Math Into Latex

Math into LATEX
An Introduction to LATEX and AMS-LATEX

This book is dedicated to those who worked so hard
and for so long to bring these important tools to us:
The LATEX3 team
and in particular
Frank Mittelbach (project leader) and David Carlisle
The AMS team
and in particular
Michael J. Downes (project leader) and David M. Jones

George Gr¨atzer
Math into LATEX
An Introduction to LATEX and AMS-LATEX
B I R K H ¨
A U S E R
B O S T O N • B A S E L • B E R L I N

George Gr¨atzer
Department of Mathematics
University of Manitoba
Winnipeg, Manitoba
Canada R3T 2N2
Library of Congress Cataloging-in-Publication Data
Gr¨atzer, George A.
Math into LaTeX : an introduction to LaTeX and AMS-LaTeX /
George Gr¨atzer
p.
cm.
Includes index.
ISBN 0-8176-3805-9 (acid-free paper) (pbk. : alk. paper)
1.
AMS-LaTeX.
2.
Mathematics printing–Computer programs.
3.
Computerized typesetting.
I.
Title.
Z253.4A65G69
1995
95-36881
688.2 2544536–dc20
CIP
Printed on acid-free paper
c Birkh¨auser Boston 1996
All rights reserved.
Typeset by the Author in LATEX
Design, layout, and typography by Mery Sawdey, Minneapolis, MN

Short contents
Preface
xviii
Introduction
xix
I
A short course
1
1 Typing your ﬁrst article
3
II
Text and math
59
2 Typing text
61
3 Text environments
111
4 Typing math
140
5 Multiline math displays
180
III
Document structure
209
6 LATEX documents
211
7 Standard LATEX document classes
235
8 AMS-LATEX documents
243
v

vi
Short contents
IV
Customizing
265
9 Customizing LATEX
267
V
Long bibliographies and indexes
309
10 BIBTEX
311
11 MakeIndex
332
A Math symbol tables
345
B Text symbol tables
356
C The AMS-LATEX sample article
360
D Sample article with user-deﬁned commands
372
E Background
379
F PostScript fonts
387
G Getting it
392
H Conversions
402
I
Final word
410
Bibliography
413
Afterword
416
Index
419

Contents
Preface
xviii
Introduction
xix
Typographical conventions . . . . . . . . . . . . . . . . . . . . . . . .
xxvi
I
A short course
1
1 Typing your ﬁrst article
3
1.1 Typing a very short “article” . . . . . . . . . . . . . . . . . . . .
4
1.1.1 The keyboard . . . . . . . . . . . . . . . . . . . . . . . .
4
1.1.2 Your ﬁrst note . . . . . . . . . . . . . . . . . . . . . . . .
5
1.1.3 Lines too wide . . . . . . . . . . . . . . . . . . . . . . .
7
1.1.4 More text features
. . . . . . . . . . . . . . . . . . . . .
9
1.2 Typing math . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.2.1 The keyboard . . . . . . . . . . . . . . . . . . . . . . . .
10
1.2.2 A note with math . . . . . . . . . . . . . . . . . . . . . .
10
1.2.3 Building blocks of a formula . . . . . . . . . . . . . . . .
14
1.2.4 Building a formula step-by-step . . . . . . . . . . . . . .
20
1.3 Formula gallery . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
1.4 Typing equations and aligned formulas . . . . . . . . . . . . . .
29
1.4.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . .
29
1.4.2 Aligned formulas . . . . . . . . . . . . . . . . . . . . . .
31
1.5 The anatomy of an article . . . . . . . . . . . . . . . . . . . . . .
33
1.5.1 The typeset article
. . . . . . . . . . . . . . . . . . . . .
38
1.6 Article templates
. . . . . . . . . . . . . . . . . . . . . . . . . .
41
1.7 Your ﬁrst article . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
1.7.1 Editing the top matter . . . . . . . . . . . . . . . . . . .
42
vii

