crc standard mathematical tables and formulae

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Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Editorial Board . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Chapter 1
Numbers and Elementary Mathematics . . . . . . . . . . . . . . . . . . . . 1
1.1 Proofs without words . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Special numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5 Series and products . . . . . . . . . . . . . . . . . . . . . . . . . 42
Chapter 2
Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.1 Elementary algebra . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.2 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.3 Vector algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.4 Linear and matrix algebra . . . . . . . . . . . . . . . . . . . . . 77
2.5 Abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Chapter 3
Discrete Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.1 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.2 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.3 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
3.4 Combinatorial design theory . . . . . . . . . . . . . . . . . . . . 164

3.5 Difference equations . . . . . . . . . . . . . . . . . . . . . . . . 177
Chapter 4
Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
4.1 Euclidean geometry . . . . . . . . . . . . . . . . . . . . . . . . 187
4.2 Coordinate systems in the plane . . . . . . . . . . . . . . . . . . 188
4.3 Plane symmetries or isometries . . . . . . . . . . . . . . . . . . 194
4.4 Other transformations of the plane . . . . . . . . . . . . . . . . . 201
4.5 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
4.6 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
4.7 Surfaces of revolution: the torus . . . . . . . . . . . . . . . . . . 213
4.8 Quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
4.9 Spherical geometry and trigonometry . . . . . . . . . . . . . . . 218
4.10 Conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
4.11 Special plane curves . . . . . . . . . . . . . . . . . . . . . . . . 233
4.12 Coordinate systems in space . . . . . . . . . . . . . . . . . . . . 242
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4.13 Space symmetries or isometries . . . . . . . . . . . . . . . . . . 245
4.14 Other transformations of space . . . . . . . . . . . . . . . . . . . 248
4.15 Direction angles and direction cosines . . . . . . . . . . . . . . . 249
4.16 Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
4.17 Lines in space . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
4.18 Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
4.19 Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
4.20 Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
4.21 Differential geometry . . . . . . . . . . . . . . . . . . . . . . . . 259
Chapter 5
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
5.1 Differential calculus . . . . . . . . . . . . . . . . . . . . . . . . 269
5.2 Differential forms . . . . . . . . . . . . . . . . . . . . . . . . . 279

5.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
5.4 Table of indefinite integrals . . . . . . . . . . . . . . . . . . . . 294
5.5 Table of definite integrals . . . . . . . . . . . . . . . . . . . . . 330
5.6 Ordinary differential equations . . . . . . . . . . . . . . . . . . . 337
5.7 Partial differential equations . . . . . . . . . . . . . . . . . . . . 349
5.8 Integral equations . . . . . . . . . . . . . . . . . . . . . . . . . . 358
5.9 Tensor analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
5.10 Orthogonal coordinate systems . . . . . . . . . . . . . . . . . . . 370
5.11 Interval analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 375
5.12 Real analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
5.13 Generalized functions . . . . . . . . . . . . . . . . . . . . . . . 386
5.14 Complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 388
Chapter 6
Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
6.1 Ceiling and floor functions . . . . . . . . . . . . . . . . . . . . . 401
6.2 Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
6.3 Logarithmic functions . . . . . . . . . . . . . . . . . . . . . . . 402
6.4 Exponential function . . . . . . . . . . . . . . . . . . . . . . . . 403
6.5 Trigonometric functions . . . . . . . . . . . . . . . . . . . . . . 404
6.6 Circular functions and planar triangles . . . . . . . . . . . . . . . 412
6.7 Tables of trigonometric functions . . . . . . . . . . . . . . . . . 416
6.8 Angle conversion . . . . . . . . . . . . . . . . . . . . . . . . . . 419
6.9 Inverse circular functions . . . . . . . . . . . . . . . . . . . . . . 420
6.10 Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . 422
6.11 Inverse hyperbolic functions . . . . . . . . . . . . . . . . . . . . 426
6.12 Gudermannian function . . . . . . . . . . . . . . . . . . . . . . 428
6.13 Orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . 430
6.14 Gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . 437
6.15 Beta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
6.16 Error functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

