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CRC
STANDARD
MATHEMATICAL
TABLES AND
FORMULAS
33RD EDITION

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Advances in Applied Mathematics
Series Editor: Daniel Zwillinger

Published Titles
Advanced Engineering Mathematics with MATLAB, Fourth Edition
Dean G. Duffy
CRC Standard Curves and Surfaces with Mathematica®, Third Edition
David H. von Seggern
CRC Standard Mathematical Tables and Formulas, 33rd Edition
Dan Zwillinger
Dynamical Systems for Biological Modeling: An Introduction
Fred Brauer and Christopher Kribs
Fast Solvers for Mesh-Based Computations Maciej Paszy´nski
Green’s Functions with Applications, Second Edition Dean G. Duffy
Handbook of Peridynamic Modeling Floriin Bobaru, John T. Foster,
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Introduction to Financial Mathematics Kevin J. Hastings
Linear and Complex Analysis for Applications John P. D’Angelo
Linear and Integer Optimization: Theory and Practice, Third Edition
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Markov Processes James R. Kirkwood
Pocket Book of Integrals and Mathematical Formulas, 5th Edition
Ronald J. Tallarida
Stochastic Partial Differential Equations, Second Edition Pao-Liu Chow
Quadratic Programming with Computer Programs Michael J. Best

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Advances in Applied Mathematics

CRC
STANDARD
MATHEMATICAL
TABLES AND
FORMULAS
33RD EDITION

Edited by

Dan Zwillinger

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Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 1
Numbers and Elementary Mathematics . . . . . . . . . . . . . . . . . . .
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9


Proofs without words . . . . . .
Constants . . . . . . . . . . . .
Special numbers . . . . . . . .
Interval analysis . . . . . . . .
Fractal Arithmetic . . . . . . .
Max-Plus Algebra . . . . . . .
Coupled-analogues of Functions
Number theory . . . . . . . . .
Series and products . . . . . . .

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1
3
5
13
24
25
26
27
28
47

Chapter 2
Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.1
2.2
2.3
2.4
2.5


Elementary algebra . . . .
Polynomials . . . . . . .
Vector algebra . . . . . .
Linear and matrix algebra
Abstract algebra . . . . .

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Chapter 3
Discrete Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.1
3.2
3.3
3.4
3.5

Sets . . . . . . . . . . . . .
Combinatorics . . . . . . .
Graphs . . . . . . . . . . .
Combinatorial design theory
Difference equations . . . .

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140
151
172

184

Chapter 4
Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
4.1
4.2
4.3
4.4
4.5
4.6
4.7

Euclidean geometry . . . . . . .
Grades and Degrees . . . . . . .
Coordinate systems in the plane .
Plane symmetries or isometries .
Other transformations of the plane
Lines . . . . . . . . . . . . . . .
Polygons . . . . . . . . . . . . .

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193
194
200
207
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vi

Table of Contents

4.8
4.9
4.10
4.11
4.12
4.13
4.14

4.15
4.16
4.17
4.18
4.19
4.20
4.21
4.22

Surfaces of revolution: the torus . . .
Quadrics . . . . . . . . . . . . . . .
Spherical geometry and trigonometry
Conics . . . . . . . . . . . . . . . .
Special plane curves . . . . . . . . .
Coordinate systems in space . . . . .
Space symmetries or isometries . . .
Other transformations of space . . . .
Direction angles and direction cosines
Planes . . . . . . . . . . . . . . . . .
Lines in space . . . . . . . . . . . .
Polyhedra . . . . . . . . . . . . . . .
Cylinders . . . . . . . . . . . . . . .
Cones . . . . . . . . . . . . . . . . .
Differential geometry . . . . . . . . .

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219
219
224
229
240

249
252
255
257
257
259
261
265
265
267

Chapter 5
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14

Differential calculus . . . . . . . .
Differential forms . . . . . . . . .

