD & EVEN FUNCTIONS: 155 DEFINITIONS: 155 COMPOSITE FUNCTIONS: 155 BASIC INTEGRATION: 155 14.2 MULTIVARIABLE FUNCTIONS: 155 LIMIT: 155 DISCRIMINANT: 155 CRITICAL POINTS: 155 14.3 FIRST ORDER, FIRST DEGREE, DIFFERENTIAL EQUATIONS: 156 SEPARABLE 156 LINEAR 156 HOMOGENEOUS 156 EXACT 157 BERNOULLI FORM: 157 14.4 SECOND ORDE, FIRST DEGREE, DIFFERENTIAL EQUATIONS: 158 GREG’S LEMMA: 158 HOMOGENEOUS 158 UNDETERMINED COEFFICIENTS 158 VARIATION OF PARAMETERS 159 EULER TYPE 160 REDUCTION OF ORDER 160 POWER SERIES SOLUTIONS: 161 14.5 ORDINARY DIFFERENTIAL EQUATIONS USING MATRICES: 163 DERIVATION OF METHODS: 163 FUNDAMENTAL MATRIX: 163 HOMOGENEOUS SOLUTION: 163 INHOMOGENEOUS SOLUTION: 164 N TH ORDER LINEAR, CONSTANT COEFFICIENT ODE: 164 14.6 APPLICATIONS OF FUNCTIONS 166 TERMINOLOGY: 166 GRADIENT VECTOR OF A SCALAR FIELD: 166 DIRECTIONAL DERIVATIVES: 166 OPTIMISING THE DIRECTIONAL DERIVATIVE: 166 14.7 ANALYTIC FUNCTIONS 166 PART 15: MATRICIES 167 15.1 BASIC PRINICPLES: 167 SIZE 167 15.2 BASIC OPERTAIONS: 167 ADDITION: 167 SUBTRACTION: 167 SCALAR MULTIPLE: 167 TRANSPOSE: 167 SCALAR PRODUCT: 167 SYMMETRY: 167 CRAMER’S RULE: 167 Page 20 of 330 TORTION: 184 ABBREVIATIONS 184 PART 17: SERIES 185 17.1 MISCELLANEOUS 185 GENERAL FORM: 185 INFINITE FORM: 185 PARTIAL SUM OF A SERIES: 185 0.99…=1: 185 17.2 TEST FOR CONVERGENCE AND DIVERGENCE 185 TEST FOR CONVERGENCE: 185 TEST FOR DIVERGENCE: 185 GEOMETRIC SERIES 185 P SERIES 185 THE SANDWICH THEOREM 185 THE INTEGRAL TEST 185 THE DIRECT COMPARISON TEST 186 THE LIMIT COMPARISON TEST 186 D’ALMBERT’S RATIO COMPARISON TEST 186 THE NTH ROOT TEST 186 ABEL’S TEST: 186 NEGATIVE TERMS 186 ALTERNATING SERIES TEST 186 ALTERNATING SERIES ERROR 187 17.3 ARITHMETIC PROGRESSION: 187 DEFINITION: 187 NTH TERM: 187 SUM OF THE FIRST N TERMS: 187 17.4 GEOMETRIC PROGRESSION: 187 DEFINITION: 187 NTH TERM: 187 SUM OF THE FIRST N TERMS: 187 SUM TO INFINITY: 187 GEOMETRIC MEAN: 187 17.5 SUMMATION SERIES 187 LINEAR: 187 QUADRATIC: 187 CUBIC: 187 17.6 APPROXIMATION SERIES 187 TAYLOR SERIES 187 MACLAURUN SERIES 188 LINEAR APPROXIMATION: 188 QUADRATIC APPROXIMATION: 188 CUBIC APPROXIMATION: 188 17.7 MONOTONE SERIES 188 STRICTLY INCREASING: 188 NON-DECREASING: 188 STRICTLY DECREASING: 188 NON-INCREASING: 188 CONVERGENCE: 188 17.8 RIEMANN ZETA FUNCTION 188 FORM: 188 EULER’S TABLE: 188 ALTERNATING SERIES: 189 Page 21 of 330 PROOF FOR N=2: 189 17.9 SUMMATIONS OF POLYNOMIAL EXPRESSIONS 190 17.10 SUMMATIONS INVOLVING EXPONENTIAL TERMS 190 17.11 SUMMATIONS INVOLVING TRIGONOMETRIC TERMS 191 17.12 INFINITE SUMMATIONS TO PI 193 17.13 LIMITS INVOLVING TRIGONOMETRIC TERMS 193 ABBREVIATIONS 193 17.14 POWER SERIES EXPANSION 193 EXPONENTIAL: 193 TRIGONOMETRIC: 194 EXPONENTIAL AND LOGARITHM SERIES: 196 FOURIER SERIES: 197 17.15 BERNOULLI EXPANSION: 197 FUNDAMENTALLY: 197 EXPANSIONS: 198 LIST OF BERNOULLI NUMBERS: 198 PART 18: ELECTRICAL 200 18.1 FUNDAMENTAL THEORY 200 CHARGE: 200 CURRENT: 200 RESISTANCE: 200 OHM’S LAW: 200 POWER: 200 CONSERVATION OF POWER: 200 ELECTRICAL ENERGY: 200 KIRCHOFF’S VOLTAGE LAW: 200 KIRCHOFF’S CURRENT LAW: 200 AVERAGE CURRENT: 200 RMS CURRENT: 200 ∆ TO Y CONVERSION: 200 18.2 COMPONENTS 201 RESISTANCE IN SERIES: 201 RESISTANCE IN PARALLEL: 201 INDUCTIVE IMPEDANCE: 201 CAPACITOR IMPEDANCE: 201 CAPACITANCE IN SERIES: 201 CAPACITANCE IN PARALLEL: 201 VOLTAGE, CURRENT & POWER SUMMARY: 201 18.3 THEVENIN’S THEOREM 201 THEVENIN’S THEOREM: 201 MAXIMUM POWER TRANSFER THEOREM: 202 18.4 FIRST ORDER RC CIRCUIT 202 18.5 FIRST ORDER RL CIRCUIT 202 18.6 SECOND ORDER RLC SERIES CIRCUIT 202 CALCULATION USING KVL: 202 IMPORTANT VARIABLES 202 SOLVING: 203 MODE 1: 203 MODE 2: 203 MODE 3: 204 MODE 4: 204 CURRENT THROUGH INDUCTOR: 205 PLOTTING MODES: 205 Page 22 of 330 18.7 SECOND ORDER RLC PARALLEL CIRCUIT 206 CALCULATION USING KCL: 206 IMPORTANT VARIABLES 206 SOLVING: 207 18.8 LAPLANCE TRANSFORMATIONS 207 IDENTITIES: 207 PROPERTIES: 208 18.9 THREE PHASE – Y 209 LINE VOLTAGE: 209 PHASE VOLTAGE: 209 LINE CURRENT: 209 PHASE CURRENT: 209 POWER: 209 18.10 THREE PHASE – DELTA 209 LINE VOLTAGE: 209 PHASE VOLTAGE: 209 LINE CURRENT: 209 PHASE CURRENT: 209 POWER: 209 18.11 POWER 209 INSTANTANEOUS: 209 AVERAGE: 210 MAXIMUM POWER: 210 TOTAL POWER: 210 COMPLEX POWER: 210 18.12 ELECTROMAGNETICS 210 DEFINITIONS: 210 PERMEABILITY OF FREE SPACE: 210 MAGNETIC FIELD INTENSITY: 210 RELUCTANCE: 210 OHM’S LAW: 210 MAGNETIC FORCE ON A CONDUCTOR: 210 ELECTROMAGNETIC INDUCTION: 210 MAGNETIC FLUX: 210 ELECTRIC FIELD: 210 MAGNETIC FORCE ON A PARTICLE: 210 PART 19: GRAPH THEORY 211 19.1 FUNDAMENTAL EXPLANATIONS: 211 LIST OF VERTICES: 211 LIST OF EDGES: 211 SUBGAPHS: 211 TREE: 211 DEGREE OF VERTEX: 211 DISTANCE: 211 DIAMETER: 211 TOTAL EDGES IN A SIMPLE BIPARTITE GRAPH: 211 TOTAL EDGES IN K-REGULAR GRAPH: 211 19.2 FACTORISATION: 211 1 FACTORISATION: 211 1 FACTORS OF A nnK , BIPARTITE GRAPH: 211 1 FACTORS OF A nK2 GRAPH: 211 19.3 VERTEX COLOURING: 211 Page 25 of 330 PHASE-TYPE DISTRIBUTION 306 TRUNCATED DISTRIBUTION 306 PART 99: CONVERSIONS 308 99.1 LENGTH: 308 99.2 AREA: 310 99.3 VOLUME: 311 99.4 PLANE ANGLE: 314 99.5 SOLID ANGLE: 315 99.6 MASS: 315 99.7 DENSITY: 317 99.8 TIME: 317 99.9 FREQUENCY: 319 99.10 SPEED OR VELOCITY: 319 99.11 FLOW (VOLUME): 320 99.12 ACCELERATION: 320 99.13 FORCE: 321 99.14 PRESSURE OR MECHANICAL STRESS: 321 99.15 TORQUE OR MOMENT OF FORCE: 322 99.16 ENERGY, WORK, OR AMOUNT OF HEAT: 322 99.17 POWER OR HEAT FLOW RATE: 324 99.18 ACTION: 325 99.19 DYNAMIC VISCOSITY: 325 99.20 KINEMATIC VISCOSITY: 326 99.21 ELECTRIC CURRENT: 326 99.22 ELECTRIC CHARGE: 326 99.23 ELECTRIC DIPOLE: 327 99.24 ELECTROMOTIVE FORCE, ELECTRIC POTENTIAL DIFFERENCE: 327 99.25 ELECTRICAL RESISTANCE: 327 99.26 CAPACITANCE: 327 99.27 MAGNETIC FLUX: 327 99.28 MAGNETIC FLUX DENSITY: 328 99.29 INDUCTANCE: 328 99.30 TEMPERATURE: 328 99.31 INFORMATION ENTROPY: 328 99.32 LUMINOUS INTENSITY: 329 99.33 LUMINANCE: 329 99.34 LUMINOUS FLUX: 329 99.35 ILLUMINANCE: 329 99.36 RADIATION - SOURCE ACTIVITY: 329 99.37 RADIATION – EXPOSURE: 330 99.38 RADIATION - ABSORBED DOSE: 330 99.39 RADIATION - EQUIVALENT DOSE: 330 Page 26 of 330 PART 1: PHYSICAL CONSTANTS 1.1 SI PREFIXES: Prefix Symbol 1000m 10n Decimal Scale yotta Y 1000 8 1024 1000000000000000000000000 Septillion zetta Z 1000 7 1021 1000000000000000000000 Sextillion exa E 1000 6 1018 1000000000000000000 Quintillion peta P 1000 5 1015 1000000000000000 Quadrillion tera T 1000 4 1012 1000000000000 Trillion giga G 1000 3 109 1000000000 Billion mega M 1000 2 106 1000000 Million kilo k 1000 1 103 1000 Thousand hecto h 1000 2⁄ 3 10 2 100 Hundred deca da 1000 1⁄ 3 10 1 10 Ten 10000 100 1 One deci d 1000 −1⁄ 3 10 −1 0.1 Tenth centi c 1000 −2⁄ 3 10 −2 0.01 Hundredth milli m 1000 −1 10−3 0.001 Thousandth micro µ 1000 −2 10−6 0.000001 Millionth nano n 1000 −3 10−9 0.000000001 Billionth pico p 1000 −4 10−12 0.000000000001 Trillionth femto f 1000 −5 10−15 0.000000000000001 Quadrillionth atto a 1000 −6 10−18 0.000000000000000001 Quintillionth zepto z 1000 −7 10−21 0.000000000000000000001 Sextillionth yocto y 1000 −8 10−24 0.000000000000000000000001 Septillionth 1.2 SI BASE UNITS: Quantity Unit Symbol length meter m mass kilogram kg time second s electric current ampere A thermodynamic temperature kelvin K amount of substance mole mol luminous intensity candela cd Page 27 of 330 1.3 SI DERIVED UNITS: Quantity Unit Symbol Expression in terms of other SI units angle, plane radian* rad m/m = 1 angle, solid steradian* sr m2/m2 = 1 Celsius temperature degree Celsius °C K electric capacitance farad F C/V electric charge, quantity of