6.17 Fresnel integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 443
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6.18 Sine, cosine, and exponential integrals . . . . . . . . . . . . . . . 445
6.19 Polylogarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
6.20 Hypergeometric functions . . . . . . . . . . . . . . . . . . . . . 448
6.21 Legendre functions . . . . . . . . . . . . . . . . . . . . . . . . . 449
6.22 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . 454
6.23 Elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 463
6.24 Jacobian elliptic functions . . . . . . . . . . . . . . . . . . . . . 466
6.25 Clebsch–Gordan coefficients . . . . . . . . . . . . . . . . . . . . 468
6.26 Integral transforms: Preliminaries . . . . . . . . . . . . . . . . . 470
6.27 Fourier integral transform . . . . . . . . . . . . . . . . . . . . . 470
6.28 Discrete Fourier transform (DFT) . . . . . . . . . . . . . . . . . 476
6.29 Fast Fourier transform (FFT) . . . . . . . . . . . . . . . . . . . . 478
6.30 Multidimensional Fourier transforms . . . . . . . . . . . . . . . 478
6.31 Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . 479
6.32 Hankel transform . . . . . . . . . . . . . . . . . . . . . . . . . . 483
6.33 Hartley transform . . . . . . . . . . . . . . . . . . . . . . . . . . 484
6.34 Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . 484
6.35 Hilbert transform . . . . . . . . . . . . . . . . . . . . . . . . . . 485
6.36 Z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488
6.37 Tables of transforms . . . . . . . . . . . . . . . . . . . . . . . . 492
Chapter 7
Probability and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
7.1 Probability theory . . . . . . . . . . . . . . . . . . . . . . . . . 509
7.2 Classical probability problems . . . . . . . . . . . . . . . . . . . 519
7.3 Probability distributions . . . . . . . . . . . . . . . . . . . . . . 524
7.4 Queuing theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

7.5 Markov chains . . . . . . . . . . . . . . . . . . . . . . . . . . . 536
7.6 Random number generation . . . . . . . . . . . . . . . . . . . . 539
7.7 Control charts and reliability . . . . . . . . . . . . . . . . . . . . 545
7.8 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
7.9 Confidence intervals . . . . . . . . . . . . . . . . . . . . . . . . 558
7.10 Tests of hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . 565
7.11 Linear regression . . . . . . . . . . . . . . . . . . . . . . . . . . 579
7.12 Analysis of variance (ANOVA) . . . . . . . . . . . . . . . . . . 583
7.13 Sample size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
7.14 Contingency tables . . . . . . . . . . . . . . . . . . . . . . . . . 595
7.15 Probability tables . . . . . . . . . . . . . . . . . . . . . . . . . . 598
Chapter 8
Scientific Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
8.1 Basic numerical analysis . . . . . . . . . . . . . . . . . . . . . . 616
8.2 Numerical linear algebra . . . . . . . . . . . . . . . . . . . . . . 629
8.3 Numerical integration and differentiation . . . . . . . . . . . . . 638
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Chapter 9
Mathematical Formulas from the Sciences . . . . . . . . . . . . . . . . . . . 659
9.1 Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661
9.2 Astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662
9.3 Atmospheric physics . . . . . . . . . . . . . . . . . . . . . . . . 664
9.4 Atomic Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 665
9.5 Basic mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 666
9.6 Beam dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 668
9.7 Classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . 669
9.8 Coordinate systems – Astronomical . . . . . . . . . . . . . . . . 670
9.9 Coordinate systems – Terrestrial . . . . . . . . . . . . . . . . . . 671

9.10 Earthquake engineering . . . . . . . . . . . . . . . . . . . . . . 672
9.11 Electromagnetic Transmission . . . . . . . . . . . . . . . . . . . 673
9.12 Electrostatics and magnetism . . . . . . . . . . . . . . . . . . . 674
9.13 Electronic circuits . . . . . . . . . . . . . . . . . . . . . . . . . 675
9.14 Epidemiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676
9.15 Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677
9.16 Fluid mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 678
9.17 Fuzzy logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679
9.18 Human body . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680
9.19 Image processing matrices . . . . . . . . . . . . . . . . . . . . . 681
9.20 Macroeconomics . . . . . . . . . . . . . . . . . . . . . . . . . . 682
9.21 Modeling physical systems . . . . . . . . . . . . . . . . . . . . . 683
9.22 Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684
9.23 Population genetics . . . . . . . . . . . . . . . . . . . . . . . . . 685
9.24 Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . 686
9.25 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688
9.26 Relativistic mechanics . . . . . . . . . . . . . . . . . . . . . . . 689
9.27 Solid mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 690
9.28 Statistical mechanics . . . . . . . . . . . . . . . . . . . . . . . . 691
9.29 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 692
Chapter 10
Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693
10.1 Calendar computations . . . . . . . . . . . . . . . . . . . . . . . 695
10.2 Cellular automata . . . . . . . . . . . . . . . . . . . . . . . . . . 696
10.3 Communication theory . . . . . . . . . . . . . . . . . . . . . . . 697
10.4 Control theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 702
10.5 Computer languages . . . . . . . . . . . . . . . . . . . . . . . . 704
10.6 Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705
10.7 Discrete dynamical systems and chaos . . . . . . . . . . . . . . . 706
10.8 Electronic resources . . . . . . . . . . . . . . . . . . . . . . . . 709

