All in one mathematics cheatsheet

ALL IN ONE
MATHEMATICS CHEAT SHEET
V2.6

Euler’s Identity:

iπ

e +1=0
CONTAINING FORMULAE FOR ELEMENTARY, HIGH SCHOOL
AND UNIVERSITY MATHEMATICS
COMPILED FROM MANY SOURCES BY ALEX SPARTALIS
2009-2012

Page 1 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 2 of 286


CONTENTS
REVISION HISTORY

2

CONTENTS

3

PART 1: PHYSICAL CONSTANTS

20

1.1
1.2
1.3
1.4
1.5
1.6

1.7
1.8
1.9

20
20
21
22
22
22
23
24
25

SI PREFIXES:
SI BASE UNITS:
SI DERIVED UNITS:
UNIVERSAL CONSTANTS:
ELECTROMAGNETIC CONSTANTS:
ATOMIC AND NUCLEAR CONSTANTS:
PHYSICO-CHEMICAL CONSTANTS:
ADOPTED VALUES:
NATURAL UNITS:

PART 2: MATHEMTAICAL SYMBOLS

26

2.1
2.2

2.3
2.4
2.5
2.6
2.7
2.8
2.9

26
26
27
28
28
30
30
31
32

BASIC MATH SYMBOLS
GEOMETRY SYMBOLS
ALGEBRA SYMBOLS
LINEAR ALGEBRA SYMBOLS
PROBABILITY AND STATISTICS SYMBOLS
COMBINATORICS SYMBOLS
SET THEORY SYMBOLS
LOGIC SYMBOLS
CALCULUS & ANALYSIS SYMBOLS

PART 3: AREA, VOLUME AND SURFACE AREA

33

3.1 AREA
TRIANGLE:
RECTANGLE:
SQUARE:
PARALLELOGRAM:
RHOMBUS:
TRAPEZIUM:
QUADRILATERAL:
RECTANGLE WITH ROUNDED CORNERS:
REGULAR HEXAGON:
REGULAR OCTAGON:
REGULAR POLYGON:
3.2 VOLUME
CUBE:
CUBOID:
PYRAMID:

33
33
33
33
33
33
33
33
33
33
33

33
33
33
33
33
Page 3 of 286


TETRAHEDRON:
OCTAHEDRON:
DODECAHEDRON:
ICOSAHEDRON:
3.3 SURFACE AREA:
CUBE:
CUBOIDS:
TETRAHEDRON:
OCTAHEDRON:
DODECAHEDRON:
ICOSAHEDRON:
CYLINDER:
3.4 MISELANIOUS
DIAGONAL OF A RECTANGLE
DIAGONAL OF A CUBOID
LONGEST DIAGONAL (EVEN SIDES)
LONGEST DIAGONAL (ODD SIDES)
TOTAL LENGTH OF EDGES (CUBE):
TOTAL LENGTH OF EDGES (CUBOID):
CIRCUMFERENCE
PERIMETER OF RECTANGLE
SEMI PERIMETER

EULER’S FORMULA
3.5 ABBREVIATIONS (3.1, 3.2, 3.3, 3.4)

33
33
33
33
33
33
33
33
34
34
34
34
34
34
34
34
34
34
34
34
34
34
34
34

PART 4: ALGEBRA

36

4.1 POLYNOMIAL FORMULA:
QUDARATIC:
CUBIC:
4.2 ALGEBRAIC EXPANSION:
BABYLONIAN IDENTITY:
COMMON PRODUCTS AND FACTORS:
BINOMIAL THEOREM:
BINOMIAL EXPANSION:
DIFFERENCE OF TWO SQUARES:
BRAHMAGUPTA–FIBONACCI IDENTITY:
DEGEN'S EIGHT-SQUARE IDENTITY:
4.3 LIMIT MANIPULATIONS:
4.4 SUMATION MANIPULATIONS:
4.5 COMMON FUNCTIONS:
CONSTANT FUNCTION:
LINE/LINEAR FUNCTION:
PARABOLA/QUADRATIC FUNCTION:
CIRCLE:
ELLIPSE:
HYPERBOLA:
4.6 LINEAR ALGEBRA:
VECTOR SPACE AXIOMS:
SUBSPACE:
COMMON SPACES:
ROWSPACE OF A SPANNING SET IN RN
COLUMNSPACE OF A SPANNING SET IN RN
NULLSPACE:
NULLITY:

36
36
36
38
38
38
38
38
39
39
39
40
40
40
40
41
41
41
41
42
42
42
42
42
43
43
43
43

Page 4 of 286


LINEAR DEPENDENCE:
BASIS:
STANDARD BASIS:
ORTHOGONAL COMPLEMENT:
ORTHONORMAL BASIS:
GRAM-SCHMIDT PROCESS:
COORDINATE VECTOR:
DIMENSION:
4.7 COMPLEX VECTOR SPACES:
FORM:
DOT PRODUCT:
INNER PRODUCT:
4.8 LINEAR TRANSITIONS & TRANSFORMATIONS:
TRANSITION MATRIX:
CHANGE OF BASIS TRANSITION MATRIX:
TRANSFORMATION MATRIX:
4.9 INNER PRODUCTS:
DEFINITION:
AXIOMS:
UNIT VECTOR:
CAVCHY-SCHUARZ INEQUALITY:
INNER PRODUCT SPACE:
ANGLE BETWEEN TWO VECTORS:
DISTANCE BETWEEN TWO VECTORS:
GENERALISED PYTHAGORAS FOR ORTHOGONAL VECTORS:
4.10 PRIME NUMBERS:
DETERMINATE:

LIST OF PRIME NUMBERS:
PERFECT NUMBERS:
LIST OF PERFECT NUMBERS:
AMICABLE NUMBERS:
LIST OF AMICABLE NUMBERS:
SOCIABLE NUMBERS:
LIST OF SOCIABLE NUMBERS:
4.11 GOLDEN RATIO & FIBONACCI SEQUENCE:
RELATIONSHIP:
INFINITE SERIES:
CONTINUED FRACTIONS:
TRIGONOMETRIC EXPRESSIONS:
FIBONACCI SEQUENCE:
4.12 FERMAT’S LAST THEOREM:

43
43
44
44
44
45
45
45
45
45
45
46
46
46
46

46
46
46
46
47
47
47
47
47
47
47
47
47
48
48
49
49
50
50
53
53
53
53
54
54
54

PART 5: COUNTING TECHNIQUES & PROBABILITY

55

5.1 2D
TRIANGLE NUMBER
SQUARE NUMBER
PENTAGONAL NUMBER
5.2 3D
TETRAHEDRAL NUMBER
SQUARE PYRAMID NUMBER
5.3 PERMUTATIONS
PERMUTATIONS:
PERMUTATIONS (WITH REPEATS):
5.4 COMBINATIONS

55
55
55
55
55
55
55
55
55
55
55
Page 5 of 286


ORDERED COMBINATIONS:
UNORDERED COMBINATIONS:
ORDERED REPEATED COMBINATIONS:

UNORDERED REPEATED COMBINATIONS:
GROUPING:
5.5 MISCELLANEOUS:
TOTAL NUMBER OF RECTANGLES AND SQUARES FROM A A X B RECTANGLE:
NUMBER OF INTERPRETERS:
MAX NUMBER OF PIZZA PIECES:
MAX PIECES OF A CRESCENT:
MAX PIECES OF CHEESE:
CARDS IN A CARD HOUSE:
DIFFERENT ARRANGEMENT OF DOMINOS:
UNIT FRACTIONS:
ANGLE BETWEEN TWO HANDS OF A CLOCK:
WINNING LINES IN NOUGHTS AND CROSSES:
BAD RESTAURANT SPREAD:
FIBONACCI SEQUENCE:
ABBREVIATIONS (5.1, 5.2, 5.3, 5.4, 5.5)
5.6 FACTORIAL:
DEFINITION:
TABLE OF FACTORIALS:
APPROXIMATION:
5.7 THE DAY OF THE WEEK:
5.8 BASIC PROBABILITY:
5.9 VENN DIAGRAMS:
COMPLEMENTARY EVENTS:
TOTALITY:
CONDITIONAL PROBABILITY:
UNION :
INDEPENDENT EVENTS:
MUTUALLY EXCLUSIVE:
BAYE’S THEOREM:

5.11 BASIC STATISTICAL OPERATIONS:
VARIANCE:
MEAN:
STANDARDIZED SCORE:
5.12 DISCRETE RANDOM VARIABLES:
STANDARD DEVIATION:
EXPECTED VALUE:
VARIANCE:
PROBABILITY MASS FUNCTION:
CUMULATIVE DISTRIBUTION FUNCTION:
5.13 COMMON DRVS:
BERNOULLI TRIAL:
BINOMIAL TRIAL:
GEOMETRIC TRIAL:
NEGATIVE BINOMIAL TRIAL:
5.14 CONTINUOUS RANDOM VARIABLES:
PROBABILITY DENSITY FUNCTION:
CUMULATIVE DISTRIBUTION FUNCTION:
INTERVAL PROBABILITY:
EXPECTED VALUE:
VARIANCE:
5.15 COMMON CRVS:
UNIFORM DISTRIBUTION:
Page 6 of 286

55
55
55
55
55

55
55
55
55
55
55
56
56
56
56
56
56
56
56
56
56
56
57
57
57
57
57
57
57
57
57
57
57
58
58

58
58
58
58
58
58
58
58
58
58
58
59
59
59
59
59
59
59
59
59
59


EXPONENTIAL DISTRIBUTION:
NORMAL DISTRIBUTION:
5.16 MULTIVARIABLE DISCRETE:
PROBABILITY:
MARGINAL DISTRIBUTION:
EXPECTED VALUE:
INDEPENDENCE:

COVARIANCE:
5.17 MULTIVARIABLE CONTINUOUS:
PROBABILITY:
MARGINAL DISTRIBUTION:
EXPECTED VALUE:
INDEPENDENCE:
COVARIANCE:
CORRELATION COEFFICIENT:
ABBREVIATIONS

60
61
61
61
61
61
61
62
62
62
62
62
62
62
62
62

PART 6: FINANCIAL

64

6.1 GENERAL FORMUALS:
PROFIT:
PROFIT MARGIN:
SIMPLE INTEREST:
COMPOUND INTEREST:
CONTINUOUS INTEREST:
ABBREVIATIONS (6.1):
6.2 MACROECONOMICS:
GDP:
RGDP:
NGDP:
GROWTH:
NET EXPORTS:
WORKING AGE POPULATION:
LABOR FORCE:
UNEMPLOYMENT:
NATURAL UNEMPLOYMENT:
UNEMPLOYMENT RATE:
EMPLOYMENT RATE:
PARTICIPATION RATE:
CPI:
INFLATION RATE:
ABBREVIATIONS (6.2)

64
64
64
64
64

64
64
64
64
64
64
64
64
64
64
64
64
64
64
64
64
64
64

PART 7: PI

66

7.1 AREA:
CIRCLE:
CYCLIC QUADRILATERAL:
AREA OF A SECTOR (DEGREES)
AREA OF A SECTOR (RADIANS)
AREA OF A SEGMENT (DEGREES)
AREA OF AN ANNULUS:

ELLIPSE :
7.2 VOLUME:

66
66
66
66
66
66
66
66
66
Page 7 of 286


SPHERE:
CAP OF A SPHERE:
CONE:
ICE-CREAM & CONE:
DOUGHNUT:
SAUSAGE:
ELLIPSOID:
7.3 SURFACE AREA:
SPHERE:
HEMISPHERE:
DOUGHNUT:
SAUSAGE:
CONE:
7.4 MISELANIOUS:
LENGTH OF ARC (DEGREES)