electricity coulomb C A·s electric conductance siemens S A/V electric inductance henry H Wb/A electric potential difference, electromotive force volt V W/A electric resistance ohm Ω V/A energy, work, quantity of heat joule J N·m force newton N kg·m/s2 frequency (of a periodic phenomenon) hertz Hz 1/s illuminance lux lx lm/m2 luminous flux lumen lm cd·sr magnetic flux weber Wb V·s magnetic flux density tesla T Wb/m2 power, radiant flux watt W J/s pressure, stress pascal Pa N/m2 activity (referred to a radionuclide) becquerel Bq 1/s absorbed dose, specific energy imparted, kerma gray Gy J/kg dose equivalent, ambient dose equivalent, directional dose equivalent, personal dose equivalent, organ dose equivalent sievert Sv J/kg catalytic activity katal kat mol/s Page 30 of 330 constant for spectral radiance 1.191 042 82(20) × 10−16 W·m² sr−1 1.7 × 10−7 Loschmidt constant at T=273.15 K and p=101.325 kPa 2.686 777 3(47) × 1025 m−3 1.8 × 10−6 gas constant 8.314 472(15) J·K−1·mol−1 1.7 × 10−6 molar Planck constant 3.990 312 716(27) × 10−10 J·s·mol−1 6.7 × 10−9 at T=273.15 K and p=100 kPa 2.2710 981(40) × 10−2 m³·mol−1 1.7 × 10−6 molar volume of an ideal gas at T=273.15 K and p=101.325 kPa 2.2413 996(39) × 10−2 m³·mol−1 1.7 × 10−6 at T=1 K and p=100 kPa −1.151 704 7(44) 3.8 × 10−6 Sackur- Tetrode constant at T=1 K and p=101.325 kPa −1.164 867 7(44) 3.8 × 10−6 second radiation constant 1.438 775 2(25) × 10−2 m·K 1.7 × 10−6 Stefan–Boltzmann constant 5.670 400(40) × 10−8 W·m−2·K−4 7.0 × 10−6 Wien displacement law constant 4.965 114 231... 2.897 768 5(51) × 10−3 m·K 1.7 × 10−6 1.8 ADOPTED VALUES: Quantity Symbol Value (SI units) Relative Standard Uncertainty conventional value of Josephson constant 4.835 979 × 1014 Hz·V−1 defined conventional value of von Klitzing constant 25 812.807 Ω defined constant 1 × 10−3 kg·mol−1 defined molar mass of carbon-12 1.2 × 10 −2 defined Page 31 of 330 kg·mol−1 standard acceleration of gravity (gee, free-fall on Earth) 9.806 65 m·s−2 defined standard atmosphere 101 325 Pa defined 1.9 NATURAL UNITS: Name Dimension Expression Value (SI units) Planck length Length (L) 1.616 252(81) × 10−35 m Planck mass Mass (M) 2.176 44(11) × 10−8 kg Planck time Time (T) 5.391 24(27) × 10−44 s Planck charge Electric charge (Q) 1.875 545 870(47) × 10 −18 C Planck temperature Temperature (Θ) 1.416 785(71) × 1032 K 1.10 MATHEMATICAL CONSTANTS: (each to 1000 decimal places) π ≈ 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482 5342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559 6446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104 5432664821339360726024914127372458700660631558817488152092096282925409171536436789259036001 1330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074 4623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907 0217986094370277053921717629317675238467481846766940513200056812714526356082778577134275778 9609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034 4181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083 0264252230825334468503526193118817101000313783875288658753320838142061717766914730359825349 0428755468731159562863882353787593751957781857780532171226806613001927876611195909216420199 e ≈ 2.71828182845904523536028747135266249775724709369995957496696762772407663035354759457138217 8525166427427466391932003059921817413596629043572900334295260595630738132328627943490763233 8298807531952510190115738341879307021540891499348841675092447614606680822648001684774118537 4234544243710753907774499206955170276183860626133138458300075204493382656029760673711320070 9328709127443747047230696977209310141692836819025515108657463772111252389784425056953696770 7854499699679468644549059879316368892300987931277361782154249992295763514822082698951936680 3318252886939849646510582093923982948879332036250944311730123819706841614039701983767932068 3282376464804295311802328782509819455815301756717361332069811250996181881593041690351598888 5193458072738667385894228792284998920868058257492796104841984443634632449684875602336248270 4197862320900216099023530436994184914631409343173814364054625315209618369088870701676839642 4378140592714563549061303107208510383750510115747704171898610687396965521267154688957035035 φ ≈ Page 32 of 330 1.61803398874989484820458683436563811772030917980576286213544862270526046281890244970720720 4189391137484754088075386891752126633862223536931793180060766726354433389086595939582905638 3226613199282902678806752087668925017116962070322210432162695486262963136144381497587012203 4080588795445474924618569536486444924104432077134494704956584678850987433944221254487706647 8091588460749988712400765217057517978834166256249407589069704000281210427621771117778053153 1714101170466659914669798731761356006708748071013179523689427521948435305678300228785699782 9778347845878228911097625003026961561700250464338243776486102838312683303724292675263116533 9247316711121158818638513316203840052221657912866752946549068113171599343235973494985090409 4762132229810172610705961164562990981629055520852479035240602017279974717534277759277862561 9432082750513121815628551222480939471234145170223735805772786160086883829523045926478780178 8992199027077690389532196819861514378031499741106926088674296226757560523172777520353613936 Page 35 of 330 ∑∑ sigma double summation ∏ capital pi product - product of all values in range of series ∏ xi=x1·x2·...·xn e e constant / Euler's number e = 2.718281828... e = lim (1+1/x) x , x→∞ γ Euler-Mascheroni constant γ = 0.527721566... φ golden ratio golden ratio constant 2.4 LINEAR ALGEBRA SYMBOLS Symbol Symbol Name Meaning / definition Example · dot scalar product a · b × cross vector product a × b A⊗B tensor product tensor product of A and B A ⊗ B inner product [ ] brackets matrix of numbers ( ) parentheses matrix of numbers | A | determinant determinant of matrix A det(A) determinant determinant of matrix A || x || double vertical bars norm A T transpose matrix transpose (AT)ij = (A)ji A † Hermitian matrix matrix conjugate transpose (A†)ij = (A)ji A * Hermitian matrix matrix conjugate transpose (A*)ij = (A)ji A -1 inverse matrix A A-1 = I rank(A) matrix rank rank of matrix A rank(A) = 3 dim(U) dimension dimension of matrix A rank(U) = 3 2.5 PROBABILITY AND STATISTICS SYMBOLS Symbol Symbol Name Meaning / definition Example P(A) probability function probability of event A P(A) = 0.5 P(A ∩ B) probability of events intersection probability that of events A and B P(A∩B) = 0.5 P(A ∪ B) probability of events union probability that of events A or B P(A∪B) = 0.5 P(A | B) conditional probability function probability of event A given event B occured P(A | B) = 0.3 f (x) probability density function (pdf) P(a ≤ x ≤ b) = ∫ f (x) dx F(x) cumulative distribution function (cdf) F(x) = P(X ≤ x) µ population mean mean of population values µ = 10 E(X) expectation value expected value of random variable X E(X) = 10 E(X | Y) conditional expectation expected value of random variable X given Y E(X | Y=2) = 5 Page 36 of 330 var(X) variance variance of random variable X var(X) = 4 σ2 variance variance of population values σ2 = 4 std(X) standard deviation standard deviation of random variable X std(X) = 2 σX standard deviation standard deviation value of random variable X σX = 2 median middle value of random variable x cov(X,Y) covariance covariance of random variables X and Y cov(X,Y) = 4 corr(X,Y) correlation correlation of random variables X and Y corr(X,Y) = 3 ρX,Y correlation correlation of random variables X and Y ρX,Y = 3 ∑ summation summation - sum of all