ALL IN ONE MATHEMATICS CHEAT SHEET V2.10 Euler’s Identity: 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 Page 1 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 2 of 330 NEW: Mathematical Constants UPDATED: Applications of Integration UPDATED: Basic Statistical Operations UPDATED: Pi UPDATED: Identities Between Relationships UPDATED: Vector Space Axioms 2.9. 4/03/2012 UPDATED: Prime Numbers UPDATED: Martricies 2.10. 9/04/2012 NEW: Boolean Algebra NEW: Functions of Random Variables NEW: Transformation of the Joint Density UPDATED: Venn Diagrams: UPDATED: Basic Statistical Operations: UPDATED: Discrete Random Variables: UPDATED: Common DRVs: UPDATED: Undetermined Coefficients UPDATED: Variation of Parameters Page 3 of 330 CONTENTS REVISION HISTORY 2 CONTENTS 4 PART 1: PHYSICAL CONSTANTS 26 1.1 SI PREFIXES: 26 1.2 SI BASE UNITS: 26 1.3 SI DERIVED UNITS: 27 1.4 UNIVERSAL CONSTANTS: 28 1.5 ELECTROMAGNETIC CONSTANTS: 28 1.6 ATOMIC AND NUCLEAR CONSTANTS: 28 1.7 PHYSICO-CHEMICAL CONSTANTS: 29 1.8 ADOPTED VALUES: 30 1.9 NATURAL UNITS: 31 1.10 MATHEMATICAL CONSTANTS: 31 PART 2: MATHEMTAICAL SYMBOLS 33 2.1 BASIC MATH SYMBOLS 33 2.2 GEOMETRY SYMBOLS 33 2.3 ALGEBRA SYMBOLS 34 2.4 LINEAR ALGEBRA SYMBOLS 35 2.5 PROBABILITY AND STATISTICS SYMBOLS 35 2.6 COMBINATORICS SYMBOLS 37 2.7 SET THEORY SYMBOLS 37 2.8 LOGIC SYMBOLS 38 2.9 CALCULUS & ANALYSIS SYMBOLS 39 PART 3: AREA, VOLUME AND SURFACE AREA 40 3.1 AREA: 40 TRIANGLE : 40 RECTANGLE : 40 SQUARE : 40 PARALLELOGRAM : 40 RHOMBUS : 40 TRAPEZIUM : 40 QUADRILATERAL : 40 RECTANGLE WITH ROUNDED CORNERS : 40 REGULAR HEXAGON : 40 REGULAR OCTAGON : 40 REGULAR POLYGON : 40 3.2 VOLUME: 40 CUBE : 40 CUBOID : 40 Page 4 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 (E VEN SIDES ) 41 LONGEST DIAGONAL (O DD SIDES ) 41 TOTAL LENGTH OF EDGES (C UBE ): 41 TOTAL LENGTH OF EDGES (C UBOID ): 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’H OPITAL ’S RULE : 47 4.6 SUMMATION MANIPULATIONS: 48 4.7 COMMON FUNCTIONS: 48 CONSTANT FUNCTION : 48 LINE /L INEAR FUNCTION : 48 PARABOLA /Q UADRATIC FUNCTION : 49 CIRCLE : 49 ELLIPSE : 49 HYPERBOLA : 49 4.8 LINEAR ALGEBRA: 50 Page 5 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 6 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 7 of 330 SEXY PRIMES : 63 SMARANDACHE –WELLIN PRIMES : 63 SOLINAS PRIMES : 63 SOPHIE GERMAIN PRIMES : 63 STAR PRIMES : 63 STERN PRIMES : 64 SUPER -PRIMES : 64 SUPERSINGULAR PRIMES : 64 SWINGING PRIMES : 64 THABIT NUMBER PRIMES : 64 PRIME TRIPLETS : 64 TWIN PRIMES : 64 TWO -SIDED PRIMES : 64 ULAM NUMBER PRIMES : 64 UNIQUE PRIMES : 64 WAGSTAFF PRIMES : 64 WALL -SUN -SUN PRIMES : 65 WEAKLY PRIME NUMBERS 65 WIEFERICH PRIMES : 65 WIEFERICH PRIMES : BASE 3 (M IRIMANOFF PRIMES :) 65 WIEFERICH PRIMES : BASE 5 65 WIEFERICH PRIMES : BASE 6 65 WIEFERICH PRIMES : BASE 7 65 WIEFERICH PRIMES : BASE 10 65 WIEFERICH PRIMES : BASE 11 65 WIEFERICH PRIMES : BASE 12 65 WIEFERICH PRIMES : BASE 13 65 WIEFERICH PRIMES : BASE 17 65 WIEFERICH PRIMES : BASE 19 66 WILSON PRIMES : 66 WOLSTENHOLME PRIMES : 66 WOODALL PRIMES : 66 4.13 GENERALISATIONS FROM PRIME NUMBERS: 66 PERFECT NUMBERS : 66 LIST OF PERFECT NUMBERS : 66 AMICABLE NUMBERS : 67 LIST OF AMICABLE NUMBERS : 67 SOCIABLE NUMBERS : 68 LIST OF SOCIABLE NUMBERS : 68 4.14 GOLDEN RATIO & FIBONACCI SEQUENCE: 71 RELATIONSHIP : 71 INFINITE SERIES : 71 CONTINUED FRACTIONS : 71 TRIGONOMETRIC EXPRESSIONS : 72 FIBONACCI SEQUENCE : 72 4.15 FERMAT’S LAST THEOREM: 72 4.16 BOOLEAN ALGEBRA: 72 AXIOMS : 72 THEOREMS OF ONE VARIABLE : 73 THEOREMS OF SEVERAL VARIABLES : 73 PART 5: COUNTING TECHNIQUES & PROBABILITY 74 5.1 2D 74 TRIANGLE NUMBER 74 Page 8 of 330 SQUARE NUMBER 74 PENTAGONAL NUMBER 74 5.2 3D 74 TETRAHEDRAL NUMBER 74 SQUARE PYRAMID NUMBER 74 5.3 PERMUTATIONS 74 PERMUTATIONS : 74 PERMUTATIONS (WITH REPEATS ): 74 5.4 COMBINATIONS 74 ORDERED COMBINATIONS : 74 UNORDERED COMBINATIONS : 74 ORDERED REPEATED COMBINATIONS : 74 UNORDERED REPEATED COMBINATIONS : 74 GROUPING : 74 5.5 MISCELLANEOUS: 74 TOTAL NUMBER OF RECTANGLES AND SQUARES FROM A A X B RECTANGLE : 74 NUMBER OF INTERPRETERS : 74 MAX NUMBER OF PIZZA PIECES : 74 MAX PIECES OF A CRESCENT : 74 MAX PIECES OF CHEESE : 74 CARDS IN A CARD HOUSE : 75 DIFFERENT ARRANGEMENT OF DOMINOS : 75 UNIT FRACTIONS : 75 ANGLE BETWEEN TWO HANDS OF A CLOCK : 75 WINNING LINES IN NOUGHTS AND CROSSES : 75 BAD RESTAURANT SPREAD : 75 FIBONACCI SEQUENCE : 75 ABBREVIATIONS (5.1, 5.2, 5.3, 5.4, 5.5) 75 5.6 FACTORIAL: 75 DEFINITION : 75 TABLE OF FACTORIALS : 75 APPROXIMATION : 76 5.7 THE DAY OF THE WEEK: 76 5.8 BASIC PROBABILITY: 76 AXIOM ’S OF PROBABILITY : 76 COMMUTATIVE LAWS : 76 ASSOCIATIVE LAWS