viii
Contents
1.7.2 Sectioning . . . . . . . . . . . . . . . . . . . . . . . . . .
43
1.7.3 Invoking proclamations . . . . . . . . . . . . . . . . . . .
44
1.7.4 Inserting references . . . . . . . . . . . . . . . . . . . . .
44
1.8 LATEX error messages . . . . . . . . . . . . . . . . . . . . . . . .
46
1.9 Logical and visual design . . . . . . . . . . . . . . . . . . . . . .
48
1.10 A brief overview . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
1.11 Using LATEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
1.11.1 AMS-LATEX revisited . . . . . . . . . . . . . . . . . . . .
52
1.11.2 Interactive LATEX . . . . . . . . . . . . . . . . . . . . . .
54
1.11.3 Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
1.11.4 Versions . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
1.12 What’s next? . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
II
Text and math
59
2 Typing text
61
2.1 The keyboard . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
2.1.1 The basic keys . . . . . . . . . . . . . . . . . . . . . . . .
62
2.1.2 Special keys . . . . . . . . . . . . . . . . . . . . . . . . .
63
2.1.3 Prohibited keys . . . . . . . . . . . . . . . . . . . . . . .
63
2.2 Words, sentences, and paragraphs . . . . . . . . . . . . . . . . .
64
2.2.1 The spacing rules . . . . . . . . . . . . . . . . . . . . . .
64
2.2.2 The period
. . . . . . . . . . . . . . . . . . . . . . . . .
66
2.3 Instructing LATEX . . . . . . . . . . . . . . . . . . . . . . . . . .
67
2.3.1 Commands and environments . . . . . . . . . . . . . . .
67
2.3.2 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
2.3.3 Types of commands . . . . . . . . . . . . . . . . . . . . .
72
2.4 Symbols not on the keyboard . . . . . . . . . . . . . . . . . . . .
73
2.4.1 Quotes
. . . . . . . . . . . . . . . . . . . . . . . . . . .
73
2.4.2 Dashes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
2.4.3 Ties or nonbreakable spaces . . . . . . . . . . . . . . . .
74
2.4.4 Special characters . . . . . . . . . . . . . . . . . . . . . .
74
2.4.5 Ligatures . . . . . . . . . . . . . . . . . . . . . . . . . .
75
2.4.6 Accents and symbols in text . . . . . . . . . . . . . . . .
75
2.4.7 Logos and numbers . . . . . . . . . . . . . . . . . . . . .
76
2.4.8 Hyphenation . . . . . . . . . . . . . . . . . . . . . . . .
78
2.5 Commenting out . . . . . . . . . . . . . . . . . . . . . . . . . .
81
2.6 Changing font characteristics . . . . . . . . . . . . . . . . . . . .
83
2.6.1 The basic font characteristics . . . . . . . . . . . . . . . .
83
2.6.2 The document font families . . . . . . . . . . . . . . . .
84
2.6.3 Command pairs . . . . . . . . . . . . . . . . . . . . . . .
85
2.6.4 Shape commands . . . . . . . . . . . . . . . . . . . . . .
85

Contents
ix
2.6.5 Italic correction . . . . . . . . . . . . . . . . . . . . . . .
86
2.6.6 Two-letter commands . . . . . . . . . . . . . . . . . . .
87
2.6.7 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
2.6.8 Size changes . . . . . . . . . . . . . . . . . . . . . . . . .
88
2.6.9 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . .
89
2.6.10 Boxed text . . . . . . . . . . . . . . . . . . . . . . . . . .
89
2.7 Lines, paragraphs, and pages . . . . . . . . . . . . . . . . . . . .
90
2.7.1 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
2.7.2 Paragraphs . . . . . . . . . . . . . . . . . . . . . . . . . .
93
2.7.3 Pages . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
2.7.4 Multicolumn printing . . . . . . . . . . . . . . . . . . . .
95
2.8 Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
2.8.1 Horizontal spaces . . . . . . . . . . . . . . . . . . . . . .
96
2.8.2 Vertical spaces . . . . . . . . . . . . . . . . . . . . . . . .
97
2.8.3 Relative spaces
. . . . . . . . . . . . . . . . . . . . . . .
99
2.8.4 Expanding spaces . . . . . . . . . . . . . . . . . . . . . .
99
2.9 Boxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
100
2.9.1 Line boxes . . . . . . . . . . . . . . . . . . . . . . . . . .
100
2.9.2 Paragraph boxes . . . . . . . . . . . . . . . . . . . . . . .
103
2.9.3 Marginal comments . . . . . . . . . . . . . . . . . . . . .
104
2.9.4 Solid boxes . . . . . . . . . . . . . . . . . . . . . . . . .
105
2.9.5 Fine-tuning boxes . . . . . . . . . . . . . . . . . . . . . .
106
2.10 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
107
2.10.1 Fragile commands
. . . . . . . . . . . . . . . . . . . . .
107
2.11 Splitting up the ﬁle . . . . . . . . . . . . . . . . . . . . . . . . .
108
2.11.1 Input and include . . . . . . . . . . . . . . . . . . . . . .
108
2.11.2 Combining ﬁles . . . . . . . . . . . . . . . . . . . . . . .
109
3 Text environments
111
3.1 List environments . . . . . . . . . . . . . . . . . . . . . . . . . .
112
3.1.1 Numbered lists: enumerate . . . . . . . . . . . . . . . .
112
3.1.2 Bulleted lists: itemize . . . . . . . . . . . . . . . . . . .
112
3.1.3 Captioned lists: description . . . . . . . . . . . . . . .
113
3.1.4 Rule and combinations . . . . . . . . . . . . . . . . . . .
114
3.2 Tabbing environment . . . . . . . . . . . . . . . . . . . . . . . .
116
3.3 Miscellaneous displayed text environments . . . . . . . . . . . .
118
3.4 Proclamations (theorem-like structures) . . . . . . . . . . . . . .
123
3.4.1 The full syntax
. . . . . . . . . . . . . . . . . . . . . . .
127
3.4.2 Proclamations with style . . . . . . . . . . . . . . . . . .
127
3.5 Proof environment . . . . . . . . . . . . . . . . . . . . . . . . .
130
3.6 Some general rules for displayed text environments . . . . . . . .
131
3.7 Tabular environment . . . . . . . . . . . . . . . . . . . . . . . .
132