Integration . . . . . . . . . . . . .
Table of indefinite integrals . . . .
Table of definite integrals . . . . .
Ordinary differential equations . . .
Partial differential equations . . . .
Integral equations . . . . . . . . . .
Tensor analysis . . . . . . . . . . .
Orthogonal coordinate systems . . .
Real analysis . . . . . . . . . . . .
Generalized functions . . . . . . .
Complex analysis . . . . . . . . . .
Significant Mathematical Equations

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277
288
291
305
343
350
362
375
378
388
393
403
405
417

Chapter 6
Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9

Ceiling and floor functions . . . . . . .
Exponentiation . . . . . . . . . . . . .
Exponential function . . . . . . . . . .
Logarithmic functions . . . . . . . . .
Trigonometric functions . . . . . . . .
Circular functions and planar triangles .
Tables of trigonometric functions . . .
Angle conversion . . . . . . . . . . . .
Inverse circular functions . . . . . . . .

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421
422
422
424
433
437
440
441


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Table of Contents

6.10
6.11
6.12

6.13
6.14
6.15
6.16
6.17
6.18
6.19
6.20
6.21
6.22
6.23
6.24
6.25
6.26
6.27
6.28
6.29
6.30
6.31
6.32
6.33
6.34
6.35
6.36
6.37
6.38
6.39
6.40
6.41


Hyperbolic functions . . . . . . . . .
Inverse hyperbolic functions . . . . .
Gudermannian function . . . . . . .
Orthogonal polynomials . . . . . . .
Clebsch–Gordan coefficients . . . . .
Bessel functions . . . . . . . . . . .
Beta function . . . . . . . . . . . . .
Elliptic integrals . . . . . . . . . . .
Jacobian elliptic functions . . . . . .
Error functions . . . . . . . . . . . .
Fresnel integrals . . . . . . . . . . .
Gamma function . . . . . . . . . . .
Hypergeometric functions . . . . . .
Lambert Function . . . . . . . . . . .
Legendre functions . . . . . . . . . .
Polylogarithms . . . . . . . . . . . .
Prolate Spheroidal Wave Functions .
Sine, cosine, and exponential integrals
Weierstrass Elliptic Function . . . . .
Integral transforms: List . . . . . . .
Integral transforms: Preliminaries . .
Fourier integral transform . . . . . .
Discrete Fourier transform (DFT) . .
Fast Fourier transform (FFT) . . . . .
Multidimensional Fourier transforms
Hankel transform . . . . . . . . . . .
Hartley transform . . . . . . . . . . .
Hilbert transform . . . . . . . . . . .
Laplace transform . . . . . . . . . .
Mellin transform . . . . . . . . . . .

Z-Transform . . . . . . . . . . . . .
Tables of transforms . . . . . . . . .

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443
447
449
451
458
460
469
470
473
475

476
478
481
483
484
488
489
490
492
493
494
494
500
502
502
503
504
505
508
512
512
517

Chapter 7
Probability and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 533
7.1
7.2
7.3
7.4
7.5

7.6
7.7
7.8
7.9

Probability theory . . . . . .
Classical probability problems
Probability distributions . . .
Queuing theory . . . . . . . .
Markov chains . . . . . . . .
Random number generation .
Random matrices . . . . . . .
Control charts and reliability .
Statistics . . . . . . . . . . .

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535
545
553

562
565
568
574
575
580


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viii

Table of Contents

7.10
7.11
7.12
7.13
7.14
7.15
7.16
7.17

Confidence intervals . . . . . .
Tests of hypotheses . . . . . . .
Linear regression . . . . . . . .
Analysis of variance (ANOVA)
Sample size . . . . . . . . . . .
Contingency tables . . . . . . .
Acceptance sampling . . . . . .

Probability tables . . . . . . . .

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588
595
609
613
620
623
626
628

Chapter 8
Scientific Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
8.1
8.2
8.3
8.4


Basic numerical analysis . . . . . . . . .
Numerical linear algebra . . . . . . . . .
Numerical integration and differentiation
Programming techniques . . . . . . . . .

