Mathematics cheat sheet, Cheat Sheet of Analytical Geometry

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Page 1 of 286 ALL IN ONE MATHEMATICS CHEAT SHEET V2.6 eiπ + 1 = 0 CONTAINING FORMULAE FOR ELEMENTARY, HIGH SCHOOL AND UNIVERSITY MATHEMATICS COMPILED FROM MANY SOURCES BY ALEX SPARTALIS 2009-2012 Euler’s Identity: Page 2 of 286 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. 2012 TO DO: USV Decomposition Page 5 of 286 LINEAR DEPENDENCE: 43 BASIS: 43 STANDARD BASIS: 44 ORTHOGONAL COMPLEMENT: 44 ORTHONORMAL BASIS: 44 GRAM-SCHMIDT PROCESS: 45 COORDINATE VECTOR: 45 DIMENSION: 45 4.7 COMPLEX VECTOR SPACES: 45 FORM: 45 DOT PRODUCT: 45 INNER PRODUCT: 46 4.8 LINEAR TRANSITIONS & TRANSFORMATIONS: 46 TRANSITION MATRIX: 46 CHANGE OF BASIS TRANSITION MATRIX: 46 TRANSFORMATION MATRIX: 46 4.9 INNER PRODUCTS: 46 DEFINITION: 46 AXIOMS: 46 UNIT VECTOR: 47 CAVCHY-SCHUARZ INEQUALITY: 47 INNER PRODUCT SPACE: 47 ANGLE BETWEEN TWO VECTORS: 47 DISTANCE BETWEEN TWO VECTORS: 47 GENERALISED PYTHAGORAS FOR ORTHOGONAL VECTORS: 47 4.10 PRIME NUMBERS: 47 DETERMINATE: 47 LIST OF PRIME NUMBERS: 47 PERFECT NUMBERS: 48 LIST OF PERFECT NUMBERS: 48 AMICABLE NUMBERS: 49 LIST OF AMICABLE NUMBERS: 49 SOCIABLE NUMBERS: 50 LIST OF SOCIABLE NUMBERS: 50 4.11 GOLDEN RATIO & FIBONACCI SEQUENCE: 53 RELATIONSHIP: 53 INFINITE SERIES: 53 CONTINUED FRACTIONS: 53 TRIGONOMETRIC EXPRESSIONS: 54 FIBONACCI SEQUENCE: 54 4.12 FERMAT’S LAST THEOREM: 54 PART 5: COUNTING TECHNIQUES & PROBABILITY 55 5.1 2D 55 TRIANGLE NUMBER 55 SQUARE NUMBER 55 PENTAGONAL NUMBER 55 5.2 3D 55 TETRAHEDRAL NUMBER 55 SQUARE PYRAMID NUMBER 55 5.3 PERMUTATIONS 55 PERMUTATIONS: 55 PERMUTATIONS (WITH REPEATS): 55 5.4 COMBINATIONS 55 Page 6 of 286 ORDERED COMBINATIONS: 55 UNORDERED COMBINATIONS: 55 ORDERED REPEATED COMBINATIONS: 55 UNORDERED REPEATED COMBINATIONS: 55 GROUPING: 55 5.5 MISCELLANEOUS: 55 TOTAL NUMBER OF RECTANGLES AND SQUARES FROM A A X B RECTANGLE: 55 NUMBER OF INTERPRETERS: 55 MAX NUMBER OF PIZZA PIECES: 55 MAX PIECES OF A CRESCENT: 55 MAX PIECES OF CHEESE: 55 CARDS IN A CARD HOUSE: 56 DIFFERENT ARRANGEMENT OF DOMINOS: 56 UNIT FRACTIONS: 56 ANGLE BETWEEN TWO HANDS OF A CLOCK: 56 WINNING LINES IN NOUGHTS AND CROSSES: 56 BAD RESTAURANT SPREAD: 56 FIBONACCI SEQUENCE: 56 ABBREVIATIONS (5.1, 5.2, 5.3, 5.4, 5.5) 56 5.6 FACTORIAL: 56 DEFINITION: 56 TABLE OF FACTORIALS: 56 APPROXIMATION: 57 5.7 THE DAY OF THE WEEK: 57 5.8 BASIC PROBABILITY: 57 5.9 VENN DIAGRAMS: 57 COMPLEMENTARY EVENTS: 57 TOTALITY: 57 CONDITIONAL PROBABILITY: 57 UNION : 57 INDEPENDENT EVENTS: 57 MUTUALLY EXCLUSIVE: 57 BAYE’S THEOREM: 57 5.11 BASIC STATISTICAL OPERATIONS: 58 VARIANCE: 58 MEAN: 58 STANDARDIZED SCORE: 58 5.12 DISCRETE RANDOM VARIABLES: 58 STANDARD DEVIATION: 58 EXPECTED VALUE: 58 VARIANCE: 58 PROBABILITY MASS FUNCTION: 58 CUMULATIVE DISTRIBUTION FUNCTION: 58 5.13 COMMON DRVS: 58 BERNOULLI TRIAL: 58 BINOMIAL TRIAL: 58 GEOMETRIC TRIAL: 59 NEGATIVE BINOMIAL TRIAL: 59 5.14 CONTINUOUS RANDOM VARIABLES: 59 PROBABILITY DENSITY FUNCTION: 59 CUMULATIVE DISTRIBUTION FUNCTION: 59 INTERVAL PROBABILITY: 59 EXPECTED VALUE: 59 VARIANCE: 59 5.15 COMMON CRVS: 59 UNIFORM DISTRIBUTION: 59 Page 7 of 286 EXPONENTIAL DISTRIBUTION: 60 NORMAL DISTRIBUTION: 61 5.16 MULTIVARIABLE DISCRETE: 61 PROBABILITY: 61 MARGINAL DISTRIBUTION: 61 EXPECTED VALUE: 61 INDEPENDENCE: 61 COVARIANCE: 62 5.17 MULTIVARIABLE CONTINUOUS: 62 PROBABILITY: 62 MARGINAL DISTRIBUTION: 62 EXPECTED VALUE: 62 INDEPENDENCE: 62 COVARIANCE: 62 CORRELATION COEFFICIENT: 62 ABBREVIATIONS 62 PART 6: FINANCIAL 64 6.1 GENERAL FORMUALS: 64 PROFIT: 64 PROFIT MARGIN: 64 SIMPLE INTEREST: 64 COMPOUND INTEREST: 64 CONTINUOUS INTEREST: 64 ABBREVIATIONS (6.1): 64 6.2 MACROECONOMICS: 64 GDP: 64 RGDP: 64 NGDP: 64 GROWTH: 64 NET EXPORTS: 64 WORKING AGE POPULATION: 64 LABOR FORCE: 64 UNEMPLOYMENT: 64 NATURAL UNEMPLOYMENT: 64 UNEMPLOYMENT RATE: 64 EMPLOYMENT RATE: 64 PARTICIPATION RATE: 64 CPI: 64 INFLATION RATE: 64 ABBREVIATIONS (6.2) 64 PART 7: PI 66 7.1 AREA: 66 CIRCLE: 66 CYCLIC QUADRILATERAL: 66 AREA OF A SECTOR (DEGREES) 66 AREA OF A SECTOR (RADIANS) 66 AREA OF A SEGMENT (DEGREES) 66 AREA OF AN ANNULUS: 66 ELLIPSE : 66 7.2 VOLUME: 66 Page 10 of 286 9.21 ABBREVIATIONS (9.1-9.19) 88 PART 10: EXPONENTIALS & LOGARITHIMS 90 10.1 FUNDAMENTAL THEORY: 90 10.2 IDENTITIES: 90 10.3 CHANGE OF BASE: 90 10.4 LAWS FOR LOG TABLES: 90 10.5 COMPLEX NUMBERS: 91 10.6 LIMITS INVOLVING LOGARITHMIC TERMS 91 PART 11: COMPLEX NUMBERS 92 11.1 GENERAL: 92 FUNDAMENTAL: 92 STANDARD FORM: 92 POLAR FORM: 92 ARGUMENT: 92 MODULUS: 92 CONJUGATE: 92 EXPONENTIAL: 92 DE MOIVRE’S FORMULA: 92 EULER’S IDENTITY: 92 11.2 OPERATIONS: 92 ADDITION: 92 SUBTRACTION: 92 MULTIPLICATION: 92 DIVISION: 92 SUM OF SQUARES: 92 11.3 IDENTITIES: 92 EXPONENTIAL: 92 LOGARITHMIC: 92 