x
Contents
3.8 Style and size environments
. . . . . . . . . . . . . . . . . . . .
138
4 Typing math
140
4.1 Math environments . . . . . . . . . . . . . . . . . . . . . . . . .
141
4.2 The spacing rules . . . . . . . . . . . . . . . . . . . . . . . . . .
143
4.3 The equation environment . . . . . . . . . . . . . . . . . . . . .
144
4.4 Basic constructs . . . . . . . . . . . . . . . . . . . . . . . . . . .
146
4.4.1 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . .
146
4.4.2 Subscripts and superscripts . . . . . . . . . . . . . . . . .
147
4.4.3 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . .
148
4.4.4 Binomial coefﬁcients . . . . . . . . . . . . . . . . . . . .
149
4.4.5 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . .
149
4.4.6 Ellipses
. . . . . . . . . . . . . . . . . . . . . . . . . . .
150
4.5 Text in math . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
151
4.6 Delimiters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
152
4.6.1 Delimiter tables . . . . . . . . . . . . . . . . . . . . . . .
153
4.6.2 Delimiters of ﬁxed size . . . . . . . . . . . . . . . . . . .
153
4.6.3 Delimiters of variable size
. . . . . . . . . . . . . . . . .
154
4.6.4 Delimiters as binary relations . . . . . . . . . . . . . . . .
155
4.7 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
155
4.7.1 Operator tables . . . . . . . . . . . . . . . . . . . . . . .
156
4.7.2 Declaring operators . . . . . . . . . . . . . . . . . . . . .
157
4.7.3 Congruences . . . . . . . . . . . . . . . . . . . . . . . .
158
4.8 Sums and products . . . . . . . . . . . . . . . . . . . . . . . . .
159
4.8.1 Large operators . . . . . . . . . . . . . . . . . . . . . . .
159
4.8.2 Multiline subscripts and superscripts . . . . . . . . . . . .
160
4.9 Math accents . . . . . . . . . . . . . . . . . . . . . . . . . . . .
161
4.10 Horizontal lines that stretch . . . . . . . . . . . . . . . . . . . .
162
4.10.1 Horizontal braces . . . . . . . . . . . . . . . . . . . . . .
162
4.10.2 Over and underlines . . . . . . . . . . . . . . . . . . . .
163
4.10.3 Stretchable arrow math symbols . . . . . . . . . . . . . .
164
4.11 The spacing of symbols . . . . . . . . . . . . . . . . . . . . . . .
164
4.12 Building new symbols . . . . . . . . . . . . . . . . . . . . . . . .
166
4.12.1 Stacking symbols . . . . . . . . . . . . . . . . . . . . . .
167
4.12.2 Declaring the type . . . . . . . . . . . . . . . . . . . . .
168
4.13 Vertical spacing . . . . . . . . . . . . . . . . . . . . . . . . . . .
169
4.14 Math alphabets and symbols . . . . . . . . . . . . . . . . . . . .
170
4.14.1 Math alphabets . . . . . . . . . . . . . . . . . . . . . . .
171
4.14.2 Math alphabets of symbols . . . . . . . . . . . . . . . . .
172
4.14.3 Bold math symbols . . . . . . . . . . . . . . . . . . . . .
173
4.14.4 Size changes . . . . . . . . . . . . . . . . . . . . . . . . .
175
4.14.5 Continued fractions . . . . . . . . . . . . . . . . . . . . .
175