10.9 Elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 711
10.10 Financial formulas . . . . . . . . . . . . . . . . . . . . . . . . . 714
10.11 Game theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719
10.12 Knot theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722
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10.13 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724
10.14 Moments of inertia . . . . . . . . . . . . . . . . . . . . . . . . . 726
10.15 Music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727
10.16 Operations research . . . . . . . . . . . . . . . . . . . . . . . . 729
10.17 Recreational mathematics . . . . . . . . . . . . . . . . . . . . . 741
10.18 Risk analysis and decision rules . . . . . . . . . . . . . . . . . . 742
10.19 Signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . 744
10.20 Symbolic logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 750
10.21 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753
10.22 Voting power . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760
10.23 Greek alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . 762
10.24 Braille code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762
10.25 Morse code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762
List of References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767
List of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777
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Preface
It has long been the established policy of CRC Press to publish, in handbook form,
the most up-to-date, authoritative, logically arranged, and readily usable reference

material available.
Just as pocket calculators obviated the need for tables of square roots and
trigonometric functions; the internet has made many other tables and formulas un-
necessary. Prior to the preparation of this 32
nd
Edition of the CRC Standard Mathe-
matical Tables and Formulae, the content has been reconsidered. The criteria estab-
lished for inclusion in this edition are:
• information that is immediately useful as a reference (e.g., names of powers of
10, addition in hexadecimal);
• information about which many readers may be unaware and should know about
(e.g., visual proofs, sequences);
• information that is more complete or concise than that which can be found on
the internet (e.g., table of conformal mappings);
• information that cannot be found on the internet due to the difficulty of entering
a query (e.g., integral tables);
• illustrations of how mathematical information is interpreted.
Using these criteria, the previous edition has been carefully analyzed by practition-
ers from mathematics, engineering, and the sciences. As a result, numerous changes
have been made in several sections, and several new areas were added. These im-
provements include:
• There is a new chapter entitled “Mathematical Formulas from the Sciences.” It
contains, in concise form, the most important formulas from a variety of fields
(including: acoustics, astrophysics, . . .); a total of 26 topics.
• New material on contingency tables, estimators, process capability, runs test,
and sample sizes has been added to the statistics chapter.
• New material on cellular automata, knot theory, music, quaternions, and ratio-
nal trigonometry has been added.
• In many places, tables have been updated and streamlined. For example, the
prime number table now only goes to 8,000. Also, many of the tables in the

section on financial computations have been updated (while the examples illus-
trating those tables remained).
Of course, the same successful format which has characterized earlier editions of the
Handbook has been retained, while its presentation has been updated and made more
consistent from page to page. Material is presented in a multi-sectional format, with
each section containing a valuable collection of fundamental reference material—
tabular and expository.
xi
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In line with the established policy of CRC Press, the Handbook will be updated
in as current and timely a manner as is possible. Suggestions for the inclusion of new
material in subsequent editions and comments regarding the present edition are wel-
comed. The home page for this book, which will include errata, will be maintained
at .
This new edition of the Handbook will continue to support the needs of practi-
tioners of mathematics in the mathematical and scientific fields, as it has for over 80
years. Even as the internet becomes more ubiquitous, it is this editor’s opinion that
the new edition will continue to be a valued reference.
Updating this edition and making it a useful tool has been exciting. It would not have
been possible without the loving support of my family, Janet Taylor and Kent Taylor
Zwillinger.
Daniel Zwillinger

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Editor-in-Chief
Daniel Zwillinger
Rensselaer Polytechnic Institute
Troy, New York

Editorial Advisory Board
J. Douglas Faires
Youngstown State University
Youngstown, Ohio
Gerald B. Folland
University of Washington
Seattle, Washington
Contributors
Karen Bolinger
Clarion University
Clarion, Pennsylvania
Lawrence Glasser
Clarkson University
Potsdam, New York
Ray McLenaghan
University of Waterloo
Waterloo, Ontario, Canada
Roger B. Nelsen
Lewis & Clark College
Portland, Oregon
Joseph J. Rushanan
MITRE Corporation
Bedford, Massachusetts
Les Servi
MITRE Corporation
Bedford, Massachusetts
Neil J. A. Sloane
AT&T Bell Labs
Murray Hill, New Jersey
Gary L. Stanek