values in range of series ∑∑ double summation double summation Mo mode value that occurs most frequently in population MR mid-range MR = (xmax+xmin)/2 Md sample median half the population is below this value Q1 lower / first quartile 25% of population are below this value Q2 median / second quartile 50% of population are below this value = median of samples Q3 upper / third quartile 75% of population are below this value x sample mean average / arithmetic mean x = (2+5+9) / 3 = 5.333 s 2 sample variance population samples variance estimator s 2 = 4 s sample standard deviation population samples standard deviation estimator s = 2 zx standard score zx = (x-x) / sx X ~ distribution of X distribution of random variable X X ~ N(0,3) N(µ,σ2) normal distribution gaussian distribution X ~ N(0,3) U(a,b) uniform distribution equal probability in range a,b X ~ U(0,3) exp(λ) exponential distribution f (x) = λe-λx , x≥0 gamma(c, λ) gamma distribution f (x) = λ c xc-1e-λx / Γ(c), x≥0 χ 2(k) chi-square distribution f (x) = xk/2-1e-x/2 / ( 2k/2 Γ(k/2) ) F (k1, k2) F distribution Bin(n,p) binomial distribution f (k) = nCk p k(1-p)n-k Poisson(λ) Poisson distribution f (k) = λke-λ / k! Geom(p) geometric distribution f (k) = p (1-p) k HG(N,K,n) hyper-geometric distribution Bern(p) Bernoulli distribution Page 37 of 330 2.6 COMBINATORICS SYMBOLS Symbol Symbol Name Meaning / definition Example n! factorial n! = 1·2·3·...·n 5! = 1·2·3·4·5 = 120 nPk permutation 5P3 = 5! / (5-3)! = 60 nCk combination 5C3 = 5!/[3!(5-3)!]=10 2.7 SET THEORY SYMBOLS Symbol Symbol Name Meaning / definition Example { } set a collection of elements A={3,7,9,14}, B={9,14,28} A ∩ B intersection objects that belong to set A and set B A ∩ B = {9,14} A ∪ B union objects that belong to set A or set B A ∪ B = {3,7,9,14,28} A ⊆ B subset subset has less elements or equal to the set {9,14,28} ⊆ {9,14,28} A ⊂ B proper subset / strict subset subset has less elements than the set {9,14} ⊂ {9,14,28} A ⊄ B not subset left set not a subset of right set {9,66} ⊄ {9,14,28} A ⊇ B superset set A has more elements or equal to the set B {9,14,28} ⊇ {9,14,28} A ⊃ B proper superset / strict superset set A has more elements than set B {9,14,28} ⊃ {9,14} A ⊅ B not superset set A is not a superset of set B {9,14,28} ⊅ {9,66} 2A power set all subsets of A Ƅ (A) power set all subsets of A A = B equality both sets have the same members A={3,9,14}, B={3,9,14}, A=B Ac complement all the objects that do not belong to set A A \ B relative complement objects that belong to A and not to B A={3,9,14}, B={1,2,3}, A-B={9,14} A - B relative complement objects that belong to A and not to B A={3,9,14}, B={1,2,3}, A-B={9,14} A ∆ B symmetric difference objects that belong to A or B but not to their intersection A={3,9,14}, B={1,2,3}, A ∆ B={1,2,9,14} A ⊖ B symmetric difference objects that belong to A or B but not to their intersection A={3,9,14}, B={1,2,3}, A ⊖ B={1,2,9,14} a∈A element of set membership A={3,9,14}, 3 ∈ A x∉A not element of no set membership A={3,9,14}, 1 ∉ A (a,b) ordered pair collection of 2 elements A×B cartesian product set of all ordered pairs from A and B |A| cardinality the number of elements of set A A={3,9,14}, |A|=3 #A cardinality the number of elements of set A A={3,9,14}, #A=3 Page 40 of 330 PART 3: AREA, VOLUME AND SURFACE AREA 3.1 AREA: Triangle: ( )( )( )csbsass A CBa CabbhA −−−==== sin2 sinsin sin 2 1 2 1 2 Rectangle: lwA = Square: 2aA = Parallelogram: AabbhA sin== Rhombus: AaA sin2= Trapezium:       += s ba hA Quadrilateral: ( )( )( )( )       ∠+∠×−−−−−= 2 cos2 CDAB abcddscsbsasA 2 sin21 IddA = Rectangle with rounded corners: ( )π−−= 42rlwA Regular Hexagon: 2 33 2a A ×= Regular Octagon: ( ) 2212 aA ×+= Regular Polygon:       = n na A 180 tan4 2 3.2 VOLUME: Cube: 3aV = Cuboid: abcV = Prisim: ( ) hbAV ×= Pyramid: ( ) hbAV ××= 3 1 Tetrahedron: 3 12 2 aV ×= Octahedron: 3 3 2 aV ×= Dodecahedron: 3 4 5715 aV ×+= Icosahedron: ( ) 3 12 535 aV ×+= 3.3 SURFACE AREA: Cube: 26aSA = Cuboids: ( )cabcabSA ++= 2 Page 41 of 330 Tetrahedron: 23 aSA ×= Octahedron: 232 aSA ××= Dodecahedron: 2510253 aSA ×+×= Icosahedron: 235 aSA ××= Cylinder: ( )rhrSA += π2 3.4 MISCELLANEOUS: Diagonal of a Rectangle 22 wld += Diagonal of a Cuboid 222 cbad ++= Longest Diagonal (Even Sides)       = n a 180 sin Longest Diagonal (Odd Sides)       = n a 90 sin2 Total Length of Edges (Cube): a12= Total Length of Edges (Cuboid): ( )cba ++= 4 Circumference drC ππ == 2 Perimeter of rectangle ( )baP += 2 Semi perimeter 2 P s = Euler’s Formula 2+=+ EdgesVerticiesFaces 3.5 ABBREVIATIONS (3.1, 3.2, 3.3, 3.4) A=area a=side ‘a’ b=base b=side ‘b’ C=circumference C=central angle c=side ‘c’ d=diameter d=diagonal d1=diagonal 1 d2=diagonal 2 E=external angle h=height I=internal angle l=length n=number of sides P=perimeter r=radius Page 42 of 330 r1=radius 1 s=semi-perimeter SA=Surface Area V=Volume w=width Page 45 of 330 4.2 FUNDAMENTALS OF ARITHMETIC: Rational Numbers: Every rational number can be written as ( ) ( ) 2 11 ++−= rrr Irrational Numbers: ( )( )    =×× ∞→∞→ irrational isx 0 rational isx 1 !coslimlim 2 xmn nm π 4.3 ALGEBRAIC EXPANSION: Babylonian Identity: (c1800BC) Common Products And Factors: Binomial Theorem: For any value of n, whether positive, negative, integer or non-integer, the value of the nth power of a binomial is given by: Binomial Expansion: For any power of n, the binomial (a + x) can be expanded Page 46 of 330 This is particularly useful when x is very much less than a so that the first few terms provide a good approximation of the value of the expression. There will always be n+1 terms and the general form is: Difference of two squares: Brahmagupta–Fibonacci Identity: Also, Degen's eight-square identity: Note that: Page 47 of 330 and, 4.4 ROOT EXPANSIONS: ( ) ( ) ( ) ( ) 2 2 2 2 22 1 1 1 1         ±=± ±=±         ±=± ±=± ±=± k y k x kyx xyx x yx y x yyx kykx k yx yxkykx k 4.5 LIMIT MANIPULATIONS: ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( )n n n n n n n n nn n n n n n n n n n nn n afaf baba akka baba →∞→∞ →∞→∞→∞ →∞→∞ →∞→∞→∞ = = = ±=± limlim limlimlim limlim limlimlim L’Hopital’s Rule: If ( ) ( ) ,or 0)(lim)(lim ∞±== →→ xgxf axax and       → )(' )(' lim xg xf ax exists (ie axxg =≠ ,0)(' ), then it follows that       =      →→ )(' )(' lim )( )( lim xg xf xg xf axax Proof: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) )(' )(' )(' )(' lim )()( lim )()( lim )()( )()( lim )( )( lim )( )( lim ag af xg xf ax agxg ax afxf axagxg axafxf axxg axxf xg xf ax ax ax axaxax =      =       − −       − − =       −÷− −÷−=      −÷ −÷=      → → → →→→ Page 50 of 330 Graph is a hyperbola that opens up and down, has a center at (h,k) , vertices b units up/down from the center and asymptotes that pass through center with slope a b± . 4.8 LINEAR ALGEBRA: Vector Space Axioms: A real vector space is a set X with a special element 0, and three operations: • Addition: Given two elements x, y in X, one can form the sum x+y, which is also an element of X. • Inverse: Given an element x in X, one can form the inverse -x, which is also an element of X. • Scalar multiplication: Given an element x in X and a real number c, one can form the product cx, which is also an element of X. These operations must satisfy the following axioms: Additive axioms. For every x,y,z in X, we have 1. x+y = y+x. 2. (x+y)+z = x+(y+z). 3. 0+x = x+0 = x. 4. (-x) + x = x + (-x) = 0. Multiplicative axioms. For every x in X and real numbers c,d, we have 5. 0x = 0 6. 