Contents
xi
4.15 Tagging and grouping . . . . . . . . . . . . . . . . . . . . . . .
176
4.16 Generalized fractions . . . . . . . . . . . . . . . . . . . . . . . .
178
4.17 Boxed formulas . . . . . . . . . . . . . . . . . . . . . . . . . . .
179
5 Multiline math displays
180
5.1 Gathering formulas . . . . . . . . . . . . . . . . . . . . . . . . .
181
5.2 Splitting a long formula . . . . . . . . . . . . . . . . . . . . . . .
182
5.3 Some general rules . . . . . . . . . . . . . . . . . . . . . . . . .
184
5.3.1 The subformula rule . . . . . . . . . . . . . . . . . . . .
185
5.3.2 Group numbering
. . . . . . . . . . . . . . . . . . . . .
186
5.4 Aligned columns . . . . . . . . . . . . . . . . . . . . . . . . . .
187
5.4.1 The subformula rule revisited
. . . . . . . . . . . . . . .
188
5.4.2 Align variants . . . . . . . . . . . . . . . . . . . . . . . .
189
5.4.3 Intertext . . . . . . . . . . . . . . . . . . . . . . . . . . .
192
5.5 Aligned subsidiary math environments . . . . . . . . . . . . . . .
193
5.5.1 Subsidiary variants of aligned math environments . . . . .
193
5.5.2 Split . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
195
5.6 Adjusted columns . . . . . . . . . . . . . . . . . . . . . . . . . .
198
5.6.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . .
198
5.6.2 Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . .
201
5.6.3 Cases
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
203
5.7 Commutative diagrams . . . . . . . . . . . . . . . . . . . . . . .
204
5.8 Pagebreak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
205
III
Document structure
209
6 LATEX documents
211
6.1 The structure of a document . . . . . . . . . . . . . . . . . . . .
212
6.2 The preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . .
213
6.3 Front matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
214
6.3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . .
214
6.3.2 Table of contents . . . . . . . . . . . . . . . . . . . . . .
215
6.4 Main matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
217
6.4.1 Sectioning . . . . . . . . . . . . . . . . . . . . . . . . . .
217
6.4.2 Cross-referencing . . . . . . . . . . . . . . . . . . . . . .
220
6.4.3 Tables and ﬁgures . . . . . . . . . . . . . . . . . . . . . .
223
6.5 Back matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
227
6.5.1 Bibliography in an article . . . . . . . . . . . . . . . . . .
227
6.5.2 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . .
231
6.6 Page style . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
232

xii
Contents
7 Standard LATEX document classes
235
7.1 The article, report, and book document classes . . . . . . . .
235
7.1.1 More on sectioning . . . . . . . . . . . . . . . . . . . . .
236
7.1.2 Options . . . . . . . . . . . . . . . . . . . . . . . . . . .
237
7.2 The letter document class . . . . . . . . . . . . . . . . . . . .
239
7.3 The LATEX distribution . . . . . . . . . . . . . . . . . . . . . . . 240
7.3.1 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . .
241
8 AMS-LATEX documents
243
8.1 The three AMS document classes . . . . . . . . . . . . . . . . . 243
8.1.1 Font size commands . . . . . . . . . . . . . . . . . . . .
244
8.2 The top matter . . . . . . . . . . . . . . . . . . . . . . . . . . .
244
8.2.1 Article info . . . . . . . . . . . . . . . . . . . . . . . . .
245
8.2.2 Author info . . . . . . . . . . . . . . . . . . . . . . . . .
246
8.2.3 AMS info . . . . . . . . . . . . . . . . . . . . . . . . . . 249
8.2.4 Multiple authors . . . . . . . . . . . . . . . . . . . . . .
250
8.2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . .
250
8.3 AMS article template . . . . . . . . . . . . . . . . . . . . . . . . 253
8.4 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
257
8.4.1 Math options . . . . . . . . . . . . . . . . . . . . . . . .
260
8.5 The AMS-LATEX packages . . . . . . . . . . . . . . . . . . . . . 261
IV
Customizing
265
9 Customizing LATEX
267
9.1 User-deﬁned commands . . . . . . . . . . . . . . . . . . . . . .
268
9.1.1 Commands as shorthand . . . . . . . . . . . . . . . . . .
268
9.1.2 Arguments
. . . . . . . . . . . . . . . . . . . . . . . . .
271
9.1.3 Redeﬁning commands . . . . . . . . . . . . . . . . . . .
274
9.1.4 Optional arguments . . . . . . . . . . . . . . . . . . . . .
275
9.1.5 Redeﬁning names . . . . . . . . . . . . . . . . . . . . . .
276
9.1.6 Showing the meaning of commands . . . . . . . . . . . .
276
9.2 User-deﬁned environments . . . . . . . . . . . . . . . . . . . . .
279
9.2.1 Short arguments . . . . . . . . . . . . . . . . . . . . . .
282
9.3 Numbering and measuring . . . . . . . . . . . . . . . . . . . . .
282
9.3.1 Counters
. . . . . . . . . . . . . . . . . . . . . . . . . .
283
9.3.2 Length commands . . . . . . . . . . . . . . . . . . . . .
287
9.4 Delimited commands . . . . . . . . . . . . . . . . . . . . . . . .
290
9.5 A custom command ﬁle . . . . . . . . . . . . . . . . . . . . . . .
292
9.6 Custom lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
297
9.6.1 Length commands for the list environment . . . . . . .
297
9.6.2 The list environment . . . . . . . . . . . . . . . . . . .
299