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646
659
668
688

Chapter 9
Mathematical Formulas from the Sciences . . . . . . . . . . . . . . . . . 689
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
9.10
9.11
9.12
9.13

9.14
9.15
9.16
9.17
9.18
9.19
9.20
9.21
9.22
9.23
9.24
9.25
9.26

Acoustics . . . . . . . . . . . . . .
Astrophysics . . . . . . . . . . . .
Atmospheric physics . . . . . . . .
Atomic Physics . . . . . . . . . . .
Basic mechanics . . . . . . . . . .
Beam dynamics . . . . . . . . . . .
Biological Models . . . . . . . . .
Chemistry . . . . . . . . . . . . . .
Classical mechanics . . . . . . . .
Coordinate systems – Astronomical
Coordinate systems – Terrestrial . .
Earthquake engineering . . . . . .
Economics (Macro) . . . . . . . .
Electromagnetic Transmission . . .
Electrostatics and magnetism . . .
Electromagnetic Field Equations . .

Electronic circuits . . . . . . . . .
Epidemiology . . . . . . . . . . . .
Fluid mechanics . . . . . . . . . .
Human body . . . . . . . . . . . .
Modeling physical systems . . . . .
Optics . . . . . . . . . . . . . . . .
Population genetics . . . . . . . . .
Quantum mechanics . . . . . . . .
Quaternions . . . . . . . . . . . . .
Radar . . . . . . . . . . . . . . . .

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691
692
694
695
696

698
699
700
701
702
703
704
705
707
708
709
710
711
712
713
714
715
716
717
719
720


“smtf33” — 2017/12/6 — 19:00 — page ix — #5

Table of Contents

9.27
9.28
9.29

9.30

Relativistic mechanics
Solid mechanics . . .
Statistical mechanics .
Thermodynamics . . .

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721
722
723
724

Chapter 10
Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725
10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8
10.9
10.10
10.11
10.12
10.13
10.14
10.15
10.16
10.17
10.18
10.19

10.20
10.21
10.22
10.23
10.24
10.25
10.26
10.27
10.28

Calendar computations . . . . . . . . .
Cellular automata . . . . . . . . . . . .
Communication theory . . . . . . . . .
Control theory . . . . . . . . . . . . .
Computer languages . . . . . . . . . .
Compressive Sensing . . . . . . . . . .
Constrained Least Squares . . . . . . .
Cryptography . . . . . . . . . . . . . .
Discrete dynamical systems and chaos .
Elliptic curves . . . . . . . . . . . . .
Financial formulas . . . . . . . . . . .
Game theory . . . . . . . . . . . . . .
Knot theory . . . . . . . . . . . . . . .
Lattices . . . . . . . . . . . . . . . . .
Logic . . . . . . . . . . . . . . . . . .
Moments of inertia . . . . . . . . . . .
Music . . . . . . . . . . . . . . . . . .
Operations research . . . . . . . . . .
Proof Methods . . . . . . . . . . . . .
Recreational mathematics . . . . . . .

Risk analysis and decision rules . . . .
Signal processing . . . . . . . . . . . .
Units . . . . . . . . . . . . . . . . . .
Voting power . . . . . . . . . . . . . .
Greek alphabet . . . . . . . . . . . . .
Braille code . . . . . . . . . . . . . . .
Morse code . . . . . . . . . . . . . . .
Bar Codes . . . . . . . . . . . . . . . .

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727
728
729
734
736

737
738
739
740
743
746
754
757
759
761
766
767
769
781
782
783
785
794
801
803
803
803
804

List of References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809
List of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819

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“smtf33” — 2017/12/6 — 19:00 — page xi — #7

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 have replaced tables of square roots and trig functions;
the internet has made printed tabulation of many tables and formulas unnecessary.
As the content and capabilities of the internet continue to grow, the content of this
book also evolves. For this edition of Standard Mathematical Tables and Formulae
the content was reconsidered and reviewed. The criteria for inclusion in this edition
includes:
• information that is immediately useful as a reference (e.g., interpretation of
powers of 10);
• information that is useful and not commonly known (e.g., proof methods);

• information that is more complete or concise than that which can be easily
found on the internet (e.g., table of conformal mappings);
• information difficult to find on the internet due to the challenges of entering an
appropriate query (e.g., integral tables).