Youngstown State University
Youngstown, Ohio
Michael T. Strauss
HME
Newburyport, Massachusetts
Nico M. Temme
CWI
Amsterdam, The Netherlands
xiii
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Chapter 1
Numbers and
Elementary
Mathematics
1.1 PROOFS WITHOUT WORDS . . . . . . . . . . . . . . . . . . . . 3
1.2 CONSTANTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Binary prefixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Decimal multiples and prefixes . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 Interpretations of powers of 10 . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.4 Roman numerals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.5 Types of numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.6 DeMoivre’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.7 Representation of numbers . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.8 Symmetric base three representation . . . . . . . . . . . . . . . . . . . . . 9
1.2.9 Hexadecimal addition and subtraction table . . . . . . . . . . . . . . . . . 10
1.2.10 Hexadecimal multiplication table . . . . . . . . . . . . . . . . . . . . . . 10
1.2.11 Hexadecimal–decimal fraction conversion . . . . . . . . . . . . . . . . . . 11
1.3 SPECIAL NUMBERS . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1 Powers of 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.2 Powers of 16 in decimal scale . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.3 Powers of 10 in hexadecimal scale . . . . . . . . . . . . . . . . . . . . . . 13
1.3.4 Special constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.5 Factorials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.6 Bernoulli polynomials and numbers . . . . . . . . . . . . . . . . . . . . . 17
1.3.7 Euler polynomials and numbers . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.8 Fibonacci numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.9 Sums of powers of integers . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.10 Negative integer powers . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3.11 Integer sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.12 p-adic Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.13 de Bruijn sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4 NUMBER THEORY . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.4.1 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.4.2 Chinese remainder theorem . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4.3 Continued fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.4.4 Diophantine equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.4.5 Greatest common divisor . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.4.6 Least common multiple . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.4.7 M¨obius function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.4.8 Prime numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1
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2 CHAPTER 1. NUMBERS AND ELEMENTARY MATHEMATICS
1.4.9 Prime numbers of special forms . . . . . . . . . . . . . . . . . . . . . . . 35
1.4.10 Prime numbers less than 8,000 . . . . . . . . . . . . . . . . . . . . . . . 38
1.4.11 Factorization table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.4.12 Euler totient function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.5 SERIES AND PRODUCTS . . . . . . . . . . . . . . . . . . . . . 42

1.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.5.2 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.5.3 Convergence tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.5.4 Types of series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.5.5 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.5.6 Series expansions of special functions . . . . . . . . . . . . . . . . . . . . 54
1.5.7 Summation formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
1.5.8 Faster convergence: Shanks transformation . . . . . . . . . . . . . . . . . 58
1.5.9 Summability methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
1.5.10 Operations with power series . . . . . . . . . . . . . . . . . . . . . . . . 59
1.5.11 Miscellaneous sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
1.5.12 Infinite products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
1.5.13 Infinite products and infinite series . . . . . . . . . . . . . . . . . . . . . . 60
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1.1. PROOFS WITHOUT WORDS 3
1.1 PROOFS WITHOUT WORDS
—the Chou pei suan ching
(author unknown, circa
B.C. 200?)
The Pythagorean Theorem
A Property of the Sequence of Odd
Integers (Galileo, 1615)
1
3
1+3
5+7
1+3+5
7+9+11
=

=
. . .
=
1+3+
. . .
+(2n–1)
(2n+1)+(2n+3)+
. . .
+(4n–1)
1
3
=
1+2+
. . .
+ n =
n(n+1)
2
1+2+
. . .
+n =
n
2
2
n
2
+
1 1
. .
=
n(n+1)

2
—Ian Richards
1 + 3 + 5 +
. . .
+ (2n–1) = n
2
1+3+
. . .
+ (2n–1) =
1
4
(2n) = n
2 2
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4 CHAPTER 1. NUMBERS AND ELEMENTARY MATHEMATICS
Geometric Series
—Rick Mabry
2 3
1
4
1
3
+
1
4
+
+
. . .
=
1

4
1
1
r
1–r
2
r
r
2
r

1
Geometric Series
1 + r + r +

2
1
1
1 – r
=
—Benjamin G. Klein
and Irl C. Bivens
sinxsiny
cos
x
sin
y
sin
y
cosxcosy

sin
x
cos
y
1
x
y
x
cos
y
Addition Formulae for the Sine
and Cosine
sin(x + y) = sinxcosy + cosxsiny
cos(x + y) = cosxcosy – sinxsiny
d
(a,b)
(a,ma + c)
x
y
y = mx + c
|ma + c – b|
1 +
m
2
1
m
d |ma + c – b|
1
=
1 + m

2
The Distance Between a Point and a Line
—R. L. Eisenman
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1.2. CONSTANTS 5
The Arithmetic Mean-Geometric Mean
Inequality
a b
ab
a+b
2
—Charles D. Gallant
a,b > 0
a+b
ab
2
The Mediant Property
—Richard A. Gibbs
a
b
c
d
a + c
b + d
<<<
a
b
c
d
a

b
c
d
a
d
Reprinted from “Proofs Without Words: Exercises in Visual Thinking,” by
Roger B. Nelsen, 1997, MAA, pages: 3, 40, 49, 60, 70, 72, 115, 120. Copyright
The Mathematical Association of America. All rights reserved.
Reprinted from “Proofs Without Words II: More Exercises in Visual Thinking,”
by Roger B. Nelsen, 2001, MAA, pages 46, 111. Copyright The Mathematical As-
sociation of America. All rights reserved.
1.2 CONSTANTS
1.2.1 BINARY PREFIXES
A byte is 8 bits. A kibibyte is 2
10
= 1024 bytes. Other prefixes for power of 2 are:
Factor Prefix Symbol
2
10
kibi Ki
2
20
mebi Mi
2
30
gibi Gi
2
40
tebi Ti
2