1x = x 7. (cd)x = c(dx) Distributive axioms. For every x,y in X and real numbers c,d, we have 8. c(x+y) = cx + cy. 9. (c+d)x = cx +dx. A normed real vector space is a real vector space X with an additional operation: • Norm: Given an element x in X, one can form the norm ||x||, which is a non-negative number. This norm must satisfy the following axioms, for any x,y in X and any real number c: 10. ||x|| = 0 if and only if x = 0. 11. || cx || = |c| ||x||. 12. || x+y || <= ||x|| + ||y|| A complex vector space consists of the same set of axioms as the real case, but elements within the vector space are complex. The axioms are adjusted to suit. Subspace: When the subspace is a subset of another vector space, only axioms (a) and (b) need to be proved to show that the subspace is also a vector space. Common Spaces: Page 51 of 330 Real Numbers nℜℜℜℜ ,...,,, 32 (n denotes dimension) Complex Numbers: nCC ,...,,CC, 32 (n denotes dimension) Polynomials nPPPP ,...,,, 321 (n denotes the highest order of x) All continuous functions [ ]baC , (a & b denote the interval) (This is never a vector space as it has infinite dimensions) Rowspace of a spanning set in Rn Stack vectors in a matrix in rows Use elementary row operations to put matrix into row echelon form The non zero rows form a basis of the vector space Columnspace of a spanning set in Rn Stack vectors in a matrix in columns Use elementary row operations to put matrix into row echelon form Columns with leading entries correspond to the subset of vectors in the set that form a basis Nullspace: Solutions to 0=xA A Using elementary row operations to put matrix into row echelon form, columns with no leading entries are assigned a constant and the remaining variables are solved with respect to these constants. Nullity: The dimension of the nullspace )()()( ARankANullityAColumns += Linear Dependence: 0...2211 =+++ nnrcrcrc Then, 021 === nccc If the trivial solution is the only solution, nrrr ,..., 21 are independent. )|()( bArAr ≠ : No Solution nbArAr == )|()( : Unique Solution nbArAr <= )|()( : Infinite Solutions Basis: S is a basis of V if: S spans V S is linearly dependant { }nuuuuS ,...,,, 321= Page 52 of 330 The general vector within the vector space is:             = ... z y x w Therefore, [ ]                                 = ++++= nnnnnn n n n nn c c c c uuuu uuuu uuuu uuuu w ucucucucw ... ... ............... ... ... ... ... 3 2 1 321 3332313 2322212 1312111 332211 If the determinant of the square matrix is not zero, the matrix is invertible. Therefore, the solution is unique. Hence, all vectors in w are linear combinations of S. Because of this, S spans w. Standard Basis: Real Numbers                                                                                 =ℜ 1 ... 0 0 0 ,..., 0 ... 1 0 0 , 0 ... 0 1 0 , 0 ... 0 0 1 )( nS Polynomials { }nn xxxxPS ,...,,,,1)( 32= Any set the forms the basis of a vector space must contain the same number of linearly independent vectors as the standard basis. Orthogonal Complement: ⊥W is the nullspace of A, where A is the matrix that contains{ }nvvvv ,...,,, 321 in rows. )()dim( AnullityW =⊥ Orthonormal Basis: A basis of mutually orthogonal vectors of length 1. Basis can be found with the Gram- Schmidt process outline below.    = ≠ >=< ji ji vv ji 1 0 , In an orthonormal basis: )... ),...,,, 332211 332211 nn nn vcvcvcvcu vvuvvuvvuvvuu ++++= ><++><+><+>=< Gram-Schmidt Process: This finds an orthonormal basis recursively. Page 55 of 330 4. 0 iff 0, 0, =>=< >≥< uuu uu Unit Vector: u u u = ^ Cavchy-Schuarz Inequality: ><×>≤<>< vvuuvu ,,, 2 Inner Product Space: vuvu ukku uuu vu vu vu vu vuvu uuu uuuuu +=+ = ==≥ ≤><≤−⇒≤       >< ⇒×≤>< >=< ><=>=< 0 iff 0,0 1 , 11 , , , ,, 2 222 2 2 1 Angle between two vectors: As defined by the inner product, ( ) vu vu ><= ,cos θ Orthogonal if: 0, >=< vu Distance between two vectors: As defined by the inner product, vuvud −=),( Generalised Pythagoras for orthogonal vectors: 222 vuvu +=+ 4.12 PRIME NUMBERS: Determinate:    =                             + ×++    + =∆ ∑         + = composite and odd is N if0 prime and odd is N if1 12 12 1 3 1 )( 2 1 1 N k k N N k N N List of Prime Numbers: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 Page 56 of 330 947 953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049 1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237 1249 1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361 1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459 1471 1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 1549 1553 1559 1567 1571 1579 1583 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657 1663 1667 1669 1693 1697 1699 1709 1721 1723 1733 1741 1747 1753 1759 1777 1783 1787 1789 1801 1811 1823 1831 1847 1861 1867 1871 1873 1877 1879 1889 1901 1907 1913 1931 1933 1949 1951 1973 1979 1987 1993 1997 1999 2003 2011 2017 2027 2029 2039 2053 2063 2069 2081 2083 2087 2089 2099 2111 2113 2129 2131 2137 2141 2143 2153 2161 2179 2203 2207 2213 2221 2237 2239 2243 2251 2267 2269 2273 2281 2287 2293 2297 2309 2311 2333 2339 2341 2347 2351 2357 2371 2377 2381 2383 2389 2393 2399 2411 2417 2423 2437 2441 2447 2459 2467 2473 2477 2503 2521 2531 2539 2543 2549 2551 2557 2579 2591 2593 2609 2617 2621 2633 2647 2657 2659 2663 2671 2677 2683 2687 2689 2693 2699 2707 2711 2713 2719 2729 2731 2741 2749 2753 2767 2777 2789 2791 2797 2801 2803 2819 2833 2837 2843 2851 2857 2861 2879 2887 2897 2903 2909 2917 2927 2939 2953 2957 2963 2969 2971 2999 3001 3011 3019 3023 3037 3041 3049 3061 3067 3079 3083 3089 3109 3119 3121 3137 3163 3167 3169 3181 3187 3191 3203 3209 3217 3221 3229 3251 3253 3257 3259 3271 3299 3301 3307 3313 3319 3323 3329 3331 3343 3347 3359 3361 3371 3373 3389 3391 3407 3413 3433 3449 3457 3461 3463 3467 3469 3491 3499 3511 3517 3527 3529 3533 3539 3541 3547 3557 3559 3571 Fundamental Theory of Arithmetic: That every integer greater than 1 is either prime itself or is the product of a finite number of prime numbers. Laprange’s Theorem: That every natural number can be written as the sum of four square integers. Eg: 2222 013759 +++= Ie: 0 2222 ,,,,; Nxdcbadcbax ∈+++= Additive primes: Primes: such that the sum of digits is a prime. 2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 113, 131… Annihilating primes: Primes: such that d(p) = 0, where d(p) is the shadow of a sequence of natural numbers 3, 7, 11, 17, 47, 53, 61, 67, 73, 79, 89, 101, 139, 151, 157, 191, 199 Bell number primes: Primes: that are the number of partitions of a set with n members. 2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837. The next term has 6,539 digits. Carol primes: Of the form (2n−1)2 − 2. 7, 47, 223, 3967, 16127, 1046527, 16769023, 1073676287, 68718952447, 274876858367, 4398042316799, 1125899839733759, 18014398241046527, 1298074214633706835075030044377087 Centered decagonal primes: Of the form 5(n2 − n) + 1. 11, 31, 61, 101, 151, 211, 281, 661, 911, 1051, 1201, 1361, 1531, 1901, 2311, 2531, 3001, 3251, 3511, 4651, 5281, 6301, 6661, 7411, 9461, 9901, 12251, 13781, 14851, 15401, 18301, 18911, 19531, 20161, 22111, 24151, 24851, 25561, 27011, 27751 Centered heptagonal primes: Of the form (7n2 − 7n + 2) / 2. Page 57 of 330 43, 71, 197, 463, 547, 953, 1471, 1933, 2647, 2843, 3697, 4663, 5741, 8233, 9283, 10781, 11173, 12391, 14561, 18397, 20483, 29303, 29947, 34651, 37493, 41203, 46691, 50821, 54251, 56897, 57793, 65213, 68111, 72073, 76147, 84631, 89041, 93563 Centered square primes: Of the form n2 + (n+1)2. 