Applying these criteria, practitioners from mathematics, engineering, and the sciences have made changes in several sections and have added new material.
• The “Mathematical Formulas from the Sciences” chapter now includes topics

from biology, chemistry, and radar.
• Material has been augmented in many areas, including: acceptance sampling,
card games, lattices, and set operations.
• New material has been added on the following topics: continuous wavelet transform, contour integration, coupled analogues, financial options, fractal arithmetic, generating functions, linear temporal logic, matrix pseudospectra, max
plus algebra, proof methods, and two dimensional integrals.
• Descriptions of new functions have been added: Lambert, prolate spheroidal,
and Weierstrass.
Of course, the same successful format which has characterized earlier editions of the
Handbook has been retained. 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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xii

Preface

In line with the established policy of CRC Press, the Handbook will be updated
in as current and timely manner as is possible. Suggestions for the inclusion of new
material in subsequent editions and comments regarding the present edition are welcomed. 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 practitioners of mathematics in the mathematical and scientific fields, as it has for almost
90 years. Even as the internet becomes more powerful, it is this editor’s opinion that
the new edition will continue to be a valued reference.


MATLAB R is a registered trademark of The MathWorks, Inc.
For product information please contact:
The MathWorks, Inc.
3 Apple Hill Drive
Natick, MA, 01760-2098 USA
Tel: 508-647-7000
Fax: 508-647-7001
E-mail:
Web: www.mathworks.com

Every book takes time and care. This book would not have been possible without the
loving support of my wife, Janet Taylor, and my son, Kent Zwillinger.

May 2017
Daniel Zwillinger


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“smtf33” — 2017/12/6 — 19:00 — page xiii — #9

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

Contributors
George E. Andrews
Evan Pugh University Professor in

Mathematics
The Pennsylvania State University
University Park, Pennsylvania

Roger B. Nelsen
Professor Emeritus of Mathematics
Lewis & Clark College
Portland, Oregon
Dr. Joseph J. Rushanan
MITRE Corporation
Bedford, Massachusetts

Lawrence Glasser
Professor of Physics Emeritus
Clarkson University
Potsdam, New York

Dr. Les Servi
MITRE Corporation
Bedford, Massachusetts

Michael Mascagni
Professor of Computer Science
Professor of Mathematics
Professor of Scientific Computing
Florida State University
Tallahassee, Florida

Dr. Michael T. Strauss
President HME

Newburyport, Massachusetts

Ray McLenaghan
Adjunct Professor in Department of
Applied Mathematics
University of Waterloo
Waterloo, Ontario, Canada

Ahmed I. Zayed
Professor in Department of
Mathematical Sciences
DePaul University
Chicago, Illinois

Dr. Nico M. Temme
Centrum Wiskunde & Informatica
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
1.2.2
1.2.3
1.2.4
1.2.5
1.2.6
1.2.7
1.2.8
1.2.9
1.2.10
1.2.11
1.2.12
1.2.13
1.2.14

1.3


Divisibility tests . . . . . . . . . . . . . . . . . . . . .
Decimal multiples and prefixes . . . . . . . . . . . . .
Binary prefixes . . . . . . . . . . . . . . . . . . . . .
Interpretations of powers of 10 . . . . . . . . . . . . .
Numerals in different languages . . . . . . . . . . . . .
Roman numerals . . . . . . . . . . . . . . . . . . . .
Types of numbers . . . . . . . . . . . . . . . . . . . .
Representation of numbers . . . . . . . . . . . . . . .
Representation of complex numbers – DeMoivre’s theorem
Arrow notation . . . . . . . . . . . . . . . . . . . . .
Ones and Twos Complement . . . . . . . . . . . . . . .
Symmetric base three representation . . . . . . . . . . .
Hexadecimal addition and subtraction table . . . . . . .
Hexadecimal multiplication table . . . . . . . . . . . .