50
pebi Pi
2
60
exbi Ei
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6 CHAPTER 1. NUMBERS AND ELEMENTARY MATHEMATICS
1.2.2 DECIMAL MULTIPLES AND PREFIXES
The prefix names and symbols below are taken from Conference G´en´erale des Poids
et Mesures, 1991. The common names are for the United States.
Factor Prefix Symbol Common name
10
(10
100
)
googolplex
10
100
googol
10
24
yotta Y heptillion
10
21
zetta Z hexillion
1 000 000 000 000 000 000 = 1 0
18
exa E quintillion
1 000 000 000 000 000 = 10
15

peta P quadrillion
1 000 000 000 000 = 10
12
tera T trillion
1 000 000 000 = 10
9
giga G billion
1 000 000 = 10
6
mega M million
1 000 = 10
3
kilo k thousand
100 = 10
2
hecto H hundred
10 = 10
1
deka da ten
0.1 = 10
−1
deci d tenth
0.01 = 10
−2
centi c hundredth
0.001 = 10
−3
milli m thousandth
0.000 001 = 10
−6

micro µ millionth
0.000 000 001 = 10
−9
nano n billionth
0.000 000 000 001 = 10
−12
pico p trillionth
0.000 000 000 000 001 = 10
−15
femto f quadrillionth
0.000 000 000 000 000 001 = 10
−18
atto a quintillionth
10
−21
zepto z hexillionth
10
−24
yocto y heptillionth
1.2.3 INTERPRETATIONS OF POWERS OF 10
10
−15
the radius of the hydrogen nucleus (a proton) in meters
10
−11
the likelihood of being dealt 13 top honors in bridge
10
−10
the radius of a hydrogen atom in meters
10

−9
the number of seconds it takes light to travel one foot
10
−6
the likelihood of being dealt a royal flush in poker
10
0
the density of water is 1 gram per milliliter
10
1
the number of fingers that people have
10
2
the number of stable elements in the periodic table
10
5
the number of hairs on a human scalp
10
6
the number of words in the English language
10
7
the number of seconds in a year
10
8
the speed of light in meters per second
10
9
the number of heartbeats in a lifetime for most mammals
10

10
the number of people on the earth
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1.2. CONSTANTS 7
10
15
the surface area of the earth in square meters
10
16
the age of the universe in seconds
10
18
the volume of water in the earth’s oceans in cubic meters
10
19
the number of possible positions of Rubik’s cube
10
21
the volume of the earth in cubic meters
10
24
the number of grains of sand in the Sahara desert
10
28
the mass of the earth in grams
10
33
the mass of the solar system in grams
10
50

the number of atoms in the earth
10
78
the volume of the universe in cubic meters
(Note: these numbers have been rounded to the nearest power of ten.)
1.2.4 ROMAN NUMERALS
The major symbols in Roman numerals are I = 1, V = 5, X = 10, L = 50, C = 1 00,
D = 500, and M = 1,000. The rules for constructing Roman numerals are:
1. A symbol following one of equal or greater value adds its value. (For example,
II = 2, XI = 11, and DV = 505.)
2. A symbol following one of lesser value has the lesser value subtracted from
the larger value. An I is only allowed to precede a V or an X, an X is only
allowed to precede an L or a C, and a C is only allowed to precede a D or
an M. (For example IV = 4, IX = 9, and XL = 40.)
3. When a symbol stands between two of greater value, its value is subtracted
from the second and the result is added to the first. (For example, XIV=
10+(5−1) = 14, CIX= 1 00 +(10−1) = 109, DXL= 500+(50−10) = 540.)
4. When two ways exist for representing a number, the one in which the symbol
of larger value occurs earlier in the string is preferred. (For example, 14 is
represented as XIV, not as VIX.)
Decimal number 1 2 3 4 5 6 7 8 9
Roman numeral I II III IV V VI VII VIII IX
10 14 50 200 400 500 600 999 1000
X XIV L CC CD D DC CMXCIX M
1950 1960 1970 1980 1990
MCML MCMLX MCMLXX MCMLXXX MCMXC
1995 1999 2000 2001 2011 2012
MCMXCV MCMXCIX MM MMI MMXI MMXII
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8 CHAPTER 1. NUMBERS AND ELEMENTARY MATHEMATICS
1.2.5 TYPES OF NUMBERS
1. Natural numbers The set of natural numbers, {0, 1, 2, . . .}, is customarily
denoted by N. Many authors do not consider 0 to be a natural number.
2. Integers The set of integers, {0, ±1, ±2, . . .}, is customarily denoted by Z.
3. Rational numbers The set of rational numbers, {
p
q
| p, q ∈ Z, q = 0}, is
customarily denoted by Q.
(a) Two fractions
p
q
and
r
s
are equal if and only if ps = qr.
(b) Addition of fractions is defined by
p
q
+
r
s
=
ps+qr
qs
.
(c) Multiplication of fractions is defined by
p
q