TRIGONOMETRIC: 92 HYPERBOLIC: 93 PART 12: DIFFERENTIATION 94 12.1 GENERAL RULES: 94 PLUS OR MINUS: 94 PRODUCT RULE: 94 QUOTIENT RULE: 94 POWER RULE: 94 CHAIN RULE: 94 BLOB RULE: 94 BASE A LOG: 94 NATURAL LOG: 94 EXPONENTIAL (X): 94 FIRST PRINCIPLES: 94 12.2 EXPONETIAL FUNCTIONS: 94 12.3 LOGARITHMIC FUNCTIONS: 95 12.4 TRIGONOMETRIC FUNCTIONS: 95 12.5 HYPERBOLIC FUNCTIONS: 95 12.5 PARTIAL DIFFERENTIATION: 96 Page 11 of 286 FIRST PRINCIPLES: 96 GRADIENT: 96 TOTAL DIFFERENTIAL: 96 CHAIN RULE: 96 IMPLICIT DIFFERENTIATION: 97 HIGHER ORDER DERIVATIVES: 98 PART 13: INTEGRATION 99 13.1 GENERAL RULES: 99 POWER RULE: 99 BY PARTS: 99 CONSTANTS: 99 13.2 RATIONAL FUNCTIONS: 99 13.3 TRIGONOMETRIC FUNCTIONS (SINE): 100 13.4 TRIGONOMETRIC FUNCTIONS (COSINE): 101 13.5 TRIGONOMETRIC FUNCTIONS (TANGENT): 102 13.6 TRIGONOMETRIC FUNCTIONS (SECANT): 102 13.7 TRIGONOMETRIC FUNCTIONS (COTANGENT): 103 13.8 TRIGONOMETRIC FUNCTIONS (SINE & COSINE): 103 13.9 TRIGONOMETRIC FUNCTIONS (SINE & TANGENT): 105 13.10 TRIGONOMETRIC FUNCTIONS (COSINE & TANGENT): 105 13.11 TRIGONOMETRIC FUNCTIONS (SINE & COTANGENT): 105 13.12 TRIGONOMETRIC FUNCTIONS (COSINE & COTANGENT): 105 13.13 TRIGONOMETRIC FUNCTIONS (ARCSINE): 105 13.14 TRIGONOMETRIC FUNCTIONS (ARCCOSINE): 106 13.15 TRIGONOMETRIC FUNCTIONS (ARCTANGENT): 106 13.16 TRIGONOMETRIC FUNCTIONS (ARCCOSECANT): 106 13.17 TRIGONOMETRIC FUNCTIONS (ARCSECANT): 107 13.18 TRIGONOMETRIC FUNCTIONS (ARCCOTANGENT): 107 13.19 EXPONETIAL FUNCTIONS 107 13.20 LOGARITHMIC FUNCTIONS 109 13.21 HYPERBOLIC FUNCTIONS 111 13.22 INVERSE HYPERBOLIC FUNCTIONS 112 13.23 ABSOLUTE VALUE FUNCTIONS 113 13.24 SUMMARY TABLE 113 13.25 SQUARE ROOT PROOFS 114 13.26 CARTESIAN APPLICATIONS 117 AREA UNDER THE CURVE: 117 VOLUME: 117 VOLUME ABOUT X AXIS: 117 VOLUME ABOUT Y AXIS: 118 SURFACE AREA ABOUT X AXIS: 118 LENGTH WRT X-ORDINATES: 118 LENGTH WRT Y-ORDINATES: 118 LENGTH PARAMETRICALLY: 118 PART 14: FUNCTIONS 119 14.1 COMPOSITE FUNCTIONS: 119 14.2 MULTIVARIABLE FUNCTIONS: 119 LIMIT: 119 DISCRIMINANT: 119 CRITICAL POINTS: 119 Page 12 of 286 14.3 FIRST ORDER, FIRST DEGREE, DIFFERENTIAL EQUATIONS: 120 SEPARABLE 120 LINEAR 120 HOMOGENEOUS 120 EXACT 120 14.4 SECOND ORDER 121 HOMOGENEOUS 121 UNDETERMINED COEFFICIENTS 121 VARIATION OF PARAMETERS 121 PART 15: MATRICIES 123 15.1 BASIC PRINICPLES: 123 SIZE 123 15.2 BASIC OPERTAIONS: 123 ADDITION: 123 SUBTRACTION: 123 SCALAR MULTIPLE: 123 TRANSPOSE: 123 SCALAR PRODUCT: 123 SYMMETRY: 123 CRAMER’S RULE: 123 LEAST SQUARES SOLUTION 123 15.3 SQUARE MATRIX: 123 DIAGONAL: 124 LOWER TRIANGLE MATRIX: 124 UPPER TRIANGLE MATRIX: 124 15.4 DETERMINATE: 124 2X2 124 3X3 124 NXN 124 RULES 124 15.5 INVERSE 126 2X2: 126 3X3: 126 MINOR: 126 COFACTOR: 127 ADJOINT METHOD FOR INVERSE: 127 LEFT INVERSE: 127 RIGHT INVERSE: 127 15.6 LINEAR TRANSFORMATION 127 AXIOMS FOR A LINEAR TRANSFORMATION: 127 TRANSITION MATRIX: 127 ZERO TRANSFORMATION: 128 IDENTITY TRANSFORMATION: 128 15.7 COMMON TRANSITION MATRICIES 128 ROTATION (CLOCKWISE): 128 ROTATION (ANTICLOCKWISE): 128 SCALING: 128 SHEARING (PARALLEL TO X-AXIS): 128 SHEARING (PARALLEL TO Y-AXIS): 128 15.8 EIGENVALUES AND EIGENVECTORS 128 DEFINITIONS: 128 EIGENVALUES: 128 EIGENVECTORS: 128 Page 15 of 286 QUADRATIC APPROXIMATION: 143 CUBIC APPROXIMATION: 143 17.7 MONOTONE SERIES 143 STRICTLY INCREASING: 143 NON-DECREASING: 143 STRICTLY DECREASING: 143 NON-INCREASING: 143 CONVERGENCE: 143 17.8 RIEMANN ZETA FUNCTION 143 FORM: 143 EULER’S TABLE: 143 ALTERNATING SERIES: 144 PROOF FOR N=2: 144 17.9 SUMMATIONS OF POLYNOMIAL EXPRESSIONS 145 17.10 SUMMATIONS INVOLVING EXPONENTIAL TERMS 145 17.11 SUMMATIONS INVOLVING TRIGONOMETRIC TERMS 146 17.12 INFINITE SUMMATIONS TO PI 148 17.13 LIMITS INVOLVING TRIGONOMETRIC TERMS 148 ABBREVIATIONS 148 17.14 POWER SERIES EXPANSION 148 EXPONENTIAL: 148 TRIGONOMETRIC: 149 EXPONENTIAL AND LOGARITHM SERIES: 151 FOURIER SERIES: 152 17.15 BERNOULLI EXPANSION: 152 FUNDAMENTALLY: 152 EXPANSIONS: 153 LIST OF BERNOULLI NUMBERS: 153 PART 18: ELECTRICAL 155 18.1 FUNDAMENTAL THEORY 155 CHARGE: 155 CURRENT: 155 RESISTANCE: 155 OHM’S LAW: 155 POWER: 155 CONSERVATION OF POWER: 155 ELECTRICAL ENERGY: 155 KIRCHOFF’S VOLTAGE LAW: 155 KIRCHOFF’S CURRENT LAW: 155 AVERAGE CURRENT: 155 RMS CURRENT: 155 ∆ TO Y CONVERSION: 155 18.2 COMPONENTS 156 RESISTANCE IN SERIES: 156 RESISTANCE IN PARALLEL: 156 INDUCTIVE IMPEDANCE: 156 CAPACITOR IMPEDANCE: 156 CAPACITANCE IN SERIES: 156 CAPACITANCE IN PARALLEL: 156 VOLTAGE, CURRENT & POWER SUMMARY: 156 18.3 THEVENIN’S THEOREM 156 THEVENIN’S THEOREM: 156 MAXIMUM POWER TRANSFER THEOREM: 157 Page 16 of 286 18.4 FIRST ORDER RC CIRCUIT 157 18.5 FIRST ORDER RL CIRCUIT 157 18.6 SECOND ORDER RLC SERIES CIRCUIT 157 CALCULATION USING KVL: 157 IMPORTANT VARIABLES 157 SOLVING: 158 MODE 1: 158 MODE 2: 158 MODE 3: 159 MODE 4: 