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5
6
6
7
8
8
9
10
10
11
11
11
12
12


SPECIAL NUMBERS . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1
1.3.2
1.3.3
1.3.4
1.3.5
1.3.6
1.3.7
1.3.8
1.3.9
1.3.10
1.3.11
1.3.12

Powers of 2 . . . . . . . . . . . .
Powers of 10 in hexadecimal . . . .
Special constants . . . . . . . . .
Factorials . . . . . . . . . . . .
Bernoulli polynomials and numbers
Euler polynomials and numbers . .
Fibonacci numbers . . . . . . . .
Sums of powers of integers . . . .
Negative integer powers . . . . . .
Integer sequences . . . . . . . . .
p-adic Numbers . . . . . . . . . .
de Bruijn sequences . . . . . . . .

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13
13
14
16
17
18
18
19
20
21
23
23

1.4

INTERVAL ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . 24

1.5

FRACTAL ARITHMETIC . . . . . . . . . . . . . . . . . . . . . . 25

1

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“smtf33” — 2017/12/6 — 19:00 — page 2 — #12

2


CHAPTER 1. NUMBERS AND ELEMENTARY MATHEMATICS

1.6

MAX-PLUS ALGEBRA . . . . . . . . . . . . . . . . . . . . . . . 26

1.7

COUPLED-ANALOGUES OF FUNCTIONS . . . . . . . . . . . . 27
1.7.1

1.8

27

NUMBER THEORY . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.8.1
1.8.2
1.8.3
1.8.4
1.8.5
1.8.6
1.8.7
1.8.8
1.8.9
1.8.10
1.8.11
1.8.12


1.9

Coupled-operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Congruences . . . . . . . . .
Chinese remainder theorem . .
Continued fractions . . . . . .
Diophantine equations . . . .
Greatest common divisor . . .
Least common multiple . . . .
Möbius function . . . . . . . .
Prime numbers . . . . . . . .
Prime numbers of special forms
Prime numbers less than 7,000
Factorization table . . . . . .
Euler totient function . . . . .

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28
29
30
32
35
35
36
37
39
42
44
46

SERIES AND PRODUCTS . . . . . . . . . . . . . . . . . . . . . 47

1.9.1
1.9.2
1.9.3
1.9.4
1.9.5
1.9.6
1.9.7
1.9.8
1.9.9
1.9.10
1.9.11
1.9.12
1.9.13

Definitions . . . . . . . . . . . . . . . .
General properties . . . . . . . . . . . .
Convergence tests . . . . . . . . . . . . .
Types of series . . . . . . . . . . . . . .
Fourier series . . . . . . . . . . . . . . .
Series expansions of special functions . . .
Summation formulas . . . . . . . . . . .
Faster convergence: Shanks transformation
Summability methods . . . . . . . . . . .
Operations with power series . . . . . . .
Miscellaneous sums . . . . . . . . . . . .
Infinite products . . . . . . . . . . . . . .
Infinite products and infinite series . . . . .

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47
47

49
50
54
59
63
63
64
64
64
65
65


“smtf33” — 2017/12/6 — 19:00 — page 3 — #13

1.1. PROOFS WITHOUT WORDS

1.1

3

PROOFS WITHOUT WORDS
A Property of the Sequence of Odd
Integers (Galileo, 1615)
1
1+3
1+3+5
=
=
=...

3
5+7
7+9+11

The Pythagorean Theorem

—the Chou pei suan ching
(author unknown, circa B.C. 200?)

n(n+1)
1+2+ . . . + n =
2

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

2
1 + 3 + 5 + . . . + (2n–1) = n

1
1
n(n+1)
1+2+ . . . +n = . n 2 + n . =
2

2
2
—Ian Richards

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

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“smtf33” — 2017/12/6 — 19:00 — page 4 — #14