Contents
xiii
9.6.3 Two complete examples . . . . . . . . . . . . . . . . . .
301
9.6.4 The trivlist environment . . . . . . . . . . . . . . . .
304
9.7 Custom formats . . . . . . . . . . . . . . . . . . . . . . . . . . .
304
V
Long bibliographies and indexes
309
10 BIBTEX
311
10.1 The database
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
311
10.1.1 Entry types . . . . . . . . . . . . . . . . . . . . . . . . .
312
10.1.2 Articles
. . . . . . . . . . . . . . . . . . . . . . . . . . .
315
10.1.3 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . .
316
10.1.4 Conference proceedings and collections . . . . . . . . . .
317
10.1.5 Theses . . . . . . . . . . . . . . . . . . . . . . . . . . . .
319
10.1.6 Technical reports . . . . . . . . . . . . . . . . . . . . . .
320
10.1.7 Manuscripts . . . . . . . . . . . . . . . . . . . . . . . . .
321
10.1.8 Other entry types . . . . . . . . . . . . . . . . . . . . . .
321
10.1.9 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . .
322
10.2 Using BIBTEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
10.2.1 The sample ﬁles . . . . . . . . . . . . . . . . . . . . . . .
323
10.2.2 The setup . . . . . . . . . . . . . . . . . . . . . . . . . .
325
10.2.3 The four steps of BIBTEXing . . . . . . . . . . . . . . . . 325
10.2.4 The ﬁles of BIBTEX . . . . . . . . . . . . . . . . . . . . . 327
10.2.5 BIBTEX rules and messages . . . . . . . . . . . . . . . . . 329
10.2.6 Concluding comments . . . . . . . . . . . . . . . . . . .
331
11 MakeIndex
332
11.1 Preparing the document . . . . . . . . . . . . . . . . . . . . . .
332
11.2 Index entries
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
335
11.3 Processing the index entries . . . . . . . . . . . . . . . . . . . .
339
11.4 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
342
11.5 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
344
A Math symbol tables
345
B Text symbol tables
356
C The AMS-LATEX sample article
360
D Sample article with user-deﬁned commands
372