·
r
s
=
pr
qs
.
4. Real numbers Real numbers are defined to be converging sequences of
rational numbers or as decimals that might or might not repeat. The set of real
numbers is customarily denoted by R.
Real numbers can be divided into two subsets. One subset, the algebraic num-
bers, are real numbers which solve a polynomial equation in one variable with
integer coefficients. For example;
√
2 is an algebraic number because it solves
the polynomial equation x
2
− 2 = 0; and all rational numbers are algebraic.
Real numbers that are not algebraic numbers are called transcendental num-
bers. Examples of transcendental numbers include π and e.
5. Definition of infinity The real numbers are extended to include the symbols
+∞ and −∞ with the following definitions
(a) for x in R: −∞ < x < ∞
(b) for x in R: x + ∞ = ∞
(c) for x in R: x − ∞ = −∞
(d) for x in R:
x
∞
=
x

−∞
= 0
(e) if x > 0 then x · ∞ = ∞
(f) if x > 0 then x·(−∞) = −∞
(g) ∞ + ∞ = ∞
(h) −∞ −∞ = −∞
(i) ∞ ·∞ = ∞
(j) −∞ ·(−∞) = ∞
6. Complex numbers The set of complex numbers is customarily denoted
by C. They are numbers of the form a + bi, where i
2
= −1, and a and b are
real numbers.
Operation computation result
addition (a + bi) + (c + di) (a + c) + i(b + d)
multiplication (a + bi)(c + di) (ac − bd) + (ad + bc)i
reciprocal
1
a + bi

a
a
2
+ b
2

−

b
a

2
+ b
2

i
complex conjugate z = a + bi z = a −bi
Properties include: z + w = z + w and zw = z w.
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1.2. CONSTANTS 9
1.2.6 DEMOIVRE’S THEOREM
A complex number a + bi can be written in the form re
iθ
, where r
2
= a
2
+ b
2
and
tan θ = b/a. Because e
iθ
= cos θ + i sin θ,
(a + bi)
n
= r
n
(cos nθ + i sin nθ),
n
√

1 = co s
2kπ
n
+ i sin
2kπ
n
, k = 0, 1, . . . , n − 1.
n
√
−1 = c os
(2k + 1)π
n
+ i sin
(2k + 1)π
n
, k = 0, 1, . . . , n − 1.
(1.2.1)
1.2.7 REPRESENTATION OF NUMBERS
Numerals as usually written have radix or base 10, so the numeral a
n
a
n−1
. . . a
1
a
0
represents the number a
n
10
n

+ a
n−1
10
n−1
+ ···+ a
2
10
2
+ a
1
10 + a
0
. However,
other bases can be used, particularly bases 2, 8, and 16. When a number is written in
base 2, the number is said to be in binary notation. The names of other bases are:
2 binary
3 ternary
4 quaternary
5 quinary
6 senary
7 septenary
8 octal
9 nonary
10 decimal
11 undenary
12 duodecimal
16 hexadecimal
20 vigesimal
60 sexagesimal
When writing a number in base b, the digits used range from 0 to b − 1. If

b > 10, then the digit A stands for 10, B for 11, etc. When a base other than 10 is
used, it is indicated by a subscript:
10111
2
= 1 × 2
4
+ 0 ×2
3
+ 1 ×2
2
+ 1 ×2 + 1 = 23,
A3
16
= 10 × 16 + 3 = 163,
543
7
= 5 × 7
2
+ 4 ×7 + 3 = 276.
(1.2.2)
To convert a number from base 10 to base b, divide the number by b, and the
remainder will be the last digit. Then divide the quotient by b, using the remainder
as the previous digit. Continue this process until a quotient of 0 is obtained.
EXAMPLE To convert 573 to base 12, divide 573 by 12, yielding a quotient of 47 and a
remainder of 9; hence, “9” is the last digit. Divide 47 by 12, yielding a quotient of 3 and
a remainder of 11 (which we represent with a “B”). Divide 3 by 12 yielding a quotient
of 0 and a remainder of 3. Therefore, 573
10
= 3B9
12