4

CHAPTER 1. NUMBERS AND ELEMENTARY MATHEMATICS

Geometric Series

Geometric Series

...
r2

r2

r
r

1–r


1

1

1

1
1
+
4
4

2

+

1
4

3

2

1 + r + r + ... = 1
1
1–r

1
3


+ . .. =

—Benjamin G. Klein
and Irl C. Bivens

—Rick Mabry

Addition Formulae for the Sine
and Cosine

The Distance Between a Point and a Line
y

sinxsiny

cosxsiny
2

y
sin

x

(a,ma + c)

1+

m

1


y

|ma + c – b|

1

sinxcosy

sy
co

m

d

(a,b)

x

x
y = mx + c

cosxcosy

sin(x + y) = sinxcosy + cosxsiny
cos(x + y) = cosxcosy – sinxsiny

www.pdfgrip.com


d |ma + c – b|
=
1
1 + m2
—R. L. Eisenman


“smtf33” — 2017/12/6 — 19:00 — page 5 — #15

1.2. CONSTANTS

The Arithmetic Mean-Geometric Mean
Inequality
a,b > 0

a+b
2

ab

5

The Mediant Property
a
c
a +c
<
<
b
b +d d


a
c
<
b
d

c
a+b
2

d

ab

a

a
b

b
—Charles D. Gallant

a
d

—Richard A. Gibbs

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 Association of America. All rights reserved.

1.2

CONSTANTS

1.2.1

DIVISIBILITY TESTS

1.
2.
3.
4.
5.
6.
7.
8.

Divisibility by 2: the last digit is divisible by 2
Divisibility by 3: the sum of the digits is divisible by 3
Divisibility by 4: the number formed from the last 2 digits is divisible by 4
Divisibility by 5: the last digit is either 0 or 5
Divisibility by 6: is divisible by both 2 and 3
Divisibility by 9: the sum of the digits is divisible by 9
Divisibility by 10: the last digit is 0
Divisibility by 11: the difference between the sum of the odd digits and the
sum of the even digits is divisible by 11

EXAMPLE Consider the number N = 1036728.
• The last digit is 8, so N is divisible by 2.
• The last two digits are 28 which is divisible by 4, so N is divisible by 4.
• The sum of the digits is 27 = 1 + 0 + 3 + 6 + 7 + 2 + 8. This is divisible by 3, so N
is divisible by 3. This is also divisible by 9, so N is divisible by 9.
• The sum of the odd digits is 19 = 1 + 3 + 7 + 8 and the sum of the even digits is
8 = 6 + 2; the difference is 19 − 8 = 11. This is divisible by 11, so N is divisible
by 11.

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“smtf33” — 2017/12/6 — 19:00 — page 6 — #16

6

CHAPTER 1. NUMBERS AND ELEMENTARY MATHEMATICS

1.2.2

DECIMAL MULTIPLES AND PREFIXES

The prefix names and symbols below are taken from Conference Générale des Poids
et Mesures, 1991. The common names are for the United States.
Factor
100

10(10
10100
1024

1021
1 000 000 000 000 000 000 = 1018
1 000 000 000 000 000 = 1015
1 000 000 000 000 = 1012
1 000 000 000 = 109
1 000 000 = 106
1 000 = 103
100 = 102
10 = 101
0.1 = 10−1
0.01 = 10−2
0.001 = 10−3
0.000 001 = 10−6
0.000 000 001 = 10−9
0.000 000 000 001 = 10−12
0.000 000 000 000 001 = 10−15
0.000 000 000 000 000 001 = 10−18
10−21
10−24

1.2.3

Prefix

Symbol

Common name

Y
Z

E
P
T
G
M
k
H
da
d
c
m
µ
n
p
f
a
z
y

googolplex
googol
heptillion
hexillion
quintillion
quadrillion
trillion
billion
million
thousand
hundred

ten
tenth
hundredth
thousandth
millionth
billionth
trillionth
quadrillionth
quintillionth
hexillionth
heptillionth