xiv
Contents
E Background
379
E.1 A short history
. . . . . . . . . . . . . . . . . . . . . . . . . . .
379
E.1.1 The ﬁrst interim solution . . . . . . . . . . . . . . . . . .
381
E.1.2 The second interim solution . . . . . . . . . . . . . . . .
382
E.2 How does it work? . . . . . . . . . . . . . . . . . . . . . . . . .
382
E.2.1 The layers . . . . . . . . . . . . . . . . . . . . . . . . . .
382
E.2.2 Typesetting . . . . . . . . . . . . . . . . . . . . . . . . .
383
E.2.3 Viewing and printing . . . . . . . . . . . . . . . . . . . .
384
E.2.4 The ﬁles of LATEX . . . . . . . . . . . . . . . . . . . . . . 385
F PostScript fonts
387
F.1 The Times font and MathTıme . . . . . . . . . . . . . . . . . . .
387
F.2 LucidaBright fonts . . . . . . . . . . . . . . . . . . . . . . . . .
390
G Getting it
392
G.1 Getting TEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
G.2 Where to get it? . . . . . . . . . . . . . . . . . . . . . . . . . . .
393
G.3 Getting ready . . . . . . . . . . . . . . . . . . . . . . . . . . . .
395
G.4 Transferring ﬁles
. . . . . . . . . . . . . . . . . . . . . . . . . .
396
G.5 More advanced ﬁle transfer commands . . . . . . . . . . . . . . .
398
G.6 The sample ﬁles . . . . . . . . . . . . . . . . . . . . . . . . . . .
400
G.7 AMS and the user groups . . . . . . . . . . . . . . . . . . . . . 400
H Conversions
402
H.1 From Plain TEX . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
H.1.1 TEX code in LATEX . . . . . . . . . . . . . . . . . . . . . 403
H.2 From LATEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
H.2.1 Version 2e . . . . . . . . . . . . . . . . . . . . . . . . . .
404
H.2.2 Version 2.09
. . . . . . . . . . . . . . . . . . . . . . . .
404
H.2.3 The LATEX symbols . . . . . . . . . . . . . . . . . . . . . 405
H.3 From AMS-TEX . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
H.4 From AMS-LATEX version 1.1 . . . . . . . . . . . . . . . . . . . 406
I
Final word
410
I.1
What was left out? . . . . . . . . . . . . . . . . . . . . . . . . . .
410
I.1.1
Omitted from LATEX . . . . . . . . . . . . . . . . . . . . 410
I.1.2
Omitted from TEX . . . . . . . . . . . . . . . . . . . . . 411
I.2
Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . .
411
Bibliography
413
Afterword
416
Index
419

List of tables
2.1 Special characters . . . . . . . . . . . . . . . . . . . . . . . . . .
74
2.2 Font table for Computer Modern typewriter style font . . . . . .
76
2.3 European accents . . . . . . . . . . . . . . . . . . . . . . . . . .
76
2.4 Extra text symbols . . . . . . . . . . . . . . . . . . . . . . . . . .
77
2.5 European characters . . . . . . . . . . . . . . . . . . . . . . . . .
77
2.6 Font family switching commands . . . . . . . . . . . . . . . . . .
85
3.1 Tabular table . . . . . . . . . . . . . . . . . . . . . . . . . . . .
133
3.2 Floating table with \multicolumn . . . . . . . . . . . . . . . . .
136
3.3 Tabular table with \multicolumn and \cline . . . . . . . . . .
137
4.1 Standard delimiters . . . . . . . . . . . . . . . . . . . . . . . . .
153
4.2 Arrow delimiters
. . . . . . . . . . . . . . . . . . . . . . . . . .
153
4.3 Operators without limits . . . . . . . . . . . . . . . . . . . . . .
157
4.4 Operators with limits . . . . . . . . . . . . . . . . . . . . . . . .
157
4.5 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . .
158
4.6 Large operators . . . . . . . . . . . . . . . . . . . . . . . . . . .
159
4.7 Math accents . . . . . . . . . . . . . . . . . . . . . . . . . . . .
161
4.8 Spacing commands . . . . . . . . . . . . . . . . . . . . . . . . .
165
9.1 Table of redeﬁnable names in LATEX . . . . . . . . . . . . . . . . 277
9.2 Standard LATEX counters . . . . . . . . . . . . . . . . . . . . . . 283
A.1
Hebrew letters . . . . . . . . . . . . . . . . . . . . . . . . . . .
345
A.2
Greek characters . . . . . . . . . . . . . . . . . . . . . . . . . .
346
A.3
LATEX binary relations . . . . . . . . . . . . . . . . . . . . . . . 347
A.4
AMS binary relations . . . . . . . . . . . . . . . . . . . . . . . 348
A.5
AMS negated binary relations . . . . . . . . . . . . . . . . . . . 349
xv

