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Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Editorial Board . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 1
Numbers and Elementary Mathematics . . . . . . . . . . . . . . . . . . . .
1.1
1.2
1.3

1.4
1.5

Proofs without words
Constants . . . . . .
Special numbers . .
Number theory . . .
Series and products .

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xi
xiii
1
3
5
12
24
42

Chapter 2
Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
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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63
67
72
77
99

Chapter 3
Discrete Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.1
3.2
3.3
3.4
3.5

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

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129
133
144
164
177

Chapter 4
Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12

Euclidean geometry . . . . . . . . .
Coordinate systems in the plane . . .
Plane symmetries or isometries . . .
Other transformations of the plane . .
Lines . . . . . . . . . . . . . . . . .
Polygons . . . . . . . . . . . . . . .
Surfaces of revolution: the torus . . .
Quadrics . . . . . . . . . . . . . . .
Spherical geometry and trigonometry
Conics . . . . . . . . . . . . . . . .
Special plane curves . . . . . . . . .
Coordinate systems in space . . . . .

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187
188
194
201
203
205
213
213
218
222
233
242


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vi


4.13
4.14
4.15
4.16
4.17
4.18
4.19
4.20
4.21

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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245
248
249
250
251
253
257
257
259


Chapter 5
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
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 .
Interval analysis . . . . . . . .
Real analysis . . . . . . . . . .

Generalized functions . . . . .
Complex analysis . . . . . . . .

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269
279
282
294
330
337
349
358
361
370
375
376
386
388

Chapter 6
Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10

6.11
6.12
6.13
6.14
6.15
6.16
6.17

Ceiling and floor functions . . . . . . .
Exponentiation . . . . . . . . . . . . .
Logarithmic functions . . . . . . . . .
Exponential function . . . . . . . . . .
Trigonometric functions . . . . . . . .
Circular functions and planar triangles .
Tables of trigonometric functions . . .
Angle conversion . . . . . . . . . . . .
Inverse circular functions . . . . . . . .
Hyperbolic functions . . . . . . . . . .
Inverse hyperbolic functions . . . . . .
Gudermannian function . . . . . . . .
Orthogonal polynomials . . . . . . . .
Gamma function . . . . . . . . . . . .
Beta function . . . . . . . . . . . . . .
Error functions . . . . . . . . . . . . .
Fresnel integrals . . . . . . . . . . . .

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401
401
402
403
404
412
416
419
420
422
426
428
430
437
441
442
443


“smtf32” — 2011/5/20 — 2:09 — page vii — #3


vii

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

Sine, cosine, and exponential integrals
Polylogarithms . . . . . . . . . . . .
Hypergeometric functions . . . . . .
Legendre functions . . . . . . . . . .
Bessel functions . . . . . . . . . . .
Elliptic integrals . . . . . . . . . . .

Jacobian elliptic functions . . . . . .
Clebsch–Gordan coefficients . . . . .
Integral transforms: Preliminaries . .
Fourier integral transform . . . . . .
Discrete Fourier transform (DFT) . .
Fast Fourier transform (FFT) . . . . .
Multidimensional Fourier transforms
Laplace transform . . . . . . . . . .
Hankel transform . . . . . . . . . . .
Hartley transform . . . . . . . . . . .
Mellin transform . . . . . . . . . . .
Hilbert transform . . . . . . . . . . .
Z-Transform . . . . . . . . . . . . .
Tables of transforms . . . . . . . . .

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445
447
448
449
454
463
466
468
470
470
476
478
478
479
483
484
484
485
488
492

Chapter 7
Probability and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
7.1
7.2
7.3
7.4
7.5
7.6
7.7

7.8
7.9
7.10
7.11
7.12
7.13
7.14
7.15

Probability theory . . . . . . .
Classical probability problems .
Probability distributions . . . .
Queuing theory . . . . . . . . .
Markov chains . . . . . . . . .
Random number generation . .
Control charts and reliability . .
Statistics . . . . . . . . . . . .
Confidence intervals . . . . . .
Tests of hypotheses . . . . . . .
Linear regression . . . . . . . .
Analysis of variance (ANOVA)
Sample size . . . . . . . . . . .
Contingency tables . . . . . . .
Probability tables . . . . . . . .