LENGTH OF CHORD (DEGREES)
PERIMETER OF AN ELLIPSE
7.6 PI:
JOHN WALLIS:
ISAAC NEWTON:
JAMES GREGORY:
LEONARD EULER:
JOZEF HOENE-WRONSKI:
FRANCISCUS VIETA:
INTEGRALS:
INFINITE SERIES:
CONTINUED FRACTIONS:
7.7 CIRCLE GEOMETRY:
RADIUS OF CIRCUMSCRIBED CIRCLE FOR RECTANGLES:
RADIUS OF CIRCUMSCRIBED CIRCLE FOR SQUARES:
RADIUS OF CIRCUMSCRIBED CIRCLE FOR TRIANGLES:
RADIUS OF CIRCUMSCRIBED CIRCLE FOR QUADRILATERALS:
RADIUS OF INSCRIBED CIRCLE FOR SQUARES:
RADIUS OF INSCRIBED CIRCLE FOR TRIANGLES:
RADIUS OF CIRCUMSCRIBED CIRCLE:
RADIUS OF INSCRIBED CIRCLE:
7.8 ABBREVIATIONS (7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7):
7.9 CRESCENT GEOMETRY:
AREA OF A LUNAR CRESCENT:
AREA OF AN ECLIPSE CRESCENT:
7.10 ABBREVIATIONS (7.9):

66
66
66

66
66
66
66
66
66
66
66
66
66
67
67
67
67
67
67
67
67
67
67
67
67
68
68
69
69
69
69
69
69

69
69
69
69
70
70
70
70

PART 8: PHYSICS

71

8.1 MOVEMENT:
STOPPING DISTANCE:
CENTRIPETAL ACCELERATION:
CENTRIPETAL FORCE:
DROPPING TIME :
FORCE:
KINETIC ENERGY:
MAXIMUM HEIGHT OF A CANNON:
PENDULUM SWING TIME:
POTENTIAL ENERGY:
RANGE OF A CANNON:

71
71
71
71
71

71
71
71
71
71
71
Page 8 of 286


TIME IN FLIGHT OF A CANNON:
UNIVERSAL GRAVITATION:
ABBREVIATIONS (8.1):
8.2 CLASSICAL MECHANICS:
NEWTON’S LAWS:
INERTIA:
MOMENTS OF INERTIA:
VELOCITY AND SPEED:
ACCELERATION:
TRAJECTORY (DISPLACEMENT):
KINETIC ENERGY:
CENTRIPETAL FORCE:
CIRCULAR MOTION:
ANGULAR MOMENTUM:
TORQUE:
WORK:
LAWS OF CONSERVATION:
ABBREVIATIONS (8.2)
8.3 RELATIVISTIC EQUATIONS:
KINETIC ENERGY:
MOMENTUM:

TIME DILATION:
LENGTH CONTRACTION:
RELATIVISTIC MASS:

71
71
71
72
72
72
73
76
76
76
77
77
77
77
78
78
78
78
78
78
79
79
79
79

PART 9: TRIGONOMETRY

80

9.1 CONVERSIONS:
9.2 BASIC RULES:
SIN RULE:
COS RULE:
TAN RULE:
AUXILIARY ANGLE:
PYTHAGORAS THEOREM:
9.3 RECIPROCAL FUNCTIONS
9.4 BASIC IDENTITES:
9.5 IDENTITIES (SINΘ):
9.6 IDENTITIES (COSΘ):
9.7 IDENTITIES (TANΘ):
9.8 IDENTITIES (CSCΘ):
9.9 IDENTITIES (COTΘ):
9.10 ADDITION FORMULAE:
9.11 DOUBLE ANGLE FORMULAE:
9.12 TRIPLE ANGLE FORMULAE:
9.13 HALF ANGLE FORMULAE:
9.14 POWER REDUCTION:
9.15 PRODUCT TO SUM:
9.16 SUM TO PRODUCT:
9.17 HYPERBOLIC EXPRESSIONS:
9.18 HYPERBOLIC RELATIONS:
9.19 MACHIN-LIKE FORMULAE:
FORM:
FORMULAE:
IDENTITIES:

9.20 SPHERICAL TRIANGLE IDENTITIES:

80
80
80
80
80
80
80
81
81
81
82
82
82
82
83
83
84
84
85
86
86
86
87
87
87
87
88
88

Page 9 of 286


9.21

ABBREVIATIONS (9.1-9.19)

88

PART 10: EXPONENTIALS & LOGARITHIMS

90

10.1
10.2
10.3
10.4
10.5
10.6

90
90
90
90
91
91

FUNDAMENTAL THEORY:
IDENTITIES:

CHANGE OF BASE:
LAWS FOR LOG TABLES:
COMPLEX NUMBERS:
LIMITS INVOLVING LOGARITHMIC TERMS

PART 11: COMPLEX NUMBERS

92

11.1 GENERAL:
FUNDAMENTAL:
STANDARD FORM:
POLAR FORM:
ARGUMENT:
MODULUS:
CONJUGATE:
EXPONENTIAL:
DE MOIVRE’S FORMULA:
EULER’S IDENTITY:
11.2 OPERATIONS:
ADDITION:
SUBTRACTION:
MULTIPLICATION:
DIVISION:
SUM OF SQUARES:
11.3 IDENTITIES:
EXPONENTIAL:
LOGARITHMIC:
TRIGONOMETRIC:
HYPERBOLIC:

92
92
92
92
92
92
92
92
92
92
92
92
92
92
92
92
92
92
92
92
93

PART 12: DIFFERENTIATION

94

12.1 GENERAL RULES:
PLUS OR MINUS:
PRODUCT RULE:

QUOTIENT RULE:
POWER RULE:
CHAIN RULE:
BLOB RULE:
BASE A LOG:
NATURAL LOG:
EXPONENTIAL (X):
FIRST PRINCIPLES:
12.2 EXPONETIAL FUNCTIONS:
12.3 LOGARITHMIC FUNCTIONS:
12.4 TRIGONOMETRIC FUNCTIONS:
12.5 HYPERBOLIC FUNCTIONS:
12.5 PARTIAL DIFFERENTIATION:

94
94
94
94
94
94
94
94
94
94
94
94
95
95
95
96

Page 10 of 286


FIRST PRINCIPLES:
GRADIENT:
TOTAL DIFFERENTIAL:
CHAIN RULE:
IMPLICIT DIFFERENTIATION:
HIGHER ORDER DERIVATIVES:

96
96
96
96
97
98

PART 13: INTEGRATION

99

13.1 GENERAL RULES:
POWER RULE:
BY PARTS:
CONSTANTS:
13.2 RATIONAL FUNCTIONS:
13.3 TRIGONOMETRIC FUNCTIONS (SINE):
13.4 TRIGONOMETRIC FUNCTIONS (COSINE):
13.5 TRIGONOMETRIC FUNCTIONS (TANGENT):

13.6 TRIGONOMETRIC FUNCTIONS (SECANT):
13.7 TRIGONOMETRIC FUNCTIONS (COTANGENT):
13.8 TRIGONOMETRIC FUNCTIONS (SINE & COSINE):
13.9 TRIGONOMETRIC FUNCTIONS (SINE & TANGENT):
13.10 TRIGONOMETRIC FUNCTIONS (COSINE & TANGENT):
13.11 TRIGONOMETRIC FUNCTIONS (SINE & COTANGENT):
13.12 TRIGONOMETRIC FUNCTIONS (COSINE & COTANGENT):
13.13 TRIGONOMETRIC FUNCTIONS (ARCSINE):
13.14 TRIGONOMETRIC FUNCTIONS (ARCCOSINE):
13.15 TRIGONOMETRIC FUNCTIONS (ARCTANGENT):
13.16 TRIGONOMETRIC FUNCTIONS (ARCCOSECANT):
13.17 TRIGONOMETRIC FUNCTIONS (ARCSECANT):
13.18 TRIGONOMETRIC FUNCTIONS (ARCCOTANGENT):
13.19 EXPONETIAL FUNCTIONS
13.20 LOGARITHMIC FUNCTIONS
13.21 HYPERBOLIC FUNCTIONS
13.22 INVERSE HYPERBOLIC FUNCTIONS
13.23 ABSOLUTE VALUE FUNCTIONS
13.24 SUMMARY TABLE
13.25 SQUARE ROOT PROOFS
13.26 CARTESIAN APPLICATIONS
AREA UNDER THE CURVE:
VOLUME:
VOLUME ABOUT X AXIS:
VOLUME ABOUT Y AXIS:
SURFACE AREA ABOUT X AXIS:
LENGTH WRT X-ORDINATES:
LENGTH WRT Y-ORDINATES:
LENGTH PARAMETRICALLY:

99
99
99
99
99
100
101
102
102
103
103
105
105
105
105
105
106
106
106
107
107
107
109
111
112
113
113
114
117
117

117
117
118
118
118
118
118

PART 14: FUNCTIONS

119

14.1 COMPOSITE FUNCTIONS:
14.2 MULTIVARIABLE FUNCTIONS:
LIMIT:
DISCRIMINANT:
CRITICAL POINTS:

119
119
119
119
119

Page 11 of 286


14.3 FIRST ORDER, FIRST DEGREE, DIFFERENTIAL EQUATIONS:
SEPARABLE
LINEAR

HOMOGENEOUS
EXACT
14.4 SECOND ORDER
HOMOGENEOUS
UNDETERMINED COEFFICIENTS
VARIATION OF PARAMETERS

120
120
120
120
120
121
121
121
121

PART 15: MATRICIES

123

15.1 BASIC PRINICPLES:
SIZE
15.2 BASIC OPERTAIONS:
ADDITION:
SUBTRACTION:
SCALAR MULTIPLE:
TRANSPOSE:
SCALAR PRODUCT:
SYMMETRY:

CRAMER’S RULE:
LEAST SQUARES SOLUTION
15.3 SQUARE MATRIX:
DIAGONAL:
LOWER TRIANGLE MATRIX:
UPPER TRIANGLE MATRIX:
15.4 DETERMINATE:
2X2
3X3

123
123
123
123
123
123
123
123
123
123
123
123
124
124
124
124
124
124
124
124

126
126
126
126
127
127
127
127
127
127
127
128
128
128
128
128
128
128
128
128
128
128
128

NXN
RULES

15.5 INVERSE
2X2:
3X3:

MINOR:
COFACTOR:
ADJOINT METHOD FOR INVERSE:
LEFT INVERSE:
RIGHT INVERSE:
15.6 LINEAR TRANSFORMATION
AXIOMS FOR A LINEAR TRANSFORMATION:
TRANSITION MATRIX:
ZERO TRANSFORMATION:
IDENTITY TRANSFORMATION:
15.7 COMMON TRANSITION MATRICIES
ROTATION (CLOCKWISE):
ROTATION (ANTICLOCKWISE):
SCALING:
SHEARING (PARALLEL TO X-AXIS):
SHEARING (PARALLEL TO Y-AXIS):
15.8 EIGENVALUES AND EIGENVECTORS
DEFINITIONS:
EIGENVALUES:
EIGENVECTORS:
Page 12 of 286


CHARACTERISTIC POLYNOMIAL:
ALGEBRAIC MULTIPLICITY:
GEOMETRIC MULTIPLICITY:
TRANSFORMATION:
LINEARLY INDEPENDENCE:
DIGITALIZATION:
CAYLEY-HAMILTON THEOREM:

ORTHONORMAL SET:
QR FACTORISATION:
15.9 JORDAN FORMS
GENERALISED DIAGONLISATION:
JORDAN BLOCK:
JORDAN FORM:
ALGEBRAIC MULTIPLICITY:
GEOMETRIC MULTIPLICITY:
GENERALISED CHAIN:
POWERS:
15.10 COMPLEX MATRICIS:
CONJUGATE TRANSPOSE:
HERMITIAN MATRIX:
SKEW-HERMITIAN:
UNITARY MATRIX:
NORMAL MATRIX:
DIAGONALISATION:
SPECTRAL THEOREM:
15.11 NUMERICAL COMPUTATIONS:
RAYLEIGH QUOTIENT:
POWER METHOD:

128
128
128
129
129
129
129
129

129
130
130
130
130
130
130
130
131
131
131
131
131
131
131
132
132
132
132
133

PART 16: VECTORS

135

16.1 BASIC OPERATIONS:
ADDITION:
SUBTRACTION:
EQUALITY:
SCALAR MULTIPLICATION:

PARALLEL:
MAGNITUDE:
UNIT VECTOR:
ZERO VECTOR:
DOT PRODUCT:
ANGLE BETWEEN TWO VECTORS:
ANGLE OF A VECTOR IN 3D:
PERPENDICULAR TEST:
SCALAR PROJECTION:
VECTOR PROJECTION:
CROSS PRODUCT:
16.2 LINES
16.3 PLANES
16.4 CLOSEST APPROACH
TWO POINTS:
POINT AND LINE:
POINT AND PLANE:
TWO SKEW LINES:
16.5 GEOMETRY

135
135
135
135
135
135
135
135
135
135

135
135
135
136
136
136
136
136
136
136
136
136
137
137
Page 13 of 286


AREA OF A TRIANGLE:
AREA OF A PARALLELOGRAM:
AREA OF A PARALLELEPIPED:
16.6 SPACE CURVES
WHERE:
VELOCITY:
ACCELERATION:
DEFINITION OF “S”:
UNIT TANGENT:
CHAIN RULE:
NORMAL:
CURVATURE:
UNIT BINOMIAL:

TORTION:
16.7 VECTOR SPACE
16.8 ABBREVIATIONS

137
137
137
137
137
137
137
137
137
137
138
138
138
138
138
138

PART 17: SERIES

140

17.1 MISCELLANEOUS
GENERAL FORM:
INFINITE FORM:
PARTIAL SUM OF A SERIES:
17.2 TEST FOR CONVERGENCE AND DIVERGENCE

TEST FOR CONVERGENCE:
TEST FOR DIVERGENCE:
GEOMETRIC SERIES
P SERIES
THE SANDWICH THEOREM
THE INTEGRAL TEST
THE DIRECT COMPARISON TEST
THE LIMIT COMPARISON TEST
D’ALMBERT’S RATIO COMPARISON TEST
THE NTH ROOT TEST
NEGATIVE TERMS
ALTERNATING SERIES TEST
ALTERNATING SERIES ERROR
17.3 ARITHMETIC PROGRESSION:
DEFINITION:
NTH TERM:
SUM OF THE FIRST N TERMS:
17.4 GEOMETRIC PROGRESSION:
DEFINITION:
NTH TERM:
SUM OF THE FIRST N TERMS:
SUM TO INFINITY:
GEOMETRIC MEAN:
17.5 SUMMATION SERIES
LINEAR:
QUADRATIC:
CUBIC:
17.6 APPROXIMATION SERIES
TAYLOR SERIES
MACLAURUN SERIES

LINEAR APPROXIMATION:

140
140
140
140
140
140
140
140
140
140
140
140
141
141
141
141
141
141
142
142
142
142
142
142
142
142
142
142

142
142
142
142
142
142
142
143

Page 14 of 286


QUADRATIC APPROXIMATION:
CUBIC APPROXIMATION:
17.7 MONOTONE SERIES
STRICTLY INCREASING:
NON-DECREASING:
STRICTLY DECREASING:
NON-INCREASING:
CONVERGENCE:
17.8 RIEMANN ZETA FUNCTION
FORM:
EULER’S TABLE:
ALTERNATING SERIES:
PROOF FOR N=2:
17.9 SUMMATIONS OF POLYNOMIAL EXPRESSIONS
17.10 SUMMATIONS INVOLVING EXPONENTIAL TERMS
17.11 SUMMATIONS INVOLVING TRIGONOMETRIC TERMS
17.12 INFINITE SUMMATIONS TO PI
17.13 LIMITS INVOLVING TRIGONOMETRIC TERMS

ABBREVIATIONS
17.14 POWER SERIES EXPANSION
EXPONENTIAL:
TRIGONOMETRIC:
EXPONENTIAL AND LOGARITHM SERIES:
FOURIER SERIES:
17.15 BERNOULLI EXPANSION:
FUNDAMENTALLY:
EXPANSIONS:
LIST OF BERNOULLI NUMBERS:

143
143
143
143
143
143
143
143
143
143
143
144
144
145
145
146
148
148
148

148
148
149
151
152
152
152
153
153

PART 18: ELECTRICAL

155

18.1 FUNDAMENTAL THEORY
CHARGE:
CURRENT:
RESISTANCE:
OHM’S LAW:
POWER:
CONSERVATION OF POWER:
ELECTRICAL ENERGY:
KIRCHOFF’S VOLTAGE LAW:
KIRCHOFF’S CURRENT LAW:
AVERAGE CURRENT:
RMS CURRENT:
∆ TO Y CONVERSION:
18.2 COMPONENTS
RESISTANCE IN SERIES:
RESISTANCE IN PARALLEL:

INDUCTIVE IMPEDANCE:
CAPACITOR IMPEDANCE:
CAPACITANCE IN SERIES:
CAPACITANCE IN PARALLEL:
VOLTAGE, CURRENT & POWER SUMMARY:
18.3 THEVENIN’S THEOREM
THEVENIN’S THEOREM:
MAXIMUM POWER TRANSFER THEOREM:

155
155
155
155
155
155
155
155
155
155
155
155
155
156
156
156
156
156
156
156
156

156
156
157

Page 15 of 286


18.4 FIRST ORDER RC CIRCUIT
18.5 FIRST ORDER RL CIRCUIT
18.6 SECOND ORDER RLC SERIES CIRCUIT
CALCULATION USING KVL:
IMPORTANT VARIABLES
SOLVING:
MODE 1:
MODE 2:
MODE 3:
MODE 4:
CURRENT THROUGH INDUCTOR:
PLOTTING MODES:
18.7 SECOND ORDER RLC PARALLEL CIRCUIT
CALCULATION USING KCL:
IMPORTANT VARIABLES
SOLVING:
18.8 LAPLANCE TRANSFORMATIONS
IDENTITIES:
PROPERTIES:
18.9 THREE PHASE – Y
LINE VOLTAGE:
PHASE VOLTAGE:
LINE CURRENT:

PHASE CURRENT:
POWER:
18.10 THREE PHASE – DELTA
LINE VOLTAGE:
PHASE VOLTAGE:
LINE CURRENT:
PHASE CURRENT:
POWER:
18.11 POWER
INSTANTANEOUS:
AVERAGE:
MAXIMUM POWER:
TOTAL POWER:
COMPLEX POWER:
18.12 ELECTROMAGNETICS
DEFINITIONS:
PERMEABILITY OF FREE SPACE:
MAGNETIC FIELD INTENSITY:
RELUCTANCE:
OHM’S LAW:
MAGNETIC FORCE ON A CONDUCTOR:
ELECTROMAGNETIC INDUCTION:
MAGNETIC FLUX:
ELECTRIC FIELD:
MAGNETIC FORCE ON A PARTICLE:

157
157
157
157

157
158
158
158
159
159
160
160
161
161
161
162
162
162
163
164
164
164
164
164
164
164
164
164
164
164
164
164
164
165

165
165
165
165
165
165
165
165
165
165
165
165
165
165

PART 19: GRAPH THEORY

166

19.1 FUNDAMENTAL EXPLANATIONS
LIST OF VERTICES:
LIST OF EDGES:
SUBGAPHS:

166
166
166
166

Page 16 of 286

DEGREE OF VERTEX:
DISTANCE:
DIAMETER:
TOTAL EDGES IN A SIMPLE BIPARTITE GRAPH:
TOTAL EDGES IN K-REGULAR GRAPH:
19.2 FACTORISATION:
1 FACTORISATION:
1 FACTORS OF A K n ,n BIPARTITE GRAPH:

166
166
166
166
166
166
166
166

1 FACTORS OF A K 2 n GRAPH:
19.3 VERTEX COLOURING
CHROMATIC NUMBER:
UNION/INTERSECTION:
EDGE CONTRACTION:
COMMON CHROMATIC POLYNOMIALS:
19.4 EDGE COLOURING:
COMMON CHROMATIC POLYNOMIALS:

166

166
167
167
167
167
167
167

PART 98: LIST OF DISTRIBUTION FUNCTIONS

168

5.18 FINITE DISCRETE DISTRIBUTIONS
BERNOULLI DISTRIBUTION
RADEMACHER DISTRIBUTION
BINOMIAL DISTRIBUTION
BETA-BINOMIAL DISTRIBUTION
DEGENERATE DISTRIBUTION
DISCRETE UNIFORM DISTRIBUTION
HYPERGEOMETRIC DISTRIBUTION
POISSON BINOMIAL DISTRIBUTION
FISHER'S NONCENTRAL HYPERGEOMETRIC DISTRIBUTION (UNIVARIATE)
FISHER'S NONCENTRAL HYPERGEOMETRIC DISTRIBUTION (MULTIVARIATE)
WALLENIUS' NONCENTRAL HYPERGEOMETRIC DISTRIBUTION (UNIVARIATE)
WALLENIUS' NONCENTRAL HYPERGEOMETRIC DISTRIBUTION (MULTIVARIATE)
5.19 INFINITE DISCRETE DISTRIBUTIONS
BETA NEGATIVE BINOMIAL DISTRIBUTION
MAXWELL–BOLTZMANN DISTRIBUTION
GEOMETRIC DISTRIBUTION
LOGARITHMIC (SERIES) DISTRIBUTION

NEGATIVE BINOMIAL DISTRIBUTION
POISSON DISTRIBUTION
CONWAY–MAXWELL–POISSON DISTRIBUTION
SKELLAM DISTRIBUTION
YULE–SIMON DISTRIBUTION
ZETA DISTRIBUTION
ZIPF'S LAW
ZIPF–MANDELBROT LAW
5.20 BOUNDED INFINITE DISTRIBUTIONS
ARCSINE DISTRIBUTION
BETA DISTRIBUTION
LOGITNORMAL DISTRIBUTION
CONTINUOUS UNIFORM DISTRIBUTION
IRWIN-HALL DISTRIBUTION
KUMARASWAMY DISTRIBUTION
RAISED COSINE DISTRIBUTION
TRIANGULAR DISTRIBUTION

168
168
168
169
170
171
172
174
175
175
176
176

177
177
177
178
179
181
182
183
184
185
185
187
188
189
189
189
191
193
194
195
196
197
198

Page 17 of 286


TRUNCATED NORMAL DISTRIBUTION
U-QUADRATIC DISTRIBUTION
VON MISES DISTRIBUTION

WIGNER SEMICIRCLE DISTRIBUTION
5.21 SEMI-BOUNDED CUMULATIVE DISTRIBUTIONS
BETA PRIME DISTRIBUTION
CHI DISTRIBUTION
NONCENTRAL CHI DISTRIBUTION
CHI-SQUARED DISTRIBUTION
INVERSE-CHI-SQUARED DISTRIBUTION
NONCENTRAL CHI-SQUARED DISTRIBUTION
SCALED-INVERSE-CHI-SQUARED DISTRIBUTION
DAGUM DISTRIBUTION
EXPONENTIAL DISTRIBUTION
FISHER'S Z-DISTRIBUTION
FOLDED NORMAL DISTRIBUTION
FRÉCHET DISTRIBUTION
GAMMA DISTRIBUTION
ERLANG DISTRIBUTION
INVERSE-GAMMA DISTRIBUTION
INVERSE GAUSSIAN/WALD DISTRIBUTION
LÉVY DISTRIBUTION
LOG-CAUCHY DISTRIBUTION
LOG-LOGISTIC DISTRIBUTION
LOG-NORMAL DISTRIBUTION
MITTAG–LEFFLER DISTRIBUTION
PARETO DISTRIBUTION
RAYLEIGH DISTRIBUTION
RICE DISTRIBUTION
TYPE-2 GUMBEL DISTRIBUTION
WEIBULL DISTRIBUTION
5.22 UNBOUNDED CUMULATIVE DISTRIBUTIONS
CAUCHY DISTRIBUTION