xvi
List of tables
A.6
Binary operations . . . . . . . . . . . . . . . . . . . . . . . . .
350
A.7
Arrows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
351
A.8
Miscellaneous symbols . . . . . . . . . . . . . . . . . . . . . . .
352
A.9
Math spacing commands . . . . . . . . . . . . . . . . . . . . .
353
A.10 Delimiters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
353
A.11 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
354
A.12 Math accents . . . . . . . . . . . . . . . . . . . . . . . . . . . .
355
A.13 Math font commands . . . . . . . . . . . . . . . . . . . . . . .
355
B.1 Special text characters . . . . . . . . . . . . . . . . . . . . . . . .
356
B.2 Text accents . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
357
B.3 Some European characters . . . . . . . . . . . . . . . . . . . . .
357
B.4 Extra text symbols . . . . . . . . . . . . . . . . . . . . . . . . . .
357
B.5 Text spacing commands
. . . . . . . . . . . . . . . . . . . . . .
358
B.6 Text font commands . . . . . . . . . . . . . . . . . . . . . . . .
358
B.7 Font size changes . . . . . . . . . . . . . . . . . . . . . . . . . .
359
B.8 AMS font size changes . . . . . . . . . . . . . . . . . . . . . . . 359
F.1 Lower font table for the Times font . . . . . . . . . . . . . . . .
389
F.2 Upper font table for the Times font . . . . . . . . . . . . . . . .
389
G.1 Some UNIX commands . . . . . . . . . . . . . . . . . . . . . .
395
G.2 Some ftp commands . . . . . . . . . . . . . . . . . . . . . . . .
396
H.1 TEX commands to avoid in LATEX . . . . . . . . . . . . . . . . . 404
H.2 A translation table . . . . . . . . . . . . . . . . . . . . . . . . . .
405
H.3 AMS-TEX style commands dropped in AMS-LATEX . . . . . . . 407
H.4 AMS-TEX commands to avoid . . . . . . . . . . . . . . . . . . . 408

List of ﬁgures
1.1 A schematic view of an article
. . . . . . . . . . . . . . . . . . .
34
1.2 The structure of LATEX . . . . . . . . . . . . . . . . . . . . . . .
51
1.3 Using LATEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
6.1 The structure of a document . . . . . . . . . . . . . . . . . . . .
212
6.2 Sectioning commands in the article document class . . . . . .
219
6.3 Sectioning commands in the amsart document class . . . . . . .
219
6.4 Page layout for the article document class
. . . . . . . . . . .
233
8.1 fleqn and reqno options for equations . . . . . . . . . . . . . .
258
8.2 Top-or-bottom tags option for split . . . . . . . . . . . . . . .
258
8.3 AMS-LATEX package and document class interdependency . . . . 263
9.1 The layout of a custom list . . . . . . . . . . . . . . . . . . . . .
298
10.1 Using BIBTEX, Step 2 . . . . . . . . . . . . . . . . . . . . . . . 326
10.2 Using BIBTEX, Step 3 . . . . . . . . . . . . . . . . . . . . . . . 326
11.1 A sample index . . . . . . . . . . . . . . . . . . . . . . . . . . .
335
11.2 Using MakeIndex, Step 1 . . . . . . . . . . . . . . . . . . . . . .
340
11.3 Using MakeIndex, Step 2 . . . . . . . . . . . . . . . . . . . . . .
340
xvii

Preface
It is indeed a lucky author who is given the opportunity to completely rewrite a
book barely a year after its publication. Writing about software affords such op-
portunities (especially if the original edition sold out), since the author is shooting
at a moving target.
LATEX and AMS-LATEX improved dramatically with the release of the new stan-
dard LATEX (called LATEX 2ε) in June of 1994 and the revision of AMS-LATEX (ver-
sion 1.2) in February of 1995. The change in AMS-LATEX is profound. LATEX 2ε
made it possible for AMS-LATEX to join the LATEX world. One of the main points
of the present book is to make this clear. This book introduces LATEX as a tool
for mathematical typesetting, and treats AMS-LATEX as a set of enhancements to
the standard LATEX, to be used in conjunction with hundreds of other LATEX 2ε
enhancements.
I am not a TEX expert. Learning the mysteries of the system has given me great
respect for those who crafted it: Donald Knuth, Leslie Lamport, Michael Spivak,
and others did the original work; David Carlisle, Michael J. Downes, David M.
Jones, Frank Mittelbach, Rainer Sch¨
opf, and many others built on the work of
these pioneers to create the new LATEX and AMS-LATEX.
Many of these experts and a multitude of others helped me while I was writing
this book. I would like to express my deepest appreciation and heartfelt thanks to
all who gave their time so generously. Their story is told in the Afterword.
Of course, the responsibility is mine for all the mistakes remaining in the book.
Please send corrections—and suggestions for improvements—to me at the follow-
ing address:
Department of Mathematics
University of Manitoba
Winnipeg MB, R3T 2N2
Canada
e-mail: George Gratzer@umanitoba.ca
xviii