5, 13, 41, 61, 113, 181, 313, 421, 613, 761, 1013, 1201, 1301, 1741, 1861, 2113, 2381, 2521, 3121, 3613, 4513, 5101, 7321, 8581, 9661, 9941, 10513, 12641, 13613, 14281, 14621, 15313, 16381, 19013, 19801, 20201, 21013, 21841, 23981, 24421, 26681 Centered triangular primes: Of the form (3n2 + 3n + 2) / 2. 19, 31, 109, 199, 409, 571, 631, 829, 1489, 1999, 2341, 2971, 3529, 4621, 4789, 7039, 7669, 8779, 9721, 10459, 10711, 13681, 14851, 16069, 16381, 17659, 20011, 20359, 23251, 25939, 27541, 29191, 29611, 31321, 34429, 36739, 40099, 40591, 42589 Chen primes: Where p is prime and p+2 is either a prime or semiprime. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 67, 71, 83, 89, 101, 107, 109, 113, 127, 131, 137, 139, 149, 157, 167, 179, 181, 191, 197, 199, 211, 227, 233, 239, 251, 257, 263, 269, 281, 293, 307, 311, 317, 337, 347, 353, 359, 379, 389, 401, 409 Circular primes: A circular prime number is a number that remains prime on any cyclic rotation of its digits (in base 10). 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 197, 199, 311, 337, 373, 719, 733, 919, 971, 991, 1193, 1931, 3119, 3779, 7793, 7937, 9311, 9377, 11939, 19391, 19937, 37199, 39119, 71993, 91193, 93719, 93911, 99371, 193939, 199933, 319993, 331999, 391939, 393919, 919393, 933199, 939193, 939391, 993319, 999331 Cousin primes: Where (p, p+4) are both prime. (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71), (79, 83), (97, 101), (103, 107), (109, 113), (127, 131), (163, 167), (193, 197), (223, 227), (229, 233), (277, 281) Cuban primes: Of the form ( ) 33 33 11, yyyx yx yx −+⇒+= − − 7, 19, 37, 61, 127, 271, 331, 397, 547, 631, 919, 1657, 1801, 1951, 2269, 2437, 2791, 3169, 3571, 4219, 4447, 5167, 5419, 6211, 7057, 7351, 8269, 9241, 10267, 11719, 12097, 13267, 13669, 16651, 19441, 19927, 22447, 23497, 24571, 25117, 26227, 27361, 33391, 35317 Of the form ( ) 2 2 2, 3333 yy yx yx yx −+ ⇒+= − − 13, 109, 193, 433, 769, 1201, 1453, 2029, 3469, 3889, 4801, 10093, 12289, 13873, 18253, 20173, 21169, 22189, 28813, 37633, 43201, 47629, 60493, 63949, 65713, 69313, 73009, 76801, 84673, 106033, 108301, 112909, 115249 Cullen primes: Of the form n×2n + 1. 3, 393050634124102232869567034555427371542904833 Dihedral primes: Primes: that remain prime when read upside down or mirrored in a seven-segment display. 2, 5, 11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121, 121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081 Double factorial primes: Of the form n!! + 1. Values of n: Page 60 of 330 Left-truncatable primes: Primes that remain prime when the leading decimal digit is successively removed. 2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 113, 137, 167, 173, 197, 223, 283, 313, 317, 337, 347, 353, 367, 373, 383, 397, 443, 467, 523, 547, 613, 617, 643, 647, 653, 673, 683 Leyland primes: Of the form xy + yx, with 1 < x ≤ y. 17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 Long primes: Primes p for which, in a given base b, p b p 11 −− gives a cyclic number. They are also called full reptend primes:. Primes p for base 10: 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593 Lucas primes: Primes in the Lucas number sequence L0 = 2, L1 = 1, Ln = Ln−1 + Ln−2. 2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149 Lucky primes: Lucky numbers that are prime. 3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193, 211, 223, 241, 283, 307, 331, 349, 367, 409, 421, 433, 463, 487, 541, 577, 601, 613, 619, 631, 643, 673, 727, 739, 769, 787, 823, 883, 937, 991, 997 Markov primes: Primes p for which there exist integers x and y such that x2 + y2 + p2 = 3xyp. 2, 5, 13, 29, 89, 233, 433, 1597, 2897, 5741, 7561, 28657, 33461, 43261, 96557, 426389, 514229, 1686049, 2922509, 3276509, 94418953, 321534781, 433494437, 780291637, 1405695061, 2971215073, 19577194573, 25209506681 Mersenne primes: Of the form 2n − 1. 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 Mersenne prime exponents: Primes p such that 2p − 1 is prime. 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583 Mills primes: Of the form  n3θ , where θ is Mills' constant. This form is prime for all positive integers n. 2, 11, 1361, 2521008887, 16022236204009818131831320183 Minimal primes: Primes for which there is no shorter sub-sequence of the decimal digits that form a prime. There are exactly 26 minimal primes:: 2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049 Motzkin primes: Primes that are the number of different ways of drawing non-intersecting chords on a circle between n points. 2, 127, 15511, 953467954114363 Page 61 of 330 Newman–Shanks–Williams primes: Newman–Shanks–Williams numbers that are prime. 7, 41, 239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599 Non-generous primes: Primes p for which the least positive primitive root is not a primitive root of p2. 2, 40487, 6692367337 Odd primes: Of the form 2n − 1. 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199... Padovan primes: Primes in the Padovan sequence P(0) = P(1) = P(2) = 1, P(n) = P(n−2) + P(n−3). 2, 3, 5, 7, 37, 151, 3329, 23833, 13091204281, 3093215881333057, 1363005552434666078217421284621279933627102780881053358473 Palindromic primes: Primes that remain the same when their decimal digits are read backwards. 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741 Palindromic wing primes: Primes of the form ( ) 210 9 110 mm b a ×±− 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 11311, 11411, 33533, 77377, 77477, 77977, 1114111, 1117111, 3331333, 3337333, 7772777, 7774777, 7778777, 111181111, 111191111, 777767777, 77777677777, 99999199999 Partition primes: Partition numbers that are prime. 2, 3, 5, 7, 11, 101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557 Pell primes: Primes in the Pell number sequence P0 = 0, P1 = 1, Pn = 2Pn−1 + Pn−2. 2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449 Permutable primes: Any permutation of the decimal digits is a prime. 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, 1111111111111111111, 11111111111111111111111 Perrin primes: Primes in the Perrin number sequence P(0) = 3, P(1) = 0, P(2) = 2, P(n) = P(n−2) + P(n−3). 2, 3, 5, 7, 17, 29, 277, 367, 853, 14197, 43721, 1442968193, 792606555396977, 187278659180417234321, 66241160488780141071579864797 Pierpont primes: Of the form 2u3v + 1 for some integers u,v ≥ 0. These are also class 1- primes:. 2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 65537, 139969, 147457 Page 62 of 330 Pillai primes: Primes p for which there exist n > 0 such that p divides n!+ 1 and n does not divide p−1. 23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, 227, 233, 239, 251, 257, 269, 271, 277, 293, 307, 311, 317, 359, 379, 383, 389, 397, 401, 419, 431, 449, 461, 463, 467, 479, 499 Primeval primes: Primes for which there are more prime permutations of some or all the decimal digits than for any smaller number. 2, 13, 37, 107, 113, 137, 1013, 1237, 1367, 10079 Primorial primes: Of the form pn# −1 or pn# + 1. 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 Proth primes: Of the form k×2n + 1, with odd k and k < 2n. 3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857 Pythagorean primes: Of the form 4n + 1. 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449 Prime quadruplets: Where (p, p+2, p+6, p+8) are all prime. (5, 7, 11, 13), (11, 13, 17, 19), (101, 103, 107, 109), (191, 193, 197, 199), (821, 823, 827, 829), (1481, 1483, 1487, 1489), (1871, 1873, 1877, 1879), (2081, 2083, 2087, 2089), (3251, 3253, 3257, 3259), (3461, 3463, 3467, 3469), (5651, 5653, 5657, 5659), (9431, 9433, 9437, 9439) Primes of binary quadratic form: Of the form x2 + xy + 2y2, with non-negative integers x and y. 2, 11, 23, 37, 43, 53, 71, 79, 107, 109, 127, 137, 149, 151, 163, 193, 197, 211, 233, 239, 263, 281, 317, 331, 337, 373, 389, 401, 421, 431, 443, 463, 487, 491, 499, 541, 547, 557, 569, 599, 613, 617, 641, 653, 659, 673, 683, 739, 743, 751, 757, 809, 821 Quartan primes: Of the form x4 + y4, where x,y > 0. 