159 CURRENT THROUGH INDUCTOR: 160 PLOTTING MODES: 160 18.7 SECOND ORDER RLC PARALLEL CIRCUIT 161 CALCULATION USING KCL: 161 IMPORTANT VARIABLES 161 SOLVING: 162 18.8 LAPLANCE TRANSFORMATIONS 162 IDENTITIES: 162 PROPERTIES: 163 18.9 THREE PHASE – Y 164 LINE VOLTAGE: 164 PHASE VOLTAGE: 164 LINE CURRENT: 164 PHASE CURRENT: 164 POWER: 164 18.10 THREE PHASE – DELTA 164 LINE VOLTAGE: 164 PHASE VOLTAGE: 164 LINE CURRENT: 164 PHASE CURRENT: 164 POWER: 164 18.11 POWER 164 INSTANTANEOUS: 164 AVERAGE: 165 MAXIMUM POWER: 165 TOTAL POWER: 165 COMPLEX POWER: 165 18.12 ELECTROMAGNETICS 165 DEFINITIONS: 165 PERMEABILITY OF FREE SPACE: 165 MAGNETIC FIELD INTENSITY: 165 RELUCTANCE: 165 OHM’S LAW: 165 MAGNETIC FORCE ON A CONDUCTOR: 165 ELECTROMAGNETIC INDUCTION: 165 MAGNETIC FLUX: 165 ELECTRIC FIELD: 165 MAGNETIC FORCE ON A PARTICLE: 165 PART 19: GRAPH THEORY 166 19.1 FUNDAMENTAL EXPLANATIONS 166 LIST OF VERTICES: 166 LIST OF EDGES: 166 SUBGAPHS: 166 Page 17 of 286 DEGREE OF VERTEX: 166 DISTANCE: 166 DIAMETER: 166 TOTAL EDGES IN A SIMPLE BIPARTITE GRAPH: 166 TOTAL EDGES IN K-REGULAR GRAPH: 166 19.2 FACTORISATION: 166 1 FACTORISATION: 166 1 FACTORS OF A nnK , BIPARTITE GRAPH: 166 1 FACTORS OF A nK2 GRAPH: 166 19.3 VERTEX COLOURING 166 CHROMATIC NUMBER: 167 UNION/INTERSECTION: 167 EDGE CONTRACTION: 167 COMMON CHROMATIC POLYNOMIALS: 167 19.4 EDGE COLOURING: 167 COMMON CHROMATIC POLYNOMIALS: 167 PART 98: LIST OF DISTRIBUTION FUNCTIONS 168 5.18 FINITE DISCRETE DISTRIBUTIONS 168 BERNOULLI DISTRIBUTION 168 RADEMACHER DISTRIBUTION 168 BINOMIAL DISTRIBUTION 169 BETA-BINOMIAL DISTRIBUTION 170 DEGENERATE DISTRIBUTION 171 DISCRETE UNIFORM DISTRIBUTION 172 HYPERGEOMETRIC DISTRIBUTION 174 POISSON BINOMIAL DISTRIBUTION 175 FISHER'S NONCENTRAL HYPERGEOMETRIC DISTRIBUTION (UNIVARIATE) 175 FISHER'S NONCENTRAL HYPERGEOMETRIC DISTRIBUTION (MULTIVARIATE) 176 WALLENIUS' NONCENTRAL HYPERGEOMETRIC DISTRIBUTION (UNIVARIATE) 176 WALLENIUS' NONCENTRAL HYPERGEOMETRIC DISTRIBUTION (MULTIVARIATE) 177 5.19 INFINITE DISCRETE DISTRIBUTIONS 177 BETA NEGATIVE BINOMIAL DISTRIBUTION 177 MAXWELL–BOLTZMANN DISTRIBUTION 178 GEOMETRIC DISTRIBUTION 179 LOGARITHMIC (SERIES) DISTRIBUTION 181 NEGATIVE BINOMIAL DISTRIBUTION 182 POISSON DISTRIBUTION 183 CONWAY–MAXWELL–POISSON DISTRIBUTION 184 SKELLAM DISTRIBUTION 185 YULE–SIMON DISTRIBUTION 185 ZETA DISTRIBUTION 187 ZIPF'S LAW 188 ZIPF–MANDELBROT LAW 189 5.20 BOUNDED INFINITE DISTRIBUTIONS 189 ARCSINE DISTRIBUTION 189 BETA DISTRIBUTION 191 LOGITNORMAL DISTRIBUTION 193 CONTINUOUS UNIFORM DISTRIBUTION 194 IRWIN-HALL DISTRIBUTION 195 KUMARASWAMY DISTRIBUTION 196 RAISED COSINE DISTRIBUTION 197 TRIANGULAR DISTRIBUTION 198 Page 20 of 286 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 21 of 286 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 22 of 286 1.4 UNIVERSAL CONSTANTS: Quantity Symbol Value Relative Standard Uncertainty speed of light in vacuum 299 792 458 m·s−1 defined Newtonian constant of gravitation 6.67428(67)×10−11 m3·kg−1·s−2 1.0 × 10−4 Planck constant 6.626 068 96(33) × 10 −34 J·s 5.0 × 10−8 reduced Planck constant 1.054 571 628(53) × 10−34 J·s 5.0 × 10−8 1.5 ELECTROMAGNETIC CONSTANTS: Quantity Symbol Value (SI units) Relative Standard Uncertainty magnetic constant (vacuum permeability) 4π × 10−7 N·A−2 = 1.256 637 061... × 10−6 N·A−2 defined electric constant (vacuum permittivity) 8.854 187 817... × 10−12 F·m−1 defined characteristic impedance of vacuum 376.730 313 461... Ω defined Coulomb's constant 8.987 551 787... × 109 N·m²·C−2 defined elementary charge 1.602 176 487(40) × 10−19 C 2.5 × 10−8 Bohr magneton 927.400 915(23) × 10−26 J·T−1 2.5 × 10−8 conductance quantum 7.748 091 7004(53) × 10−5 S 6.8 × 10−10 inverse conductance quantum 12 906.403 7787(88) Ω 6.8 × 10−10 Josephson constant 4.835 978 91(12) × 1014 Hz·V−1 2.5 × 10−8 magnetic flux quantum 2.067 833 667(52) × 10−15 Wb 2.5 × 10−8 nuclear magneton 5.050 783 43(43) × 10−27 J·T−1 8.6 × 10−8 von Klitzing constant 25 812.807 557(18) Ω 6.8 × 10 −10 1.6 ATOMIC AND NUCLEAR CONSTANTS: Page 25 of 286 Josephson constant 1014 Hz·V−1 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 kg·mol−1 defined 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 Page 26 of 286 PART 2: MATHEMTAICAL SYMBOLS 2.1 BASIC MATH SYMBOLS Symbol Symbol Name Meaning / definition Example = equals sign equality 5 = 2+3 ≠ not equal sign inequality 5 ≠ 4 > strict inequality greater than 5 > 4 < strict inequality less than 4 < 5 ≥ inequality greater than or equal to 5 ≥ 4 ≤ inequality less than or equal to 4 ≤ 5 ( ) parentheses calculate expression inside first 2 × (3+5) = 16 [ ] brackets calculate expression inside first [(1+2)*(1+5)] = 18 + plus sign addition 1 + 1 = 2 − minus sign subtraction 2 − 1 = 1 ± plus - minus both plus and minus operations 