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509
519
524
533
536
539
545
550
558
565
579
583
590
595
598

Chapter 8

Scientific Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
8.1
8.2
8.3

Basic numerical analysis . . . . . . . . . . . . . . . . . . . . . . 616
Numerical linear algebra . . . . . . . . . . . . . . . . . . . . . . 629
Numerical integration and differentiation . . . . . . . . . . . . . 638

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“smtf32” — 2011/5/20 — 2:09 — page viii — #4

viii

Chapter 9
Mathematical Formulas from the Sciences . . . . . . . . . . . . . . . . . . . 659
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
9.27
9.28
9.29

Acoustics . . . . . . . . . . . . . .
Astrophysics . . . . . . . . . . . .
Atmospheric physics . . . . . . . .
Atomic Physics . . . . . . . . . . .
Basic mechanics . . . . . . . . . .
Beam dynamics . . . . . . . . . . .
Classical mechanics . . . . . . . .
Coordinate systems – Astronomical
Coordinate systems – Terrestrial . .
Earthquake engineering . . . . . .
Electromagnetic Transmission . . .
Electrostatics and magnetism . . .

Electronic circuits . . . . . . . . .
Epidemiology . . . . . . . . . . . .
Finance . . . . . . . . . . . . . . .
Fluid mechanics . . . . . . . . . .
Fuzzy logic . . . . . . . . . . . . .
Human body . . . . . . . . . . . .
Image processing matrices . . . . .
Macroeconomics . . . . . . . . . .
Modeling physical systems . . . . .
Optics . . . . . . . . . . . . . . . .
Population genetics . . . . . . . . .
Quantum mechanics . . . . . . . .
Quaternions . . . . . . . . . . . . .
Relativistic mechanics . . . . . . .
Solid mechanics . . . . . . . . . .
Statistical mechanics . . . . . . . .
Thermodynamics . . . . . . . . . .

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661
662
664
665
666
668
669
670
671
672
673
674

675
676
677
678
679
680
681
682
683
684
685
686
688
689
690
691
692

Chapter 10
Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693
10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8
10.9
10.10

10.11
10.12

Calendar computations . . . . . . . . .
Cellular automata . . . . . . . . . . . .
Communication theory . . . . . . . . .
Control theory . . . . . . . . . . . . .
Computer languages . . . . . . . . . .
Cryptography . . . . . . . . . . . . . .
Discrete dynamical systems and chaos .
Electronic resources . . . . . . . . . .
Elliptic curves . . . . . . . . . . . . .
Financial formulas . . . . . . . . . . .
Game theory . . . . . . . . . . . . . .
Knot theory . . . . . . . . . . . . . . .

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695
696
697
702
704
705
706
709
711
714

719
722


“smtf32” — 2011/5/20 — 2:09 — page ix — #5

ix

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

Lattices . . . . . . . . . . . . .
Moments of inertia . . . . . . .
Music . . . . . . . . . . . . . .
Operations research . . . . . .
Recreational mathematics . . .
Risk analysis and decision rules
Signal processing . . . . . . . .
Symbolic logic . . . . . . . . .

Units . . . . . . . . . . . . . .
Voting power . . . . . . . . . .
Greek alphabet . . . . . . . . .
Braille code . . . . . . . . . . .
Morse code . . . . . . . . . . .

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724
726
727
729
741
742
744
750
753
760
762
762

762

List of References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767
List of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777

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“smtf32” — 2011/5/20 — 2:09 — 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 obviated the need for tables of square roots and
trigonometric functions; the internet has made many other tables and formulas unnecessary. Prior to the preparation of this 32nd Edition of the CRC Standard Mathematical Tables and Formulae, the content has been reconsidered. The criteria established 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 practitioners from mathematics, engineering, and the sciences. As a result, numerous changes
have been made in several sections, and several new areas were added. These improvements 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 rational 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 illustrating 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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“smtf32” — 2011/5/20 — 2:09 — page xii — #8

xii

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 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 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


www.pdfgrip.com


“smtf32” — 2011/5/20 — 2:09 — page xiii — #9

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

Les Servi
MITRE Corporation
Bedford, Massachusetts

Lawrence Glasser
Clarkson University
Potsdam, New York

Neil J. A. Sloane
AT&T Bell Labs
Murray Hill, New Jersey

Ray McLenaghan
University of Waterloo
Waterloo, Ontario, Canada

Gary L. Stanek
Youngstown State University
Youngstown, Ohio

Roger B. Nelsen
Lewis & Clark College
Portland, Oregon

Michael T. Strauss
HME

Newburyport, Massachusetts

Joseph J. Rushanan
MITRE Corporation
Bedford, 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
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

1.3

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5
6
6
7
8
9
9
9
10
10
11

SPECIAL NUMBERS . . . . . . . . . . . . . . . . . . . . . . . . 12
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
1.3.13

1.4

Binary prefixes . . . . . . . . . . . . . .
Decimal multiples and prefixes . . . . . .
Interpretations of powers of 10 . . . . . .
Roman numerals . . . . . . . . . . . . .
Types of numbers . . . . . . . . . . . . .
DeMoivre’s theorem . . . . . . . . . . . .
Representation of numbers . . . . . . . .
Symmetric base three representation . . . .
Hexadecimal addition and subtraction table
Hexadecimal multiplication table . . . . .
Hexadecimal–decimal fraction conversion .