2, 17, 97, 257, 337, 641, 881 Ramanujan primes: Integers Rn that are the smallest to give at least n primes: from x/2 to x for all x ≥ Rn (all such integers are primes:). 2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167, 179, 181, 227, 229, 233, 239, 241, 263, 269, 281, 307, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, 439, 461, 487, 491 Regular primes: Primes p which do not divide the class number of the p-th cyclotomic field. 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 239, 241, 251, 269, 277, 281 Repunit primes: Primes containing only the decimal digit 1. 11, 1111111111111111111, 11111111111111111111111 The next have 317 and 1,031 digits. Primes in residue classes: Of the form an + d for fixed a and d. Also called primes: congruent to d modulo a. Page 65 of 330 Wall-Sun-Sun primes: A prime p > 5 if p2 divides the Fibonacci number      − 5 p p F , where the Legendre symbol       5 p is defined as ( ) ( )  ±≡− ±≡ =      5mod2p if1 5mod1p if1 5 p As of 2011, no Wall-Sun-Sun primes: are known. Wedderburn-Etherington number primes: Wedderburn-Etherington numbers that are prime. 2, 3, 11, 23, 983, 2179, 24631, 3626149, 253450711, 596572387 Weakly prime numbers Primes that having any one of their (base 10) digits changed to any other value will always result in a composite number. 294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139 Wieferich primes: Primes p for which p2 divides 2p−1 − 1. 1093, 3511 Wieferich primes: base 3 (Mirimanoff primes:) Primes p for which p2 divides 3p−1 − 1. 11, 1006003 Wieferich primes: base 5 Primes p for which p2 divides 5p−1 − 1 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 Wieferich primes: base 6 Primes p for which p2 divides 6p−1 − 1. 66161, 534851, 3152573 Wieferich primes: base 7 Primes p for which p2 divides 7p−1 − 1. 5, 491531 Wieferich primes: base 10 Primes p for which p2 divides 10p−1 − 1. 3, 487, 56598313 Wieferich primes: base 11 Primes p for which p2 divides 11p−1 − 1. 71 Wieferich primes: base 12 Primes p for which p2 divides 12p−1 − 1. 2693, 123653 Wieferich primes: base 13 Primes p for which p2 divides 13p−1 − 1 863, 1747591 Wieferich primes: base 17 Primes p for which p2 divides 17p−1 − 1. 3, 46021, 48947 Page 66 of 330 Wieferich primes: base 19 Primes p for which p2 divides 19p−1 − 1 3, 7, 13, 43, 137, 63061489 Wilson primes: Primes p for which p2 divides (p−1)! + 1. 5, 13, 563 Wolstenholme primes: Primes p for which the binomial coefficient )(mod1 1 12 4p p p ≡      − − 16843, 2124679 Woodall primes: Of the form n×2n − 1. 7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319 4.13 GENERALISATIONS FROM PRIME NUMBERS: Perfect Numbers: A perfect number is a positive integer that is equal to the sum of its proper positive divisors, excluding the number itself. Even perfect numbers are of the form 2p−1(2p−1), where (2p−1) is prime and by extension p is also prime. It is unknown whether there are any odd perfect numbers. List of Perfect Numbers: Rank p Perfect number Digits Year Discoverer 1 2 6 1 Known to the Greeks 2 3 28 2 Known to the Greeks 3 5 496 3 Known to the Greeks 4 7 8128 4 Known to the Greeks 5 13 33550336 8 1456 First seen in the medieval manuscript, Codex Lat. Monac. 6 17 8589869056 10 1588 Cataldi 7 19 12 1588 Cataldi 8 31 19 1772 Euler 9 61 37 1883 Pervushin 10 89 54 1911 Powers 11 107 65 1914 Powers 12 127 77 1876 Lucas 13 521 314 1952 Robinson 14 607 366 1952 Robinson 15 1279 770 1952 Robinson 16 2203 1327 1952 Robinson 17 2281 1373 1952 Robinson 18 3217 1937 1957 Riesel 19 4253 2561 1961 Hurwitz 20 4423 2663 1961 Hurwitz 21 9689 5834 1963 Gillies 22 9941 5985 1963 Gillies Page 67 of 330 23 11213 6751 1963 Gillies 24 19937 12003 1971 Tuckerman 25 21701 13066 1978 Noll & Nickel 26 23209 13973 1979 Noll 27 44497 26790 1979 Nelson & Slowinski 28 86243 51924 1982 Slowinski 29 110503 66530 1988 Colquitt & Welsh 30 132049 79502 1983 Slowinski 31 216091 130100 1985 Slowinski 32 756839 455663 1992 Slowinski & Gage 33 859433 517430 1994 Slowinski & Gage 34 1257787 757263 1996 Slowinski & Gage 35 1398269 841842 1996 Armengaud, Woltman, et al. 36 2976221 1791864 1997 Spence, Woltman, et al. 37 3021377 1819050 1998 Clarkson, Woltman, Kurowski, et al. 38 6972593 4197919 1999 Hajratwala, Woltman, Kurowski, et al. 39 13466917 8107892 2001 Cameron, Woltman, Kurowski, et al. 40 20996011 12640858 2003 Shafer, Woltman, Kurowski, et al. 41 24036583 14471465 2004 Findley, Woltman, Kurowski, et al. 42 25964951 15632458 2005 Nowak, Woltman, Kurowski, et al. 43 30402457 18304103 2005 Cooper, Boone, Woltman, Kurowski, et al. 44 32582657 19616714 2006 Cooper, Boone, Woltman, Kurowski, et al. 45 37156667 22370543 2008 Elvenich, Woltman, Kurowski, et al. 46 42643801 25674127 2009 Strindmo, Woltman, Kurowski, et al. 47 43112609 25956377 2008 Smith, Woltman, Kurowski, et al. Amicable Numbers: Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. List of Amicable Numbers: Amicable Pairs Amicable Pairs Amicable Pairs 220 284 1,328,470 1,483,850 8,619,765 9,627,915 1,184 1,210 1,358,595 1,486,845 8,666,860 10,638,356 2,620 2,924 1,392,368 1,464,592 8,754,130 10,893,230 5,020 5,564 1,466,150 1,747,930 8,826,070 10,043,690 6,232 6,368 1,468,324 1,749,212 9,071,685 9,498,555 10,744 10,856 1,511,930 1,598,470 9,199,496 9,592,504 12,285 14,595 1,669,910 2,062,570 9,206,925 10,791,795 17,296 18,416 1,798,875 1,870,245 9,339,704 9,892,936 63,020 76,084 2,082,464 2,090,656 9,363,584 9,437,056 66,928 66,992 2,236,570 2,429,030 9,478,910 11,049,730 67,095 71,145 2,652,728 2,941,672 9,491,625 10,950,615 69,615 87,633 2,723,792 2,874,064 9,660,950 10,025,290 79,750 88,730 2,728,726 3,077,354 9,773,505 11,791,935 100,485 124,155 2,739,704 2,928,136 10,254,970 10,273,670 122,265 139,815 2,802,416 2,947,216 10,533,296 10,949,704 122,368 123,152 2,803,580 3,716,164 10,572,550 10,854,650 141,664 153,176 3,276,856 3,721,544 10,596,368 11,199,112 142,310 168,730 3,606,850 3,892,670 10,634,085 14,084,763 171,856 176,336 3,786,904 4,300,136 10,992,735 12,070,305 Page 70 of 330 C6 Needham 2006 13D 4773123705616 2^4*7*347*122816069 5826394399664 2^4*101*3605442079 5574013457296 2^4*53*677*1483*6547 5454772780208 2^4*53*239*2971*9059 5363145542992 2^4*307*353*3093047 5091331952624 2^4*318208247039 C8 Flammenkamp 1990 Brodie ? 10D 1095447416 2^3*7*313*62497 1259477224 2^3*43*3661271 1156962296 2^3*7*311*66431 1330251784 2^3*43*3867011 1221976136 2^3*41*1399*2663 1127671864 2^3*11*61*83*2531 1245926216 2^3*19*8196883 1213138984 2^3*67*2263319 C8 Flammenkamp 1990 Brodie ? 