3 ± 5 = 8 and -2 ∓ minus - plus both minus and plus operations 3 ∓ 5 = -2 and 8 * asterisk multiplication 2 * 3 = 6 × times sign multiplication 2 × 3 = 6 · multiplication dot multiplication 2 · 3 = 6 ÷ division sign / obelus division 6 ÷ 2 = 3 / division slash division 6 / 2 = 3 – horizontal line division / fraction mod modulo remainder calculation 7 mod 2 = 1 . period decimal point, decimal separator 2.56 = 2+56/100 a b power exponent 23 = 8 a^b caret exponent 2 ^ 3 = 8 √a square root √a · √a = a √9 = ±3 3√a cube root 3√8 = 2 4√a forth root 4√16 = ±2 n√a n-th root (radical) for n=3, n√8 = 2 % percent 1% = 1/100 10% × 30 = 3 ‰ per-mille 1‰ = 1/1000 = 0.1% 10‰ × 30 = 0.3 ppm per-million 1ppm = 1/1000000 10ppm × 30 = 0.0003 ppb per-billion 1ppb = 1/1000000000 10ppb × 30 = 3×10-7 ppt per-trillion 1ppb = 10-12 10ppb × 30 = 3×10-10 2.2 GEOMETRY SYMBOLS Symbol Symbol Name Meaning / definition Example ∠ angle formed by two rays ∠ABC = 30º ∡ measured angle ∡ABC = 30º ∢ spherical angle ∢AOB = 30º ∟ right angle = 90º α = 90º º degree 1 turn = 360º α = 60º ´ arcminute 1º = 60´ α = 60º59' Page 27 of 286 ´´ arcsecond 1´ = 60´´ α = 60º59'59'' AB line line from point A to point B ray line that start from point A | perpendicular perpendicular lines (90º angle) AC | BC || parallel parallel lines AB || CD ≅ congruent to equivalence of geometric shapes and size ∆ABC ≅ ∆XYZ ~ similarity same shapes, not same size ∆ABC ~ ∆XYZ ∆ triangle triangle shape ∆ABC ≅ ∆BCD | x-y | distance distance between points x and y | x-y | = 5 π pi constant π = 3.141592654... is the ratio between the circumference and diameter of a circle c = π·d = 2·π·r rad radians radians angle unit 360º = 2π rad grad grads grads angle unit 360º = 400 grad 2.3 ALGEBRA SYMBOLS Symbol Symbol Name Meaning / definition Example x x variable unknown value to find when 2x = 4, then x = 2 ≡ equivalence identical to ≜ equal by definition equal by definition := equal by definition equal by definition ~ approximately equal weak approximation 11 ~ 10 ≈ approximately equal approximation sin(0.01) ≈ 0.01 ∝ proportional to proportional to f(x) ∝ g(x) ∞ lemniscate infinity symbol ≪ much less than much less than 1 ≪ 1000000 ≫ much greater than much greater than 1000000 ≫ 1 ( ) parentheses calculate expression inside first 2 * (3+5) = 16 [ ] brackets calculate expression inside first [(1+2)*(1+5)] = 18 { } braces set ⌊x⌋ floor brackets rounds number to lower integer ⌊4.3⌋= 4 ⌈x⌉ ceiling brackets rounds number to upper integer ⌈4.3⌉= 5 x! exclamation mark factorial 4! = 1*2*3*4 = 24 | x | single vertical bar absolute value | -5 | = 5 f (x) function of x maps values of x to f(x) f (x) = 3x+5 (f ◦g) function composition (f ◦g) (x) = f (g(x)) f (x)=3x, g(x)=x-1 ⇒(f ◦g)(x)=3(x-1) (a,b) open interval (a,b) ≜ {x | a < x < b} x ∈ (2,6) [a,b] closed interval [a,b] ≜ {x | a ≤ x ≤ b} x ∈ [2,6] ∆ delta change / difference ∆t = t1 - t0 ∆ discriminant ∆ = b2 - 4ac ∑ sigma summation - sum of all values in range of series ∑ xi= x1+x2+...+xn Page 30 of 286 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 31 of 286 aleph infinite cardinality א Ø empty set Ø = { } C = {Ø} U universal set set of all possible values ℕ0 natural numbers set (with zero) ℕ0 = {0,1,2,3,4,...} 0 ∈ ℕ0 ℕ1 natural numbers set (without zero) ℕ1 = {1,2,3,4,5,...} 6 ∈ ℕ1 ℤ integer numbers set ℤ = {...-3,-2,-1,0,1,2,3,...} -6 ∈ ℤ ℚ rational numbers set ℚ = {x | x=a/b, a,b∈ℕ} 2/6 ∈ ℚ ℝ real numbers set ℝ = {x | -∞ < x <∞} 6.343434 ∈ ℝ ℂ complex numbers set ℂ = {z | z=a+bi, -∞<a<∞, -∞<b<∞} 6+2i ∈ ℂ 2.8 LOGIC SYMBOLS Symbol Symbol Name Meaning / definition Example · and and x · y ^ caret / circumflex and x ^ y & ampersand and x & y + plus or x + y ∨ reversed caret or x ∨ y | vertical line or x | y x' single quote not - negation x' x bar not - negation x ¬ not not - negation ¬ x ! exclamation mark not - negation ! x ⊕ circled plus / oplus exclusive or - xor x ⊕ y ~ tilde negation ~ x ⇒ implies ⇔ equivalent if and only if ∀ for all ∃ there exists ∄ there does not exists ∴ therefore ∵ because / since Page 32 of 286 2.9 CALCULUS & ANALYSIS SYMBOLS Symbol Symbol Name Meaning / definition Example limit limit value of a function ε epsilon represents a very small number, near zero ε → 0 e e constant / Euler's number e = 2.718281828... e = lim (1+1/x)x , x→∞ y ' derivative derivative - Leibniz's notation (3x3)' = 9x2 y '' second derivative derivative of derivative (3x3)'' = 18x y(n) nth derivative n times derivation (3x3)(3) = 18 derivative derivative - Lagrange's notation d(3x3)/dx = 9x2 second derivative derivative of derivative d2(3x3)/dx2 = 18x nth derivative n times derivation time derivative derivative by time - Newton notation time second derivative derivative of derivative partial derivative ∂(x2+y2)/∂x = 2x ∫ integral opposite to derivation ∬ double integral integration of function of 2 variables ∭ triple integral integration of function of 3 variables ∮ closed contour / line integral ∯ closed surface integral ∰ closed volume integral [a,b] closed interval [a,b] = {x | a ≤ x ≤ b} (a,b) open interval (a,b) = {x | a < x < b} i imaginary unit i ≡ √-1 z = 3 + 2i z* complex conjugate z = a+bi → z*=a-bi z* = 3 + 2i z complex conjugate z = a+bi → z = a-bi z = 3 + 2i ∇ nabla / del gradient / divergence operator ∇f (x,y,z) vector unit vector x * y convolution y(t) = x(t) * h(t) ℒ Laplace transform F(s) = ℒ{f (t)} ℱ Fourier transform X(ω) = ℱ{f (t)} δ delta function Page 35 of 286 s=semi-perimeter SA=Surface Area V=Volume w=width Page 36 of 286 PART 4: ALGEBRA 4.1 POLYNOMIAL FORMULA: Qudaratic: Where 02 =++ cbxax , a acbb x 2 42 −±−= Cubic: Where 023 =+++ dcxbxax , Let, a b yx 3 −= a a bc a b d y a a b c y a a bc a b d y a a b c y a bc a b dy a b cay d a b yc a b yb a b ya       −+ −=       − + =       −+ +       − + =      −++      −+ =+      −+      −+      −∴ 327 2 3 0 327 2 3 0 327 2 3 0 333 2 32 3 2 32 3 2 32 3 23 Let, ( )1...33 2 st a a b c A =       − = Let, ( )2...327 2 33 2 3 ts a a bc a b d B −=       −+ −= 333 3 3 tsstyy BAyy −=+ =+∴ Solution to the equation = ts − Let, tsy −= ( ) ( ) ( ) ( ) 33223223 333 3333 3 tssttststtss tstsstts −=−+−+− −=−+−∴ Solving (1) for s and substituting into (2) yields: Let, 3tu = 0 27 3 2 =−+∴ ABuu Page 37 of 286 3 3 2 3 3 2 2 3 2 2 27 4 2 27 4 2 4 27 1 0: A BB ut A BB u u A B uuie +±− ==∴ +±− = −±− = −= = = =++ α αγββ γ β α γβα Substituting into (2) yields: 3 3 3 3 2 3 3 3 2 33 2 27 4 2 27 4               +±− +=∴               +±− +=+= A BB Bs A BB BtBs Now, tsy −= 3 3 2 3 3 3 3 2 2 27 4 2 27 4 A BB A BB By +±− −               +±− +=∴ Now, a b yx 3 −= a b A BB A BB Bx 32 27 4 2 27 4 3 3 2 3 3 3 3 2 −                 +±− −               +±− += Where, a a b c A       − = 3 2 & a a bc a b d B       −+ −= 327 2 2 3 Page 40 of 286 4.3 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 4.4 SUMATION MANIPULATIONS: , where C is a constant 4.5 COMMON FUNCTIONS: Constant Function: y=a or f (x)=a Page 41 of 286 Graph is a horizontal line passing through the point (0,a) x=a Graph is a vertical line passing through the point (a,0) Line/Linear Function: cmxy += Graph is a line with point (0,c) and slope m. Where the gradient is between any two points ),(&),( 2211 yxyx 12 12 xx yy run rise m − −== Also, )( 11 xxmyy −+= The equation of the line with gradient m .and passing through the point ),( 11 yx . Parabola/Quadratic Function: khxay +−= 2)( The graph is a parabola that opens up if a > 0 or down if a < 0 and has a vertex at (h,k). cbxaxy ++= 2 The graph is a parabola that opens up if a > 0 or down if a < 0 and has a vertex at             −− a b f a b 2 , 2 . cbyayx ++= 2 The graph is a parabola that opens right if a > 0 or left if a < 0 and has a vertex at             −       − a b a b g 2 , 2 . This is not a function. Circle: ( ) ( ) 222 rkyhx =−+− Graph is a circle with radius r and center (h,k). Ellipse: ( ) ( ) 1 2 2 2 2 =−+− b ky a hx Graph is an ellipse with center (h,k) with vertices a units right/left from the center and vertices b units up/down from the center. Page 42 of 286 Hyperbola: ( ) ( ) 1 2 2 2 2 =−−− b ky a hx Graph is a hyperbola that opens left and right, has a center at (h,k) , vertices a units left/right of center and asymptotes that pass through center with slope a b± . ( ) ( ) 1 2 2 2 2 =−−− a hx b ky 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± . 4.6 LINEAR ALGEBRA: Vector Space Axioms: Let V be a set on which addition and scalar multiplication are defined (this means that if u and v are objects in V and c is a scalar then we’ve defined and cu in some way). If the following axioms are true for all objects u, v, and w in V and all scalars c and k then V is called a vector space and the objects in V are called vectors. (a) is in V This is called closed under addition. (b) cu is in V This is called closed under scalar multiplication. (c) (d) (e) There is a special object in V, denoted 0 and called the zero vector, such that for all u in V we have . (f) For every u in V there is another object in V, denoted and called the negative of u, such that . (g) (h) (i) (j) 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 45 of 286 Gram-Schmidt Process: This finds an orthonormal basis recursively. In a basis { }nuuuuB ,...,,, 321= 1 1 1 ^ 1 11 q q qv uq == = Next vector needs to be orthogonal to 1v , 11222 , vvuuq ><−= Similarly n n nn nnnn q q qv vvuvvuvvuuq vvuvvuuq == ><−−><−><−= ><−><−= ^ 3223113 22311333 ,...,, ,, Coordinate Vector: If nnecececv +++= ...2211             = n B c c c v ... 