Powers of 2 . . . . . . . . . . . .
Powers of 16 in decimal scale . . .
Powers of 10 in hexadecimal scale .
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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12
13
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14
16
17
18
18
19
20
21
23
23

NUMBER THEORY . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.4.1
1.4.2
1.4.3
1.4.4
1.4.5
1.4.6
1.4.7
1.4.8

Congruences . . . . . . .
Chinese remainder theorem
Continued fractions . . . .

Diophantine equations . .
Greatest common divisor .
Least common multiple . .
Măobius function . . . . . .
Prime numbers . . . . . .

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24
25
26
28
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31
32
33


“smtf32” — 2011/5/20 — 2:09 — page 2 — #12

2

CHAPTER 1. NUMBERS AND ELEMENTARY MATHEMATICS

1.4.9
1.4.10
1.4.11
1.4.12

1.5

Prime numbers of special forms
Prime numbers less than 8,000
Factorization table . . . . . .
Euler totient function . . . . .

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35
38
40
41

SERIES AND PRODUCTS . . . . . . . . . . . . . . . . . . . . . 42
1.5.1
1.5.2
1.5.3
1.5.4
1.5.5
1.5.6

1.5.7
1.5.8
1.5.9
1.5.10
1.5.11
1.5.12
1.5.13

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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www.pdfgrip.com

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42
42
44
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59
59
59
60
60


“smtf32” — 2011/5/20 — 2:09 — 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

The Pythagorean Theorem

3

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

1+2+ . . . + n =

1+2+ . . . +n =


n(n+1)
2

=

1+3
5+7

=

1+3+5
7+9+11

=...

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

1 + 3 + 5 + . . . + (2n–1) = n

2

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

2
2
2
—Ian Richards

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

www.pdfgrip.com


“smtf32” — 2011/5/20 — 2:09 — page 4 — #14

4

CHAPTER 1. NUMBERS AND ELEMENTARY MATHEMATICS

Geometric Series

...

Geometric Series

r2

r2

r
1–r


r

1

1

1
2

1 + r + r + ... = 1
1
1–r

1
1
1 3
1 2
+
+
+...=
3
4
4
4

—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

2

y

cosxsiny

sin

x

(a,ma + c)

1+

m

1

y

y


|ma + c – b|

1

sinxcosy

s
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


“smtf32” — 2011/5/20 — 2:09 — page 5 — #15

1.2. CONSTANTS

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

a+b
2

5

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

a
c
<
b

d

ab
c

a+b
2

d

ab

a

a
b

b

a
d

—Richard A. Gibbs

—Charles D. Gallant

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

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

www.pdfgrip.com


“smtf32” — 2011/5/20 — 2:09 — 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 G´en´erale 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

INTERPRETATIONS OF POWERS OF 10

−15

10
10−11
10−10
10−9
10−6
100
101
102
105
106
107
108
109
1010

the radius of the hydrogen nucleus (a proton) in meters
the likelihood of being dealt 13 top honors in bridge
the 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 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

www.pdfgrip.com


“smtf32” — 2011/5/20 — 2:09 — page 7 — #17

1.2. CONSTANTS

1015
1016
1018
1019
1021
1024
1028
1033
1050
1078

7

the surface area of the earth in square meters
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 grams
the mass of the solar system in grams

the number of atoms in the earth
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 = 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
Roman numeral
10
X

14

XIV

1950
MCML

1
I

50
L

2
II

3
III

4
IV

5
V

200
CC

400
CD

500

D

600
DC

1960
MCMLX

1995
MCMXCV

1970
MCMLXX

1999
MCMXCIX

2000
MM

6
VI

7
VII

999
CMXCIX

1980

MCMLXXX
2001
MMI

www.pdfgrip.com

8
VIII

2011
MMXI

9
IX

1000
M

1990
MCMXC
2012
MMXII


“smtf32” — 2011/5/20 — 2:09 — page 8 — #18

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, { 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 include the symbols
+∞ 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
∞
−∞

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

(e)
(f)
(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 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
−
a2 + b 2
a2 + b 2
z = a − bi

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

www.pdfgrip.com


i


“smtf32” — 2011/5/20 — 2:09 — page 9 — #19

1.2. CONSTANTS

1.2.6

9

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

1.2.7

k = 0, 1, . . . , n − 1. (1.2.1)
k = 0, 1, . . . , n − 1.