10D 1276254780 2^2*3*5*1973*10781 2299401444 2^2*3*991*193357 3071310364 2^2*767827591 2303482780 2^2*5*67*211*8147 2629903076 2^2*23*131*218213 2209210588 2^2*13^2*17*192239 2223459332 2^2*131*4243243 1697298124 2^2*907*467833 C9 Flammenkamp 1990 9D/10D 805984760 2^3*5*7*1579*1823 1268997640 2^3*5*17*61*30593 1803863720 2^3*5*103*367*1193 2308845400 2^3*5^2*11544227 3059220620 2^2*5*2347*65173 3367978564 2^2*841994641 2525983930 2*5*17*367*40487 2301481286 2*13*19*4658869 1611969514 2*805984757 C28 Poulet 1918 5D/6D 14316 2^2*3*1193 19116 2^2*3^4*59 31704 2^3*3*1321 47616 2^9*3*31 83328 2^7*3*7*31 177792 2^7*3*463 295488 2^6*3^5*19 629072 2^4*39317 589786 2*294893 294896 2^4*7*2633 Page 71 of 330 358336 2^6*11*509 418904 2^3*52363 366556 2^2*91639 274924 2^2*13*17*311 275444 2^2*13*5297 243760 2^4*5*11*277 376736 2^5*61*193 381028 2^2*95257 285778 2*43*3323 152990 2*5*15299 122410 2*5*12241 97946 2*48973 48976 2^4*3061 45946 2*22973 22976 2^6*359 22744 2^3*2843 19916 2^2*13*383 17716 2^2*43*103 This list is exhaustive for known social numbers where C>4 4.14 GOLDEN RATIO & FIBONACCI SEQUENCE: Relationship: 803091798056563811772204586834387498948481.61803398=ϕ Infinite Series: Continued Fractions: Page 72 of 330 Trigonometric Expressions: Fibonacci Sequence:                 −−       += −−=−−= − nn nnnn nF nF 2 51 2 51 5 1 )( 5 )( 5 )1( )( ϕϕϕϕ 4.15 FERMAT’S LAST THEOREM: nnn cba ≠+ for integers cba &, and 2>n Proposed by Fermat in 1637 as an extension of Diophantus’s explanation of the case when n=2. Case when n=3 was proved by Euler (1770) Case when n=4 was proved by Fermat Case when n=5 was proved by Legendre and Dirichlet (1885) Case when n=7 was proved by Gabriel Lamé (1840) General case when n>2 was proved by Andrew Wiles (1994). The proof is too long to be written here. See: http://www.cs.berkeley.edu/~anindya/fermat.pdf 4.16 BOOLEAN ALGEBRA: Axioms: Axiom Dual Name 0=B if 1≠B 1=B if 0≠B Binary Field 10 = 01 = NOT 000 =• 111 =+ AND/OR 111 =• 000 =+ AND/OR 00110 =•=• 10110 =+=+ AND/OR Page 75 of 330 Cards in a card house: ( ) 2 13 += ll Different arrangement of dominos: !2 ndn ×= − Unit Fractions:       +       − + +    = 11 1 a b INTb a b MODa a b INT b a Angle between two hands of a clock: hm 305.5 −=θ Winning Lines in Noughts and Crosses: ( )12 += a Bad Restaurant Spread: s P − = 1 Fibonacci Sequence:                 −−       += nn 2 51 2 51 5 1 ABBREVIATIONS (5.1, 5.2, 5.3, 5.4, 5.5) a=side ‘a’ b=side ‘b’ c=cuts d=double dominos h=hours L=Languages l=layers m=minutes n= nth term n=n number P=Premium/Starting Quantity p=number you pick r=number of roles/turns s=spread factor T=Term θ=the angle 5.6 FACTORIAL: Definition: 12...)2()1(! ×××−×−×= nnnn Table of Factorials: 0! 1 (by definition) 1! 1 11! 39916800 2! 2 12! 479001600 3! 6 13! 6227020800 4! 24 14! 87178291200 5! 120 15! 1307674368000 6! 720 16! 20922789888000 7! 5040 17! 355687428096000 Page 76 of 330 8! 40320 18! 6402373705728000 9! 362880 19! 121645100408832000 10! 3628800 20! 2432902008176640000 Approximation: n n enn − + ××≈ 2 1 2! π (within 1% for n>10) 5.7 THE DAY OF THE WEEK: This only works after 1753          +   −   +   ++= 400100412 31 7 yyym ydMOD d=day m=month y=year SQUARE BRAKETS MEAN INTEGER DIVISION INT=Keep the integer MOD=Keep the remainder 5.8 BASIC PROBABILITY: Axiom’s of Probability: 1. ( ) 1=ΩP for the eventspace Ω 2. ( ) [ ]1,0∈AP for any event A. 3. If 1A and 2A are disjoint, then ( ) ( ) ( )2121 APAPAAP +=∪ Generally, if iA are mutually disjoint, then ( )∑ ∞ = ∞ = =      11 i i i i APAP U Commutative Laws: ABBA ABBA ∩=∩ ∪=∪ Associative Laws: ( ) ( ) ( ) ( )CBACBA CBACBA ∩∩=∩∩ ∪∪=∪∪ Distributive Laws: ( ) ( ) ( ) ( ) ( ) ( )CBCACBA CBCACBA ∪∩∪=∪∩ ∩∪∩=∩∪ Indicator Function:    ∉ ∈ =Ι D ispoint if0 D ispoint if1 D 5.9 VENN DIAGRAMS: Complementary Events: ( ) ( )APAP =−1 Null Set: ( ) 0=ΦP Totality: ...)()|()()|()( )()|()( 2211 1 ++= =∑ = BPBAPBPBAPAP BPBAPAP m i ii where Φ=∩ ji BB for ji ≠ )'()()( BAPBAPAP ∩+∩= Page 77 of 330 Conditional Probability: ( ) ( )( )BP BAP BAP ∩=| ( ) ( ) ( ) ( ) ( )ABPAPBAPBPBAP || ⋅=⋅=∩ (Multiplication Law) Union: ( ) ( ) ( ) ( )BAPBPAPBAP ∩−+=∪ (Addition Law) Independent Events: ( ) ( ) ( )BPAPBAP ⋅=∩ ( ) ( ) ( ) ( ) ( )BPAPBPAPBAP ⋅−+=∪ ( ) ( )BPABP =| ( ) ( ) ( )kk APAPAPAAAP ...)...( 2121 =∩∩∩ Mutually Exclusive: ( ) 0=∩ BAP ( ) )(' APBAP =∩ ( ) ( ) ( )BPAPBAP +=∪ ( ) ( )'' BPBAP =∪ Subsets: if BA ⊂ then ( ) ( )BPAP ≤ Baye’s Theorem: )'()'|()()|( )()|( )( )()|( )|( BPBAPBPBAP BPBAP AP BPBAP ABP + == Event’s Space: ∑ = ∩= m i iBAPAP 1 )()( 5.11 BASIC STATISTICAL OPERATIONS: Variance: 2σ=v Arithmetic Mean: 2 ba b b a a b a +=⇒=== − − µ µ µ µ µ s i n x∑=µ Geometric Mean: abb aa b =⇒== − − µ µ µ µ µ Harmonic Mean: baa b a b /1/1 2 + =⇒= − − µ µ µ Standardized Score: σ µ− = i x z Confidence Interval: Quantile: The pth of quantile of the distribution F is defined to be the value xp such that pxF p =)( or pxXP p =≤ )( ( )pFx p 1−=∴ x0.25 is the lower quartile x0.5 is the median x0.75 is the upper quartile 5.12 DISCRETE RANDOM VARIABLES: Standard Deviation: ( ) s i n xx 2 ∑ −=σ Expected Value: Page 80 of 330 Expected Value: ∫ ∞ ∞− ×= dxxfxxE )()( ∫ ∞ ∞− ×= dxxfxgxgE )()())(( Variance: 22 ))(()()( XEXEXVar −= 5.15 COMMON CRVs: Uniform Distribution: Declaration: ),(~ baUniformX PDF:     ≤≤ −= otherwise bxa abxf 0 1 )( CDF: ∫ ∞−      > ≤≤ − − < == x bx bxa ab ax ax dxxfxF 1 0 )()( Expected Value: 2 ba += Variance: ( ) 12 2ab −= Exponential Distribution: Declaration: )(~ λlExponentiaX Page 81 of 330 PDF:    ≥ < = − 0 00 )( xe x xf xλλ CDF: ∫ ∞− −    ≥− < == x x xe x dxxfxF 01 00 )()( λ Expected Value: λ 1= Variance: 2 1 λ = Normal Distribution: Declaration: ),(~ 2σµNormalX Standardized Z Score: σ µ−= xZ PDF: 2 2 2 1 2 1 2 1 2 1 )( z x eexf −       −− == πσπσ σ µ CDF: ( )ZΦ (The integration is provided within statistic tables) Expected Value: µ= Variance: = 2σ 5.16 BIVARIABLE DISCRETE: Probability: ),(),( yxfyYxXP === ∑=≤≤ ),(),( yxfyYxXP over all values of x & y Page 82 of 330 Marginal Distribution: ( ) ∑ ∑ = = x iY y iX yxfyf yxfxf ),()( ),( Expected Value: ∑∑ ∑ ∑ ××= ×= ×= x y YX y Y x X yxfyxYXE yfyYE xfxXE ),(],[ )(][ )(][ , Independence: )()(),( yfxfyxf YX ×= Covariance: ][][],[ YEXEYXECov ×−= 5.17 BIVARIABLE CONTINUOUS: Conditions: ( ) ( )∫ ∫ ∞ ∞− ∞ ∞− =≥ 1,&0, ,, dxdyyxfyxf YXYX Probability: ( ) ∫ ∫ ∫ ∫ ∞− ∞− ∞− ∞− =<∞<<−∞=< =∞<<−∞<=< ==≤≤ y Y x X y x YXYX dyyfyYXPyYP dxxfYxXPxXP dxdyyxfyxFyYxXP )(),()( )(),()( ),(,),( ,, Where the domain is: ( ) ( ) ( ) ( )( ){ } ( ) ( ) ( ) ( ) ∫ ∫ ∫= ∈∈= D b a x x YXYX dydxyxfdxdyyxf xxybaxyxD β α βα ,, ,,,:, ,, Where the domain is: ( ) ( ) ( )( ) ( ){ } ( ) ( ) ( ) ( ) ∫ ∫∫ = ∈∈= d c x x YXD YX dxdyyxfdxdyyxf dcyxxxyxD δ γ δγ ,, ,,,:, ,, Where the domain is the event space: ( ) ( ) ( ) ( ) ( ) ( ) 1,, ,, == ∫ ∫∫ ∫ d c x x YX b a x x YX dxdyyxfdydxyxf δ γ β α Marginal Distribution: ( ) ∫= b a YXX dyyxfxf ),(, where a & b are bounds of y Page 85 of 330 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫∫∫ ∫ ∫∫ ∫ ∫ ∫∫ ∞ ∞− ∞ ∞−∞− ∞ ∞− ∞ ∞− ∞ ∞− ∞ = +−=       +       = +== dxzxxfxzf dxzxxxfdxzxxxfdxdyyxf dz d dxdyyxf dz d dydxyxf dz d dydxyxf dz d z z dF zf YXZ YXYX zx YX zx YX zx YX zx YX Z Z , ,,,, ,, , 0 , 0 , 0 0 ,, 0 0 ,, If X & Y are independent: ( ) ( ) ( )∫ ∞ ∞− = dxzxfxfxzf YXZ Maximum: Assuming that nXXX ,...,, 21 are independant random variables with cdf F and density f. ( )nXXXZ ,...,,max 21= zZ ≤ iff izX i ∀≤ , ( ) ( ) ( ) ( ) ( ) ( ) ( )( )nXnnZ zFzXPzXPzXPzXzXzXPzZPzF =≤××≤×≤=≤≤≤=≤= ...,...,, 2121 ( ) ( )( ) ( ) ( )zfznFzF dz d zf X n X n XZ 1−== Minimum: Assuming that nXXX ,...,, 21 are independant random variables with cdf F and density f. ( )nXXXZ ,...,,min 21= zZ > iff izX i ∀> , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )nXn nZ zFzXPzXPzXP zXzXzXPzZPzZPzF −−=>××>×>−= >>>−=>−=≤= 11...1 ,...,,11 21 21 ( ) ( )( )( ) ( )( ) ( )zfzFnzF dz d zf X n X n XZ 1111 −−=−−= Order Statistics: Assuming that nXXX ,...,, 21 are independant random variables with cdf F and density f. Sorting iX in non decreasing order: ( ) ( ) ( )nXXX ≤≤≤ ...21 The kth order statistic of a statistical sample is equal to its kth smallest value. Particularly: ( ) ( )nXXXX ,...,,min 211 = and ( ) ( )nn XXXX ,...,,max 21= Let nk <<1 . The event ( ) dxxXx k +≤< occurs if: 1. 