)

yotta
zetta
exa
peta
tera
giga
mega
kilo
hecto
deka
deci
centi
milli
micro
nano
pico

femto
atto
zepto
yocto

BINARY PREFIXES

A byte is 8 bits. A kibibyte is 210 = 1024 bytes. Other prefixes for power of 2 are:
Factor Prefix Symbol
210
220
230
240
250
260

kibi
mebi
gibi
tebi
pebi
exbi

Ki
Mi
Gi
Ti
Pi
Ei


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“smtf33” — 2017/12/6 — 19:00 — page 7 — #17

1.2. CONSTANTS

1.2.4

INTERPRETATIONS OF POWERS OF 10
10−43
10−35
10−30
10−27
10−15
10−11
10−10
10−9
10−6
100
101
102
104
105
106
107
108
109
1010
1011

1013
1014
1015
1016
1017
1018
1019
1021
1024
1025
1030
1050
1052
1054
1078

Planck time in seconds
Planck length in meters
mass of an electron in kilograms
mass of a proton in kilograms
the radius of the hydrogen nucleus (a proton) in meters
the likelihood of being dealt 13 top honors in bridge
the (Bohr) radius of a hydrogen atom in meters
the number of seconds it takes light to travel one foot
the likelihood of being dealt a royal flush in poker
the density of water is 1 gram per milliliter
the number of fingers that people have
the number of stable elements in the periodic table
the speed of the Earth around the sun in meters/second
the number of hairs on a human scalp

the number of words in the English language
the number of seconds in a year
the speed of light in meters per second
the number of heartbeats in a lifetime for most mammals
the number of people on the earth
the distance from the Earth to the sun in meters
diameter of the solar system in meters
number of cells in the human body
the surface area of the earth in square meters
the number of meters light travels in one year
the age of the universe in seconds
the volume of water in the earth’s oceans in cubic meters
the number of possible positions of Rubik’s cube
the volume of the earth in cubic meters
the number of grains of sand in the Sahara desert
the mass of the earth in kilograms
the mass of the sun in kilograms
the number of atoms in the earth
the mass of the observable universe in kilograms
the number of elements in the monster group
the volume of the universe in cubic meters

(Note: these numbers have been rounded to the nearest power of ten.)

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7


“smtf33” — 2017/12/6 — 19:00 — page 8 — #18


8

CHAPTER 1. NUMBERS AND ELEMENTARY MATHEMATICS

1.2.5

NUMERALS IN DIFFERENT LANGUAGES

1.2.6

ROMAN NUMERALS

The major symbols in Roman numerals are I = 1, V = 5, X = 10, L = 50, C = 100,
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= 100+(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
1995
1999
2000 2001
2017
2018
MCMXCV MCMXCIX MM MMI MMXVII MMXVIII

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“smtf33” — 2017/12/6 — 19:00 — page 9 — #19

1.2. CONSTANTS

1.2.7


9

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, { pq | p, q ∈ Z, q = 0}, is
customarily denoted by Q.
(a) Two fractions pq and rs are equal if and only if ps = qr.
(b) Addition of fractions is defined by pq + rs = ps+qr
qs .
(c) Multiplication of fractions is defined by pq · rs = 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 numbers, 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 x2 − 2 = 0; and all rational numbers are algebraic.
Real numbers that are not algebraic numbers are called transcendental numbers. Examples of transcendental numbers include π and e.
5. Definition of infinity
The real numbers are extended to R by the inclusion
of +∞ and −∞ with the following definitions

(a)
(b)
(c)

for x in R: −∞ < x < ∞
for x in R: x + ∞ = ∞
for x in R: x − ∞ = −∞

(d)

for x in R:

x
x
=
=0
∞
−∞

(e)
(f)
(g)
(h)
(i)
(j)

if x > 0 then x · ∞ = ∞
if x > 0 then x·(−∞) = −∞
∞+∞= ∞
−∞ − ∞ = −∞

∞·∞= ∞
−∞ · (−∞) = ∞

6. Complex numbers
The set of complex numbers is customarily denoted
by C. They are numbers of the form a + bi, where i2 = −1, and a and b are
real numbers.
Operation
addition
multiplication

computation
(a + bi) + (c + di)
(a + bi)(c + di)
1
reciprocal
a + bi
complex conjugate z = a + bi

result
(a + c) + i(b + d)
(ac − bd) + (ad + bc)i
a
b
−
2
2
2
a +b
a + b2

z = a − bi

Properties include: z + w = z + w and zw = z w.