.
Converting from base b to base r can be done by converting to and from base
10. However, it is simple to convert from base b to base b
n
. For example, to con-
vert 110111101
2
to base 16, group the digits in fours (because 16 is 2
4
), yielding
1 1011 1101
2
, and then convert each group of 4 to base 16 directly, yielding 1BD
16
.
1.2.8 SYMMETRIC BASE THREE REPRESENTATION
In this representation, powers of 3 are added and subtracted to represent numbers.
The symbols {↓, 0, ↑} are used for {−1, 0, 1}. For example “5” is written as ↑↓↓
since 5 = 9 − 3 − 1. To negate a number, turn its symbol upside down: “−5” is
written as ↓↑↑. Basic arithmetic operations are simple in this representation.
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10 CHAPTER 1. NUMBERS AND ELEMENTARY MATHEMATICS
1.2.9 HEXADECIMAL ADDITION AND SUBTRACTION TABLE
A = 10, B = 11, C = 12, D = 13, E = 14, F = 15.
Example: 6 + 2 = 8; hence 8 − 6 = 2 and 8 −2 = 6.
Example: 4 + E = 12; hence 12 − 4 = E and 12 − E = 4.
1 2 3 4 5 6 7 8 9 A B C D E F
1 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F 10
2 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F 10 11
3 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F 10 11 12

4 05 06 07 08 09 0A 0B 0C 0D 0E 0F 10 11 12 13
5 06 07 08 09 0A 0B 0C 0D 0E 0F 10 11 12 13 14
6 07 08 09 0A 0B 0C 0D 0E 0F 10 11 12 13 14 15
7 08 09 0A 0B 0C 0D 0E 0F 10 11 12 13 14 15 16
8 09 0A 0B 0C 0D 0E 0F 10 11 12 13 14 15 16 17
9 0A 0B 0C 0D 0E 0F 10 11 12 13 14 15 16 17 18
A 0B 0C 0D 0E 0F 10 11 12 13 14 15 16 17 18 19
B 0C 0D 0E 0F 10 11 12 13 14 15 16 17 18 19 1A
C 0D 0E 0F 10 11 12 13 14 15 16 17 18 19 1A 1B
D 0E 0F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C
E 0F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D
F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E
1.2.10 HEXADECIMAL MULTIPLICATION TABLE
Example: 2 ×4 = 8.
Example: 2 ×F = 1E.
1 2 3 4 5 6 7 8 9 A B C D E F
1 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F
2 02 04 06 08 0A 0C 0E 10 12 14 16 18 1A 1C 1E
3 03 06 09 0C 0F 12 15 18 1B 1E 21 24 27 2A 2D
4 04 08 0C 10 14 18 1C 20 24 28 2C 30 34 38 3C
5 05 0A 0F 14 19 1E 23 28 2D 32 37 3C 41 46 4B
6 06 0C 12 18 1E 24 2A 30 36 3C 42 48 4E 54 5A
7 07 0E 15 1C 23 2A 31 38 3F 46 4D 54 5B 62 69
8 08 10 18 20 28 30 38 40 48 50 58 60 68 70 78
9 09 12 1B 24 2D 36 3F 48 51 5A 63 6C 75 7E 87
A 0A 14 1E 28 32 3C 46 50 5A 64 6E 78 82 8C 96
B 0B 16 21 2C 37 42 4D 58 63 6E 79 84 8F 9A A5
C 0C 18 24 30 3C 48 54 60 6C 78 84 90 9C A8 B4
D 0D 1A 27 34 41 4E 5B 68 75 82 8F 9C A9 B6 C3
E 0E 1C 2A 38 46 54 62 70 7E 8C 9A A8 B6 C4 D2

F 0F 1E 2D 3C 4B 5A 69 78 87 96 A5 B4 C3 D2 E1
“smtf32” — 2011/5/20 — 2:09 — page 11 — #21
1.2. CONSTANTS 11
1.2.11 HEXADECIMAL–DECIMAL FRACTION CONVERSION
Hex Decimal
.00 0
.01 0.0039
.02 0.0078
.03 0.0117
.04 0.0156
.05 0.0195
.06 0.0234
.07 0.0273
.08 0.0313
.09 0.0352
.0A 0.0391
.0B 0.0430
.0C 0.0469
.0D 0.0508
.0E 0.0547
.0F 0.0586
.10 0.0625
.11 0.0664
.12 0.0703
.13 0.0742
.14 0.0781
.15 0.0820
.16 0.0859
.17 0.0898
.18 0.0938

.19 0.0977
.1A 0.1016
.1B 0.1055
.1C 0.1094
.1D 0.1133
.1E 0.1172
.1F 0.1210
.20 0.1250
.21 0.1289
.22 0.1328
.23 0.1367
.24 0.1406
.25 0.1445
.26 0.1484
.27 0.1523
.28 0.1563
.29 0.1602
.2A 0.1641
.2B 0.1680
.2C 0.1719
.2D 0.1758
.2E 0.1797
.2F 0.1836
Hex Decimal
.30 0.1875
.31 0.1914
.32 0.1953
.33 0.1992
.34 0.2031
.35 0.2070