1−k observations are less than x 2. one observation is in the interval ( ]dxxx +, 3. kn − observations are greater than dxx + The probability of any particular arrangment of this type is: ( ) ( ) ( )( ) dxxFxFxf knk −− −= 11 By the combination law: ( ) ( ) ( ) ( ) ( ) ( )( ) knk K xFxFxfknk n kf −− − −− = 1 !!1 ! 1 Page 86 of 330 5.19 TRANSFORMATION OF THE JOINT DENSITY: Bivariate Functions: Let X and Y have joint density ( )yxf YX ,, Let ℜ→ℜℜ→ℜ 22 2 1 :&: gg ( ) ( )YXgVYXgU ,&, 21 == Assuming that 21 & gg can be inverted. There exist ℜ→ℜℜ→ℜ 2 2 2 1 :&: hh so that ( ) ( )VUhYVUhX ,&, 21 == Multivariate Functions: Let nXX ,...,1 have joint density ( )nX xxf ,..,1 Let ℜ→ℜnig : ( )nii XXgY ,...,1= Assuming that ig can be inverted. There exist ℜ→ℜ n ih : so that ( )nii YYhX ,...,1= Jacobian: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )yx x g yx y g yx y g yx x g yx y g yx x g yx y g yx x g yxJ ,,,, ,, ,, , 2121 22 11 ∂ ∂× ∂ ∂− ∂ ∂× ∂ ∂=             ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ( ) ( ) ( ) ( ) ( )               ∂∂ ∂∂ = x x g x x g x x g x x g yxJ n nn n ... ......... ... , 1 1 1 1 There is an assumption that the derivateves exist and that ( ) yxyxJ ,0, ∀≠ Joint Density: ( ) ( ) ( )( )( ) ( )( )vuhvuhJ vuhvuhf vuf YXVU ,,, ,,, , 21 21, , = ( ) ( ) ( )( )( ) ( )( )nnn nnnX nY xxhxxhJ xxhxxhf yyf ,...,,...,,..., ,...,,...,,..., ,..., 111 111 1 = ABBREVIATIONS σ = Standard Deviation µ = mean ns = number of scores p = probability of favourable result v = variance xi = Individual x score x = mean of the x scores Page 87 of 330 z = Standardized Score Page 90 of 330 τi: i th treatment effect βj: jh block effect Test for Treatment Effect: Test for Block Effect: Relative Efficiency: ( ) ( ) ( )MSEba MSEabMSb MSE MSE RE BLOCK RBD CRD CRDRBD 1 11 / − −+−== ANOVA: Source of Variation Degrees of Freedom Sum of Squares Treatment a-1 ( ) N y y b yybSS a i i a i iTREATMENT 2 .. 1 2 . 1 2 ... 1 −     =−= ∑∑ == Block b-1 ( ) N y y a yyaSS b i j b i jBLOCK 2 .. 1 2 . 1 2 ... 1 −     =−= ∑∑ == Error (a-1)(b-1) ( ) BLOCKSTREATMENTT a i b j jiijERROR SSSSSSyyyySS −−=+−−=∑∑ = =1 1 2 .... Total ab-1 ( ) N y yyySS a i b j ij a i b j ijT 2 .. 1 1 2 1 1 2 .. −       =−= ∑∑∑∑ = == = 6.4 LATIN SQUARE DESIGN (LSD): a=4 design 1 2 3 4 1 111y 221y 331y 441y 2 212y 322y 432y 142y 3 313y 423y 133y 243y 4 414y 124y 234y 344y Treatments: = a Factors: = 1 Replications per treatment: = a Total Treatments: 2aN == Mathematical Model:     = = = ++++=     = = = += ak aj ai y ak aj ai y ijkjiij ijkijkijk ,...,2,1 ,...,2,1 ,...,2,1 , ,...,2,1 ,...,2,1 ,...,2,1 , εγβτµ εµ a: number of levels of the factor, yijk :is the ijk th observation and Page 91 of 330 εijk :random experimental error, normally independently distributed with mean 0 and variance σ2 µ: overall mean τi: i th treatment effect βj: jth row effect γ k: kth column effect Test for Treatment Effect: Relative Efficiency: ( ) ( )MSEa MSEaMSMS MSE MSE RE COLUMNSROWS LSD CRD CRDLSD 1 1 / + −++== ANOVA: Source of Variation Degrees of Freedom Sum of Squares Treatment a-1 ( )∑ = −= a i iTREATMENT yyaSS 1 2 ..... Rows a-1 ( )∑ = −= a i jROWS yyaSS 1 2 ..... Columns a-1 ( )∑ = −= a i kCOLUMNS yyaSS 1 2 ..... Error 232 +− aa ( )∑∑∑ = = = +−−−= a i a j a k kjiijkERROR yyyyySS 1 1 1 2 ......... 2 Total 12 −a ( )∑∑∑ = = = −= a i a j a k ijkT yySS 1 1 1 2 ... 6.5 ANALYSIS OF COVARIANCE: Mathematical Model: ( )    = = +−++= i ijijiij nj ai xxy ,...,2,1 ,...,2,1 ,εβτµ a: number of levels of the factor, ni :is the number of observations on the i th level of the factor, yij :is the ij th observation and εij :random experimental error, normally independently distributed with mean 0 and variance σ2 µ: overall mean τi: i th treatment effect β: linear regression coefficient indicating dependency of yij on xij xij :is the ij th covariate Assumptions: • Treatment do not influence the covariate o May be obvious from the nature of the covariates. o Test through ANOVA on Covariates. • The regression coefficient β is the same for all treatments o Perform analysis of variance on covariates • The relationship between the response y and covariate x is linear. Page 92 of 330 o For each treatment fit a linear regression model and assess its quality 6.6 RESPONSE SURFACE METHODOLOGY: Definition: Creating a design such that it will optimise the response 1st order: 2nd order: Common Designs: • Designs for first order model o 2 level factorial designs + some centre points • Designs for second order model o 2 level factorial designs + some centre points + axial runs (central composite designs) o 3 levels designs; 2 level factorial designs and incomplete block designs (Box-Behnken designs) • Latin Hypercube Designs o Space filling Multilevel designs Criterion for determining the optimatility of a design: X Matrix is determined as the coefficients of β in whatever order studied. 6.7 FACTORIAL OF THE FORM 2n: General Definition: A 2n consists of n factors each studied at 2 levels. Contrasts for a 22 design: n Ibaab M n Ibaab M n Iabab M ABBA 2 ; 2 ; 2 +−−=−+−=−+−= Sum of Squares for a 22 design: ( ) ( ) ( ) ABBATREATMENT ABBA SSSSSSSS n Ibaab SS n Ibaab SS n Iabab SS ++= +−−=−+−=−+−= 2 2 2 2 2 2 2 ; 2 ; 2 Hypothesis for a CRD 22 design: Test for interaction effect of factor AB: Page 95 of 330 • Difference is significant if SlE >)( ^^ Page 96 of 330 PART 7: PI 7.1 AREA: Circle: 44 2 2 CddrA === ππ Cyclic Quadrilateral: ( )( )( )( )dscsbsas −−−− Area of a sector (degrees) 2 360 r Q A π×= Area of a sector (radians) θ2 2 1 rA = Area of a segment (degrees)       −×= QQrA sin 1802 2 π Area of an annulus: ( ) 2 2 1 2 2 2      =−= wrrA ππ Ellipse: 214 rrlwA ππ == 7.2 VOLUME: Cylinder: hrV 2π= Sphere: 3 3 4 rV π= Cap of a Sphere: ( )22136 1 hrhV += π Cone: hrV 2 3 1 π= Ice-cream & Cone: ( )rhrV 2 3 1 2 += π Doughnut: ( )( )2 2 2 12 2 4 2 ababrrV −+== ππ Sausage:       −= 34 2 w l w V π Ellipsoid: 3213 4 rrrV π= 7.3 SURFACE AREA: Sphere: 24 rSA π= Hemisphere: 23 rSA π= Doughnut: ( )2221224 abrrSA −== ππ Sausage: wlSA π= Cone: ( )22 hrrrSA ++= π Page 97 of 330 7.4 MISELANIOUS: Length of arc (degrees) rQCQl π×=×= 180360 Length of chord (degrees) 222 2 sin2 hr Q rl −=     ×= Perimeter of an ellipse ( ) ( ) ( ) ( ) ( )               + − −+ + −+ +≈ 2 21 2 21 2 21 2 21 21 3 410 3 1 rr rr rr rr rrP π 7.6 PI: 7950288...462643383235897932383.14159265≈π d C=π Archimedes’ Bounds:      <<      k k k k nn 2 sin2 2 sin2 θπθ John Wallis: ∏ ∞ = − =××××××××= 1 2 2 14 4 ... 9 8 7 8 7 6 5 6 5 4 3 4 3 2 1 2 2 n n nπ Isaac Newton: ... 27 1 642 631 25 1 42 31 23 1 2 1 2 1 6 753 +      ××× ××+      ×× ×+      × +=π James Gregory: ... 15 1 13 1 11 1 9 1 7 1 5 1 3 1 1 4 −+−+−+−=π Schulz von Strassnitzky:      +     +     = 8 1 arctan 5 1 arctan 2 1 arctan 4 π John Machin:      −     = 239 1 arctan 5 1 arctan4 4 π Leonard Euler: ... 4 1 3 1 2 1 1 1 6 2222 2 ++++=π ... 32 31 28 29 24 23 20 19 16 17 12 13 12 11 8 7 4 5 4 3 4 ××××××××××=π where the numerators are the odd primes; each denominator is the multiple of four nearest to the numerator. ` ... 13 1 12 1 11 1 10 1 9 1 8 1 7 1 6 1 5 1 4 1 3 1 2 1 1 +−++−++++−+++=π If the denominator is a prime of the form 4m - 1, the sign is positive; if the denominator is 2 or a prime of the form 4m + 1, the sign is negative; for composite numbers, the sign is equal the product of the signs of its factors. Jozef Hoene-Wronski: = π lim → n ∞ 4 n       − ( ) + 1 i       1 n ( ) − 1 i       1 n i Franciscus Vieta: ... 2 222 2 22 2 22 ×++×+×= π