2 1 For a fixed basis (usually the standard basis) there is 1 to 1 correspondence between vectors and coordinate vectors. Hence, a basis can be found in Rn and then translated back into the general vector space. Dimension: Real Numbers nn =ℜ )dim( Polynomials 1)dim( += nPn Matricis qpM qp ×=)dim( , If you know the dimensions and you are checking if a set forms a basis of the vector space, only Linear Independence or Span needs to be checked. 4.7 COMPLEX VECTOR SPACES: Form:             + + + = nn n iba iba iba C ... 22 11 Dot Product: nn vuvuvuvu _ 2 _ 21 _ 1 ... +++=• Where: Page 46 of 286 0 iff 0 0 ),( )( ==• ≥• ∈•=• •+•=•+ •≠•=• uuu uu Csvusvsu wvwuwvu uvuvvu Inner Product: vuvud uuuuuu n −= +++=•= ),( ... 22 2 2 1 Orthogonal if 0=• vu Parallel if Cssvu ∈= , 4.8 LINEAR TRANSITIONS & TRANSFORMATIONS: Transition Matrix: From 1 vector space to another vector space )(...)()()()( )...()( 332211 332211 nn nn uTcuTcuTcuTcuT ucucucucTuT ++++= ++++= Nullity(T)+Rank(T)=Dim(V)=Columns(T) Change of Basis Transition Matrix: BBBB BBBB vCv vMMv '' 1 '' = = − For a general vector space with the standard basis: { }nsssS ,...,, 21= [ ] [ ]SmSB SnSB uuM vvM )(|...|)( )(|...|)( 1' 1 = = Transformation Matrix: From 1 basis to another basis { } { } { } { } ( ) ( ) ( )[ ] BBBB BnBB m m n n ACCA vTvTvTA uuuuB uuuuspanU vvvvB vvvvspanV ' 1 ' 22221 3212 321 3211 321 ' )(|...|)(|)( ,...,,, ),...,,,( ,...,,, ),...,,,( −= = = = = = 4.9 INNER PRODUCTS: Definition: An extension of the dot product into a general vector space. Axioms: 1. >>=<< uvvu ,, 2. ><+>>=<+< wuvuwvu ,,, Page 47 of 286 3. ><>=< vukvku ,, 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 +=+ 4.10 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 Page 50 of 286 185,368 203,432 4,238,984 4,314,616 11,252,648 12,101,272 196,724 202,444 4,246,130 4,488,910 11,498,355 12,024,045 280,540 365,084 4,259,750 4,445,050 11,545,616 12,247,504 308,620 389,924 4,482,765 5,120,595 11,693,290 12,361,622 319,550 430,402 4,532,710 6,135,962 11,905,504 13,337,336 356,408 399,592 4,604,776 5,162,744 12,397,552 13,136,528 437,456 455,344 5,123,090 5,504,110 12,707,704 14,236,136 469,028 486,178 5,147,032 5,843,048 13,671,735 15,877,065 503,056 514,736 5,232,010 5,799,542 13,813,150 14,310,050 522,405 525,915 5,357,625 5,684,679 13,921,528 13,985,672 600,392 669,688 5,385,310 5,812,130 14,311,688 14,718,712 609,928 686,072 5,459,176 5,495,264 14,426,230 18,087,818 624,184 691,256 5,726,072 6,369,928 14,443,730 15,882,670 635,624 712,216 5,730,615 6,088,905 14,654,150 16,817,050 643,336 652,664 5,864,660 7,489,324 15,002,464 15,334,304 667,964 783,556 6,329,416 6,371,384 15,363,832 16,517,768 726,104 796,696 6,377,175 6,680,025 15,938,055 17,308,665 802,725 863,835 6,955,216 7,418,864 16,137,628 16,150,628 879,712 901,424 6,993,610 7,158,710 16,871,582 19,325,698 898,216 980,984 7,275,532 7,471,508 17,041,010 19,150,222 947,835 1,125,765 7,288,930 8,221,598 17,257,695 17,578,785 998,104 1,043,096 7,489,112 7,674,088 17,754,165 19,985,355 1,077,890 1,099,390 7,577,350 8,493,050 17,844,255 19,895,265 1,154,450 1,189,150 7,677,248 7,684,672 17,908,064 18,017,056 1,156,870 1,292,570 7,800,544 7,916,696 18,056,312 18,166,888 1,175,265 1,438,983 7,850,512 8,052,488 18,194,715 22,240,485 1,185,376 1,286,744 8,262,136 8,369,864 18,655,744 19,154,336 1,280,565 1,340,235 Sociable Numbers: Sociable numbers are generalisations of amicable numbers where a sequence of numbers each of whose numbers is the sum of the factors of the preceding number, excluding the preceding number itself. The sequence must be cyclic, eventually returning to its starting point . List of Sociable Numbers: C4s 1264460 1547860 1727636 1305184 2115324 3317740 3649556 2797612 2784580 3265940 3707572 Page 51 of 286 3370604 4938136 5753864 5504056 5423384 7169104 7538660 8292568 7520432 C5 Poulet 1918 5D 12496 2^4*11*71 14288 2^4*19*47 15472 2^4*967 14536 2^3*23*79 14264 2^3*1783 C6 Moews&Moews 1992 11D 21548919483 3^5*7^2*13*19*17*431 23625285957 3^5*7^2*13*19*29*277 24825443643 3^2*7^2*13*19*11*20719 26762383557 3^4*7^2*13*19*27299 25958284443 3^2*7^2*13*19*167*1427 23816997477 3^2*7^2*13*19*218651 C6 Moews&Moews 1995 11D/12D 90632826380 2^2*5*109*431*96461 101889891700 2^2*5^2*31*193*170299 127527369100 2^2*5^2*31*181*227281 159713440756 2^2*31*991*1299709 129092518924 2^2*31*109*9551089 106246338676 2^2*17*25411*61487 C6 Needham 2006 13D 1771417411016 2^3*11*20129743307 1851936384424 2^3*7*1637*20201767 2118923133656 2^3*7*863*43844627 2426887897384 2^3*59*5141711647 2200652585816 2^3*43*1433*4464233 2024477041144 2^3*253059630143 C6 Needham 2006 13D 3524434872392 2^3*7*17*719*5149009 4483305479608 2^3*89*6296777359 4017343956392 2^3*13*17*3019*752651 4574630214808 2^3*607*6779*138967 4018261509992 2^3*31*59*274621481 3890837171608 2^3*61*22039*361769 Page 52 of 286 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 55 of 286 PART 5: COUNTING TECHNIQUES & PROBABILITY 5.1 2D Triangle Number ( ) 2 1+= nnTn 1 2 −+= nn TTn Square Number 2nTn = Pentagonal Number ( ) 2 13 −= nnTn 5.2 3D Tetrahedral Number 6 23 23 nnn Tn ++= Square Pyramid Number 6 32 23 nnn Tn ++= 5.3 PERMUTATIONS Permutations: !n= Permutations (with repeats): ( ) ( ) ...!! ! ×× = groupBgroupA n 5.4 COMBINATIONS Ordered Combinations: ( )! ! pn n − = Unordered Combinations: ( )!! ! pnp n p n − =      = Ordered Repeated Combinations: pn= Unordered Repeated Combinations: ( ) ( )!1! !1 −× −+= np np Grouping: !!...!! ! ... 