.36 0.2109
.37 0.2148
.38 0.2188
.39 0.2227
.3A 0.2266
.3B 0.2305
.3C 0.2344
.3D 0.2383
.3E 0.2422
.3F 0.2461
.40 0.2500
.41 0.2539
.42 0.2578
.43 0.2617
.44 0.2656
.45 0.2695
.46 0.2734
.47 0.2773
.48 0.2813
.49 0.2852
.4A 0.2891
.4B 0.2930
.4C 0.2969
.4D 0.3008
.4E 0.3047
.4F 0.3086
.50 0.3125
.51 0.3164
.52 0.3203
.53 0.3242

.54 0.3281
.55 0.3320
.56 0.3359
.57 0.3398
.58 0.3438
.59 0.3477
.5A 0.3516
.5B 0.3555
.5C 0.3594
.5D 0.3633
.5E 0.3672
.5F 0.3711
Hex Decimal
.60 0.3750
.61 0.3789
.62 0.3828
.63 0.3867
.64 0.3906
.65 0.3945
.66 0.3984
.67 0.4023
.68 0.4063
.69 0.4102
.6A 0.4141
.6B 0.4180
.6C 0.4219
.6D 0.4258
.6E 0.4297
.6F 0.43365
.70 0.4375

.71 0.4414
.72 0.4453
.73 0.4492
.74 0.4531
.75 0.4570
.76 0.4609
.77 0.4648
.78 0.4688
.79 0.4727
.7A 0.4766
.7B 0.4805
.7C 0.4844
.7D 0.4883
.7E 0.4922
.7F 0.4961
.80 0.5000
.81 0.5039
.82 0.5078
.83 0.5117
.84 0.5156
.85 0.5195
.86 0.5234
.87 0.5273
.88 0.5313
.89 0.5352
.8A 0.5391
.8B 0.5430
.8C 0.5469
.8D 0.5508
.8E 0.5547

.8F 0.5586
Hex Decimal
.90 0.5625
.91 0.5664
.92 0.5703
.93 0.5742
.94 0.5781
.95 0.5820
.96 0.5859
.97 0.5898
.98 0.5938
.99 0.5977
.9A 0.6016
.9B 0.6055
.9C 0.6094
.9D 0.6133
.9E 0.6172
.9F 0.6211
.A0 0.6250
.A1 0.6289
.A2 0.6328
.A3 0.6367
.A4 0.6406
.A5 0.6445
.A6 0.6484
.A7 0.6523
.A8 0.6563
.A9 0.6602
.AA 0.6641
.AB 0.6680

.AC 0.6719
.AD 0.6758
.AE 0.6797
.AF 0.68365
.B0 0.6875
.B1 0.6914
.B2 0.6953
.B3 0.6992
.B4 0.7031
.B5 0.7070
.B6 0.7109
.B7 0.7148
.B8 0.7188
.B9 0.7227
.BA 0.7266
.BB 0.7305
.BC 0.7344
.BD 0.7383
.BE 0.7422
.BF 0.7461
Hex Decimal
.C0 0.7500
.C1 0.7539
.C2 0.7578
.C3 0.7617
.C4 0.7656
.C5 0.7695
.C6 0.7734
.C7 0.7773
.C8 0.7813

.C9 0.7852
.CA 0.7891
.CB 0.7930
.CC 0.7969
.CD 0.8008
.CE 0.8047
.CF 0.8086
.D0 0.8125
.D1 0.8164
.D2 0.8203
.D3 0.8242
.D4 0.8281
.D5 0.8320
.D6 0.8359
.D7 0.8398
.D8 0.8438
.D9 0.8477
.DA 0.8516
.DB 0.8555
.DC 0.8594
.DD 0.8633
.DE 0.8672
.DF 0.8711
.E0 0.8750
.E1 0.8789
.E2 0.8828
.E3 0.8867
.E4 0.8906
.E5 0.8945
.E6 0.8984

.E7 0.9023
.E8 0.9063
.E9 0.9102
.EA 0.9141
.EB 0.9180
.EC 0.9219
.ED 0.9258
.EE 0.9297
.EF 0.9336
Hex Decimal
.F0 0.9375
.F1 0.9414
.F2 0.9453
.F3 0.9492
.F4 0.9531
.F5 0.9570
.F6 0.9609
.F7 0.9648
.F8 0.9688
.F9 0.9727
.FA 0.9766
.FB 0.9805
.FC 0.9844
.FD 0.9883
.FE 0.9922
.FF 0.9961