EXPONENTIALLY MODIFIED GAUSSIAN DISTRIBUTION
FISHER–TIPPETT/ GENERALIZED EXTREME VALUE DISTRIBUTION
GUMBEL DISTRIBUTION
FISHER'S Z-DISTRIBUTION
GENERALIZED NORMAL DISTRIBUTION
GEOMETRIC STABLE DISTRIBUTION
HOLTSMARK DISTRIBUTION
HYPERBOLIC DISTRIBUTION
HYPERBOLIC SECANT DISTRIBUTION
LAPLACE DISTRIBUTION
LÉVY SKEW ALPHA-STABLE DISTRIBUTION
LINNIK DISTRIBUTION
LOGISTIC DISTRIBUTION
NORMAL DISTRIBUTION
NORMAL-EXPONENTIAL-GAMMA DISTRIBUTION
SKEW NORMAL DISTRIBUTION
STUDENT'S T-DISTRIBUTION
NONCENTRAL T-DISTRIBUTION
VOIGT DISTRIBUTION
GENERALIZED PARETO DISTRIBUTION
TUKEY LAMBDA DISTRIBUTION
5.23 JOINT DISTRIBUTIONS
DIRICHLET DISTRIBUTION
Page 18 of 286

200
201
202
203
205

205
206
207
207
208
210
211
212
213
216
216
217
218
219
220
221
222
224
225
226
227
228
229
230
231
232
233
233
234
236

237
238
238
240
240
241
242
243
244
246
246
248
249
249
250
252
252
253
254
254
254


BALDING–NICHOLS MODEL
MULTINOMIAL DISTRIBUTION
MULTIVARIATE NORMAL DISTRIBUTION
NEGATIVE MULTINOMIAL DISTRIBUTION
WISHART DISTRIBUTION
INVERSE-WISHART DISTRIBUTION
MATRIX NORMAL DISTRIBUTION

MATRIX T-DISTRIBUTION
5.24 OTHER DISTRIBUTIONS
CATEGORICAL DISTRIBUTION
CANTOR DISTRIBUTION
PHASE-TYPE DISTRIBUTION
TRUNCATED DISTRIBUTION

255
256
256
257
258
258
258
259
259
259
260
261
261

PART 99: CONVERSIONS

263

99.1
99.2
99.3
99.4
99.5

99.6
99.7
99.8
99.9
99.10
99.11
99.12
99.13
99.14
99.15
99.16
99.17
99.18
99.19
99.20
99.21
99.22
99.23
99.24
99.25
99.26
99.27
99.28
99.29
99.30
99.31
99.32
99.33
99.34
99.35

99.36
99.37
99.38
99.39

263
265
266
270
270
270
272
273
274
274
275
276
276
277
278
278
280
281
281
281
281
282
282
282
283

283
283
283
283
283
284
284
285
285
285
285
285
285
286

LENGTH:
AREA:
VOLUME:
PLANE ANGLE:
SOLID ANGLE:
MASS:
DENSITY:
TIME:
FREQUENCY:
SPEED OR VELOCITY:
FLOW (VOLUME):
ACCELERATION:
FORCE:
PRESSURE OR MECHANICAL STRESS:
TORQUE OR MOMENT OF FORCE:

ENERGY, WORK, OR AMOUNT OF HEAT:
POWER OR HEAT FLOW RATE:
ACTION:
DYNAMIC VISCOSITY:
KINEMATIC VISCOSITY:
ELECTRIC CURRENT:
ELECTRIC CHARGE:
ELECTRIC DIPOLE:
ELECTROMOTIVE FORCE, ELECTRIC POTENTIAL DIFFERENCE:
ELECTRICAL RESISTANCE:
CAPACITANCE:
MAGNETIC FLUX:
MAGNETIC FLUX DENSITY:
INDUCTANCE:
TEMPERATURE:
INFORMATION ENTROPY:
LUMINOUS INTENSITY:
LUMINANCE:
LUMINOUS FLUX:
ILLUMINANCE:
RADIATION - SOURCE ACTIVITY:
RADIATION – EXPOSURE:
RADIATION - ABSORBED DOSE:
RADIATION - EQUIVALENT DOSE:
Page 19 of 286


PART 1: PHYSICAL CONSTANTS
1.1

SI PREFIXES:

Prefix
yotta
zetta
exa
peta
tera
giga
mega
kilo
hecto
deca

deci
centi
milli
micro
nano
pico
femto
atto
zepto
yocto

1.2

Symbol
Y
Z

E
P
T
G
M
k
h
da
d
c
m
µ
n
p
f
a
z
y

1000m
10008
10007
10006
10005
10004
10003
10002
10001
⁄
10002 3

⁄
10001 3
10000
1000−1⁄3
1000−2⁄3
1000−1
1000−2
1000−3
1000−4
1000−5
1000−6
1000−7
1000−8

10n
1024
1021
1018
1015
1012
109
106
103
102
101
100
10−1
10−2
10−3
10−6

10−9
10−12
10−15
10−18
10−21
10−24

Decimal

Scale

1000000000000000000000000
1000000000000000000000
1000000000000000000
1000000000000000
1000000000000
1000000000
1000000
1000
100
10
1
0.1
0.01
0.001
0.000001
0.000000001
0.000000000001
0.000000000000001
0.000000000000000001

0.000000000000000000001
0.000000000000000000000001

Septillion
Sextillion
Quintillion
Quadrillion
Trillion
Billion
Million
Thousand
Hundred
Ten
One
Tenth
Hundredth
Thousandth
Millionth
Billionth
Trillionth
Quadrillionth
Quintillionth
Sextillionth
Septillionth

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 20 of 286


1.3

SI DERIVED UNITS:

Quantity

Unit

Symbol

angle, plane
angle, solid

radian*
steradian*
degree
Celsius
farad

rad
sr

coulomb
siemens
henry

C
S
H

Celsius temperature
electric capacitance
electric charge, quantity
of electricity
electric conductance
electric inductance
electric potential
difference, electromotive
force
electric resistance
energy, work, quantity of
heat

force
frequency (of a periodic
phenomenon)
illuminance
luminous flux
magnetic flux
magnetic flux density
power, radiant flux
pressure, stress
activity (referred to a
radionuclide)
absorbed dose, specific
energy imparted, kerma
dose equivalent, ambient
dose equivalent,
directional dose
equivalent, personal dose
equivalent, organ dose
equivalent
catalytic activity