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,

(1.2.2)

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 .

1.2.8

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.

www.pdfgrip.com


“smtf32” — 2011/5/20 — 2:09 — page 10 — #20

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

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


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

www.pdfgrip.com

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



“smtf32” — 2011/5/20 — 2:09 — page 11 — #21

1.2. CONSTANTS

1.2.11 HEXADECIMAL–DECIMAL FRACTION CONVERSION
Hex Decimal Hex Decimal Hex Decimal Hex Decimal Hex Decimal
.00 0
.30 0.1875 .60 0.3750 .90 0.5625 .C0 0.7500
.01 0.0039 .31 0.1914 .61 0.3789 .91 0.5664 .C1 0.7539
.02 0.0078 .32 0.1953 .62 0.3828 .92 0.5703 .C2 0.7578
.03 0.0117 .33 0.1992 .63 0.3867 .93 0.5742 .C3 0.7617
.04 0.0156 .34 0.2031 .64 0.3906 .94 0.5781 .C4 0.7656
.05 0.0195 .35 0.2070 .65 0.3945 .95 0.5820 .C5 0.7695
.06 0.0234 .36 0.2109 .66 0.3984 .96 0.5859 .C6 0.7734
.07 0.0273 .37 0.2148 .67 0.4023 .97 0.5898 .C7 0.7773
.08 0.0313 .38 0.2188 .68 0.4063 .98 0.5938 .C8 0.7813
.09 0.0352 .39 0.2227 .69 0.4102 .99 0.5977 .C9 0.7852
.0A 0.0391 .3A 0.2266 .6A 0.4141 .9A 0.6016 .CA 0.7891
.0B 0.0430 .3B 0.2305 .6B 0.4180 .9B 0.6055 .CB 0.7930
.0C 0.0469 .3C 0.2344 .6C 0.4219 .9C 0.6094 .CC 0.7969
.0D 0.0508 .3D 0.2383 .6D 0.4258 .9D 0.6133 .CD 0.8008
.0E 0.0547 .3E 0.2422 .6E 0.4297 .9E 0.6172 .CE 0.8047
.0F 0.0586 .3F 0.2461 .6F 0.43365 .9F 0.6211 .CF 0.8086
.10
.11
.12
.13
.14
.15
.16

.17
.18
.19
.1A
.1B
.1C
.1D
.1E
.1F

0.0625
0.0664
0.0703
0.0742
0.0781
0.0820
0.0859
0.0898
0.0938
0.0977
0.1016
0.1055
0.1094
0.1133
0.1172
0.1210

.40
.41
.42

.43
.44
.45
.46
.47
.48
.49
.4A
.4B
.4C
.4D
.4E
.4F

0.2500
0.2539
0.2578
0.2617
0.2656
0.2695
0.2734
0.2773
0.2813
0.2852
0.2891
0.2930
0.2969
0.3008
0.3047
0.3086


.70
.71
.72
.73
.74
.75
.76
.77
.78
.79
.7A
.7B
.7C
.7D
.7E
.7F

0.4375
0.4414
0.4453
0.4492
0.4531
0.4570
0.4609
0.4648
0.4688
0.4727
0.4766
0.4805

0.4844
0.4883
0.4922
0.4961

.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

.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

.20
.21
.22
.23
.24
.25
.26
.27
.28
.29
.2A
.2B
.2C
.2D
.2E
.2F

0.1250
0.1289
0.1328
0.1367

0.1406
0.1445
0.1484
0.1523
0.1563
0.1602
0.1641
0.1680
0.1719
0.1758
0.1797
0.1836

.50
.51
.52
.53
.54
.55
.56
.57
.58
.59
.5A
.5B
.5C
.5D
.5E
.5F


0.3125
0.3164
0.3203
0.3242
0.3281
0.3320
0.3359
0.3398
0.3438
0.3477
0.3516
0.3555
0.3594
0.3633
0.3672
0.3711

.80
.81
.82
.83
.84
.85
.86
.87
.88
.89
.8A
.8B
.8C

.8D
.8E
.8F

0.5000
0.5039
0.5078
0.5117
0.5156
0.5195
0.5234
0.5273
0.5313
0.5352
0.5391
0.5430
0.5469
0.5508
0.5547
0.5586

.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

.E0
.E1
.E2
.E3
.E4
.E5
.E6
.E7
.E8
.E9
.EA
.EB
.EC
.ED
.EE
.EF

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0.8750
0.8789
0.8828

0.8867
0.8906
0.8945
0.8984
0.9023
0.9063
0.9102
0.9141
0.9180
0.9219
0.9258
0.9297
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

11