3213 21 2 1 1 rnnnn n n nnn n nn n n =      −−       −       = 5.5 MISCELLANEOUS: Total Number of Rectangles and Squares from a a x b rectangle: ∑ ×= ba TT Number of Interpreters: 1−= LT Max number of pizza pieces: ( ) 1 2 1 ++= cc Max pieces of a crescent: ( ) 1 2 3 ++= cc Max pieces of cheese: 1 6 53 ++= cc Page 56 of 286 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 57 of 286 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: ∑ = 1P 5.9 VENN DIAGRAMS: Complementary Events: ( ) ( )APAP =−1 Totality: ∑ = = m i ii BPBAPAP 1 )()|()( )'()()( BAPBAPAP ∩+∩= Conditional Probability: ( ) ( )( )BP BAP BAP ∩=| ( ) ( ) ( )BAPBPBAP |⋅=∩ Union : ( ) ( ) ( ) ( )BAPBPAPBAP ∩−+=∪ Independent Events: ( ) ( ) ( )BPAPBAP ⋅=∩ ( ) ( ) ( ) ( ) ( )BPAPBPAPBAP ⋅−+=∪ ( ) ( )BPABP =| Mutually Exclusive: ( ) 0=∩ BAP ( ) )(' APBAP =∩ ( ) ( ) ( )BPAPBAP +=∪ ( ) ( )'' BPBAP =∪ Baye’s Theorem: Page 60 of 286 CDF: ∫ ∞−      > ≤≤ − − < == x bx bxa ab ax ax dxxfxF 1 0 )()( Expected Value: 2 ba += Variance: ( ) 12 2ab −= Exponential Distribution: Declaration: )(~ λlExponentiaX PDF:    ≥ < = − 0 00 )( xe x xf xλλ Page 61 of 286 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 MULTIVARIABLE DISCRETE: Probability: ),(),( yxfyYxXP === ∑=≤≤ ),(),( yxfyYxXP over all values of x & y Marginal Distribution: ( ) ∑ ∑ = = x iY y iX yxfyf yxfxf ),()( ),( Expected Value: ∑∑ ∑ ∑ ××= ×= ×= x y YX y Y x X yxfyxYXE yfyYE xfxXE ),(],[ )(][ )(][ , Independence: )()(),( yfxfyxf YX ×= Page 62 of 286 Covariance: ][][],[ YEXEYXECov ×−= 5.17 MULTIVARIABLE CONTINUOUS: Probability: ∫ ∫ ∫ ∞− ∞− ∞− =<∞<<−∞=< =≤≤ y Y y x dyyfyYXPyYP dxdyyxfyYxXP )(),()( ),(),( Marginal Distribution: ( ) ∫= b a X dyyxfxf ),( where a & b are bounds of y ∫= b a Y dxyxfyf ),()( where a & b are bounds of x Expected Value: ∫ ∫ ∫ ∫ ∞ ∞− ∞ ∞− ∞ ∞− ∞ ∞− ××= ×= ×= dxdyyxfyxYXE dyyfyYE dxxfxXE YX Y X ),(],[ )(][ )(][ , Independence: )()(),( yfxfyxf YX ×= Covariance: ][][],[ YEXEYXECov ×−= Correlation Coefficient: YX YX YXCov σσ ρ ),(, = 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 z = Standardized Score Page 65 of 286 AE=Aggregate Expenditure C=Consumption CPI=Consumer Price Index E=Employed G=Government I=Investment LF=Labor Force M=Imports NGDP=Nominal GDP NUE=Natural Unemployment NX=Net Export P=Participation RGDP=Real GDP (Price is adjusted to base year) UE=Unemployed WAP=Working Age Population X=Exports Y=GDP Page 66 of 286 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 67 of 286 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 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 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 Integrals: Page 70 of 286 a=side ‘a’ B=Angle ‘B’ b=side ‘b’ B=Angle ‘B’ c=side ‘c’ C=circumference d=diameter d=side ‘d’ h=shortest length from the center to the chord r=radius r1=radius 1 ( 21 rr < ) r2=radius 2 ( 32 rr < ) r3=radius 3 l=length n=number of sides P=perimeter Q=central angle s=semi-perimeter w=width w=length of chord from r1 7.9 CRESCENT GEOMETRY: Area of a lunar crescent: cdA π 4 1= Area of an eclipse crescent:                             −+ +               −+ −−                             −+ +               −+ −= −− −− 2 2 cos2sin 360 2 cos2 2 2 cos2sin 360 2 cos2 222 1 222 1 2 222 1 222 1 2 wl blw wl blw b wl blw wl blw wA π π π π 7.10 ABBREVIATIONS (7.9): A=Area b=radius of black circle c=width of the crescent d=diameter l=distance between the centres of the circles w=radius of white circle Page 71 of 286 PART 8: PHYSICS 8.1 MOVEMENT: Stopping distance: a v s 2 2 − = Centripetal acceleration: r v a 2 = Centripetal force: r mv maFC 2 == Dropping time : g h t 2= Force: 2 3 2 2 1       − = c v ma F Kinetic Energy: 2 2 1 mvEk = Maximum height of a cannon: ( ) g u h 2sinθ= Pendulum swing time: g l t π2= Potential Energy: mghE p = Range of a cannon: ( ) ( )θθθ cossin2cos u g u uts ×== Time in flight of a cannon: g u t θsin2= Universal Gravitation: 2 21 r mm GF = ABBREVIATIONS (8.1): a=acceleration (negative if retarding) c=speed of light ( 8103× ms-1) Ek=Kinetic Energy Ep=potential energy F=force g=gravitational acceleration (≈9.81 on Earth) G=gravitational constant = 111067.6 −× h=height l=length of a pendulum m=mass m1=mass 1 m2=mass 2 Page 72 of 286 r=radius r=distance between two points s=distance t=time u=initial speed v=final speed θ=the angle 8.2 CLASSICAL MECHANICS: Newton’s Laws: First law: If an object experiences no net force, then its velocity is constant; the object is either at rest (if its velocity is zero), or it moves in a straight line with constant speed (if its velocity is nonzero). Second law: The acceleration a of a body is parallel and directly proportional to the net force F acting on the body, is in the direction of the net force, and is inversely proportional to the mass m of the body, i.e., F = ma. Third law: When two bodies interact by exerting force on each other, these forces (termed the action and the reaction) are equal in magnitude, but opposite in direction. Inertia: Page 75 of 286 Right circular cone with radius r, height h and mass m Torus of tube radius a, cross-sectional radius b and mass m. About a diameter: About the vertical axis: Ellipsoid (solid) of semiaxes a, b, and c with axis of rotation a and mass m Thin rectangular plate of height h and of width w and mass m (Axis of rotation at the end of the plate) Thin rectangular plate of height h and of width w and mass m Solid cuboid of height h, width w, and depth d, and mass m Page 76 of 286 Solid cuboid of height D, width W, and length L, and mass m with the longest diagonal as the axis. Plane polygon with vertices , , , ..., and mass uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin. Infinite disk with mass normally distributed on two axes around the axis of rotation (i.e. Where : is the mass-density as a function of x and y). Velocity and Speed: t P vAVE ∆ ∆= Acceleration: t V aAVE ∆ ∆= Trajectory (Displacement): Page 77 of 286 Kinetic Energy: Centripetal Force: Circular Motion: , or , Angular Momentum: Page 80 of 286 PART 9: TRIGONOMETRY 9.1 CONVERSIONS: Degrees 30° 60° 120° 150° 210° 240° 300° 330° Radians Grads 33⅓ grad 66⅔ grad 133⅓ grad 166⅔ grad 233⅓ grad 266⅔ grad 333⅓ grad 366⅔ grad Degrees 45° 90° 135° 180° 225° 270° 315° 360° Radians Grads 50 grad 100 grad 150 grad 200 grad 250 grad 300 grad 350 grad 400 grad 9.2 BASIC RULES: θ θθ cos sin tan = Sin Rule: C c B b A a sinsinsin == or c C b B a A sinsinsin == Cos Rule: bc acb A 2 cos 222 −+= or Abccba cos2222 −+= Tan Rule: Auxiliary Angle: Pythagoras Theorem: 222 cba =+ Page 81 of 286 9.3 RECIPROCAL FUNCTIONS θ θ θ θ θ θ θ θ sin cos tan 1 cot sin 1 csc cos 1 sec == = = 9.4 BASIC IDENTITES: Pythagorean Identity: 9.5 IDENTITIES (SINΘ): • • • • • Page 82 of 286 • 9.6 IDENTITIES (COSΘ): • • • • • • 9.7 IDENTITIES (TANΘ): • • • • • • 9.8 IDENTITIES (CSCΘ): • • • • • • 9.9 IDENTITIES (COTΘ): • Page 85 of 286 9.14 POWER REDUCTION: Sine: If n is even: If n is odd: Cosine: If n is even: If n is odd: Page 86 of 286 Sine & Cosine: 9.15 PRODUCT TO SUM: 9.16 SUM TO PRODUCT: 9.17 HYPERBOLIC EXPRESSIONS: Hyperbolic sine: Hyperbolic cosine: Hyperbolic tangent: Hyperbolic cotangent: Page 87 of 286 Hyperbolic secant: Hyperbolic cosecant: 9.18 HYPERBOLIC RELATIONS: 9.19 MACHIN-LIKE FORMULAE: Form: Formulae: Page 90 of 286 PART 10: EXPONENTIALS & LOGARITHIMS 10.1 FUNDAMENTAL THEORY: 10.2 IDENTITIES: 10.3 CHANGE OF BASE: 10.4 LAWS FOR LOG TABLES: Page 91 of 286 10.5 COMPLEX NUMBERS: 10.6 LIMITS INVOLVING LOGARITHMIC TERMS Page 92 of 286 PART 11: COMPLEX NUMBERS 11.1 GENERAL: Fundamental: 12 −=i Standard Form: biaz += Polar Form: ( )θθθ sincos irrcisz +== Argument: ( ) θ=zarg , where a b=θtan Modulus: ( ) 22mod babiazrz +=+=== Conjugate: biaz −= Exponential: θierz ⋅= De Moivre’s Formula:       += = n k cisrz rcisz nn πθ θ 211 , k=0,1,…,(n-1) Euler’s Identity: 01 =+πie (Special Case when n=2) 0 1 0 2 =∑ − = n k n ki e π (Generally) 11.2 OPERATIONS: Addition: d)i + (b + c) + (a = di) + (c + bi) + (a Subtraction: d)i. - (b + c) - (a = di) + (c - bi) + (a Multiplication: ad)i. + (bc + bd) - (ac = bdi + adi + bci + ac = di) + bi)(c + (a 2 Division: i. dc ad - bc + dc bd ac = di) - di)(c + (c bd + adi - bci + ac di) - di)(c + (c di) - bi)(c + (a = di) + (c bi) + (a 2222       +       + += Sum of Squares: 11.3 IDENTITIES: Exponential: Logarithmic: Trigonometric: Page 95 of 286 12.3 LOGARITHMIC FUNCTIONS: 12.4 TRIGONOMETRIC FUNCTIONS: 12.5 HYPERBOLIC FUNCTIONS: Page 96 of 286 12.5 PARTIAL DIFFERENTIATION: First Principles: ie: Gradient: Total Differential: Chain Rule: Case 1. Suppose F = F(u,v) and u = u(x,y), v = v(2,y). Then F is also a function of gv and y and ar akan, akae Go ax guaz avar ou OP oF aFau ,aFav aF _aFau , oPav ou ouady dvay aF @Fdu aF dv dé Qudt Qu dt (Note the ‘full’ derivative here) af dFdu gx duar ar dFou ay dudy Implicit Differentiation: Page 97 of 286