°C
F

Expression in terms of other SI
units
m/m = 1
m2/m2 = 1
K
C/V

A·s
A/V
Wb/A
W/A

volt
ohm

V
Ω

joule
newton

J
N

hertz
lux
lumen
weber
tesla
watt
pascal

Hz
lx
lm
Wb
T

W
Pa

becquerel

Bq

V/A
N·m
kg·m/s2
1/s
lm/m2
cd·sr
V·s
Wb/m2
J/s
N/m2
1/s
J/kg

gray

Gy

J/kg

sievert
katal

Sv

kat

Page 21 of 286

mol/s


1.4

UNIVERSAL CONSTANTS:

Quantity
speed of light in
vacuum
Newtonian constant
of gravitation
Planck constant
reduced Planck
constant

1.5

Value
299 792 458 m·s−1

Relative Standard
Uncertainty
defined

6.67428(67)×10−11 m3·kg−1·s−2 1.0 × 10−4

6.626 068 96(33) × 10−34 J·s

5.0 × 10−8

1.054 571 628(53) × 10−34 J·s 5.0 × 10−8

ELECTROMAGNETIC CONSTANTS:

Quantity
magnetic constant
(vacuum permeability)
electric constant
(vacuum permittivity)
characteristic
impedance of vacuum
Coulomb's constant
elementary charge
Bohr magneton
conductance quantum
inverse conductance
quantum
Josephson constant
magnetic flux quantum
nuclear magneton
von Klitzing constant
1.6

Symbol

Symbol

Value (SI units)

Relative Standard
Uncertainty

4π × 10−7 N·A−2 = 1.256
defined
637 061... × 10−6 N·A−2
8.854 187 817... × 10−12
defined
F·m−1
376.730 313 461... Ω
8.987 551 787... × 109
N·m²·C−2
1.602 176 487(40) ×
10−19 C
927.400 915(23) × 10−26
J·T−1
7.748 091 7004(53) ×
10−5 S

defined
defined
2.5 × 10−8
2.5 × 10−8
6.8 × 10−10

12 906.403 7787(88) Ω 6.8 × 10−10
4.835 978 91(12) × 1014

2.5 × 10−8
−1
Hz·V
2.067 833 667(52) ×
2.5 × 10−8
10−15 Wb
5.050 783 43(43) ×
8.6 × 10−8
10−27 J·T−1
25 812.807 557(18) Ω

ATOMIC AND NUCLEAR CONSTANTS:
Page 22 of 286

6.8 × 10−10


Quantity

Symbol

Relative
Standard
Uncertainty

5.291 772
108(18) ×
3.3 × 10−9
−11
10 m

2.817 940
2894(58) ×
2.1 × 10−9
−15
10 m
9.109 382
15(45) × 10−31 5.0 × 10−8
kg

Bohr radius
classical
electron
radius
electron mass
Fermi
coupling
constant
fine-structure
constant
Hartree
energy
proton mass
quantum of
circulation
Rydberg
constant
Thomson
cross section
weak mixing
angle

1.7

Value (SI
units)

1.166 39(1) ×
10−5 GeV−2

8.6 × 10−6

7.297 352 537
6(50) × 10−3
4.359 744
17(75) × 10−18
J
1.672 621
637(83) ×
10−27 kg
3.636 947
550(24) × 10−4
m² s−1
10 973
731.568
525(73) m−1
6.652 458
73(13) × 10−29
m²

6.8 × 10−10

0.222 15(76)

3.4 × 10−3

1.7 × 10−7

5.0 × 10−8

6.7 × 10−9

6.6 × 10−12

2.0 × 10−8

PHYSICO-CHEMICAL CONSTANTS:

Quantity

Symbol

Value (SI units)

Relative
Standard
Uncertainty

atomic mass unit
(unified atomic mass
unit)

1.660 538 86(28) ×
10−27 kg

1.7 × 10−7

Avogadro's number

6.022 141 5(10) ×
1023 mol−1

1.7 × 10−7

Boltzmann constant

1.380 6504(24) ×

1.8 × 10−6

Page 23 of 286


10−23 J·K−1
96
8.6 × 10−8
485.3383(83)C·mol−1
3.741 771 18(19) ×
5.0 × 10−8
10−16 W·m²

Faraday constant

first
radiation
constant

for
spectral
radiance
at
T=273.15
Loschmidt
K and
constant
p=101.325
kPa

molar Planck constant

1.8

1.7 × 10−7

2.686 777 3(47) ×
1025 m−3

1.8 × 10−6

8.314 472(15)
1.7 × 10−6
J·K−1·mol−1
3.990 312 716(27) ×

6.7 × 10−9
10−10 J·s·mol−1

gas constant

at
T=273.15
K and
molar
p=100 kPa
volume of
an ideal at
T=273.15
gas
K and
p=101.325
kPa
at T=1 K
and p=100
Sackur- kPa
Tetrode at T=1 K
constant and
p=101.325
kPa
second radiation
constant
Stefan–Boltzmann
constant
Wien displacement
law constant

1.191 042 82(20) ×
10−16 W·m² sr−1

4.965 114 231...

2.2710 981(40) ×
10−2 m³·mol−1

1.7 × 10−6

2.2413 996(39) ×
10−2 m³·mol−1

1.7 × 10−6

−1.151 704 7(44)

3.8 × 10−6

−1.164 867 7(44)

3.8 × 10−6

1.438 775 2(25) ×
1.7 × 10−6
10−2 m·K
5.670 400(40) × 10−8
7.0 × 10−6
W·m−2·K−4

2.897 768 5(51) ×
1.7 × 10−6
10−3 m·K

ADOPTED VALUES:

Quantity

Symbol

conventional value of
Page 24 of 286

Relative
Standard
Uncertainty
4.835 979 × defined
Value (SI
units)


1014 Hz·V−1
25 812.807
defined
Ω
1 × 10−3
defined
kg·mol−1
1.2 × 10−2
defined

kg·mol−1

Josephson constant
conventional value of von
Klitzing constant
constant
molar mass
of carbon-12
standard acceleration of
gravity (gee, free-fall on
Earth)
standard atmosphere
1.9

9.806 65
m·s−2

defined

101 325 Pa

defined

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 25 of 286