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“smtf33” — 2017/12/6 — 19:00 — page 10 — #20

10

CHAPTER 1. NUMBERS AND ELEMENTARY MATHEMATICS

1.2.8

REPRESENTATION OF NUMBERS

Numerals as usually written have radix or base 10, so the numeral an an−1 . . . a1 a0
represents the number an 10n + an−1 10n−1 + · · · + a2 102 + a1 10 + a0 . 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
6 senary
10 decimal
20 vigesimal
3 ternary
7 septenary
11 undenary

60 sexagesimal
4 quaternary
8 octal
12 duodecimal
5 quinary
9 nonary
16 hexadecimal
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:
101112 = 1 × 24 + 0 × 23 + 1 × 22 + 1 × 2 + 1 = 23,
A316 = 10 × 16 + 3 = 163,

(1.2.1)

2

5437 = 5 × 7 + 4 × 7 + 3 = 276.

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, 57310 = 3B912 .

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 bn . For example, to convert 1101111012 to base 16, group the digits in fours (because 16 is 24 ), yielding
1 1011 11012, and then convert each group of 4 to base 16 directly, yielding 1BD16 .

1.2.9

REPRESENTATION OF COMPLEX NUMBERS –
DEMOIVRE’S THEOREM

A complex number a + bi can be written in the form reiθ , where r2 = a2 + b2 and
tan θ = b/a. Because eiθ = cos θ + i sin θ,
(a + bi)n = rn (cos nθ + i sin nθ),
√
2kπ
2kπ
n
+ i sin
,
1 = cos
n
n
√
(2k + 1)π
(2k + 1)π
n
−1 = cos
+ i sin
,
n
n


k = 0, 1, . . . , n − 1. (1.2.2)
k = 0, 1, . . . , n − 1.

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“smtf33” — 2017/12/6 — 19:00 — page 11 — #21

1.2. CONSTANTS

11

1.2.10 ARROW NOTATION

Arrow notation is used to represent large numbers. Start with m ↑ n = m · m · · · m,
n

then (evaluation proceeds from the right)
m ↑↑ n = m ↑ m ↑ · · · ↑ m

m ↑↑↑ n = m ↑↑ m ↑↑ · · · ↑↑ m

n

n

For example, m ↑ n = m , m ↑↑ 2 = m , and m ↑↑ 3 = m
n

m


(mm )

.

1.2.11 ONES AND TWOS COMPLEMENT

One’s and two’s complement are ways to represent numbers in a computer. For
positive values the binary representation, the ones’ complement representation, and
the twos’ complement representation are the same.
• Ones’ complement represents integers from − 2N −1 − 1 to 2N −1 − 1. For
negative values, the binary representation of the absolute value is obtained, and
then all of the bits are inverted (i.e., swapping 0’s for 1’s and vice versa).
• Twos’ complement represents integers from −2N −1 to 2N −1 − 1. For negative
vales, the two’s complement representation is the same as the value one added
to the ones’ complement representation.
Number
7
6
5
4
3
2
1
0
−0
−1
−2
−3
−4

−5
−6
−7
−8

Ones’ complement
0111
0110
0101
0100
0011
0010
0001
0000
1111
1110
1101
1100
1011
1010
1001
1000

Twos’ complement
0111
0110
0101
0100
0011
0010

0001
0000
1111
1110
1101
1100
1011
1010
1001
1000

1.2.12 SYMMETRIC BASE THREE REPRESENTATION

In the symmetric base three representation, powers of 3 are added and subtracted
to represent numbers; the symbols {↓, 0, ↑} represent {−1, 0, 1}. For example, one
writes ↑↓↓ for 5 since 5 = 9−3−1. To negate a number in symmetric base three, turn
its symbol upside down, e.g., −5 =↓↑↑. Basic arithmetic operations are especially
simple in this base.

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