Introduction
Is this book for you?
This book is for the mathematician, engineer, scientist, or technical typist who
wants to write and typeset articles containing mathematical formulas but does not
want to spend much time learning how to do it.
I assume you are set up to use LATEX, and you know how to use an editor to
type a document, such as:
\documentclass{article}
\begin{document}
The square root of two: $\sqrt{2}$.
I can type math!
\end{document}
I also assume you know how to typeset a document, such as this example, with
LATEX to get the printed version:
√
The square root of two:
2. I can type math!
and you can view and print the typeset document.
And what do I promise to deliver? I hope to provide you with a solid founda-
tion in LATEX, the AMS enhancements, and some standard LATEX enhancements,
so typing a mathematical document will become second nature to you.
How to read this book?
Part I gives a short course in LATEX. Read it, work through the examples, and you
are ready to type your ﬁrst paper. Later, at your leisure, read the other parts to
become more proﬁcient.
xix

xx
Introduction
The rest of this section introduces TEX, LATEX, and AMS-LATEX, and then
outlines what is in this book. If you already know that you want to use LATEX to
typeset math, you may choose to skip it.
TEX, LATEX, and AMS-LATEX
TEX is a typesetting language created by Donald E. Knuth; it has extensive capa-
bilities to typeset math. LATEX is an extension of TEX designed by Leslie Lamport;
its major features include
a strong focus on document structure and the logical markup of text;
automatic numbering and cross-referencing.
AMS-LATEX distills the decades-long experience of the American Mathematical So-
ciety (AMS) in publishing mathematical journals and books; it adds to LATEX a host
of features related to mathematical typesetting, especially the typesetting of multi-
line formulas and the production of ﬁnely-tuned printed output.
Articles written in LATEX (and AMS-LATEX) are accepted for publication by
an increasing number of journals, including all the journals of the AMS.
Look at the typeset sample articles: sampart.tex (in Appendix C, on pages
361–363) and intrart.tex (on pages 39–40). You can begin creating such high-
quality typeset articles after completing Part I.
What is document markup?
Most word processing programs are WYSIWYG (what you see is what you get); as
you work, the text on the computer monitor is shown, more or less, as it’ll look
when printed. Different fonts, font sizes, italics, and bold face are all shown.
A different approach is taken by a markup language. It works with a text edi-
tor, an editing program that shows the text, the source ﬁle, on the computer moni-
tor with only one font, in one size and shape. To indicate that you wish to change
the font in the printed copy in some way, you must “mark up” the source ﬁle. For
instance, to typeset the phrase “Small Caps” in small caps, you type
\textsc{Small Caps}
The \textsc command is a markup command, and the printed output is
Small Caps
TEX is a markup language; LATEX is another markup language, an extension
of TEX. Actually, it’s quite easy to learn how to mark up text. For another exam-
ple, look at the abstract of the sampart.tex sample article (page 364), and the
instruction

Introduction
xxi
\emph{complete-simple distributive lattices}
to emphasize the phrase “complete-simple distributive lattices”, which
when typeset looks like
complete-simple distributive lattices
On pages 364–371 we show the source ﬁle and the typeset version of the
sampart.tex sample article together. The markup in the source ﬁle may appear
somewhat bewildering at ﬁrst, especially if you have previously worked on a WYSI-
WYG word processor. The typeset article is a rather pleasing-to-the-eye polished
version of that same marked up material.1
TEX
TEX has excellent typesetting capabilities. It deals with mathematical formulas as
√
well as text. To get
a2 + b2 in a formula, type \sqrt{a^{2} + b^{2}}. There
is no need to worry about how to construct the square root symbol that covers
a2 + b2.
A tremendous appeal of the TEX language is that a source ﬁle is plain text,
sometimes called an ASCII ﬁle.2 Therefore articles containing even the most com-
plicated mathematical expressions can be readily transmitted electronically—to col-
leagues, coauthors, journals, editors, and publishers.
TEX is platform independent. You may type the source ﬁle on a Macintosh,
and your coauthor may make improvements to the same ﬁle on an IBM compati-
ble personal computer; the journal publishing the article may use a DEC minicom-
puter. The form of TEX, a richer version, used to typeset documents is called Plain
TEX. I’ll not try to distinguish between the two.
TEX, however, is a programming language, meant to be used by programmers.
LATEX
LATEX is much easier and safer to work with than TEX; it has a number of built-in
safety features and a large set of error messages.
LATEX, building on TEX, provides the following additional features:
An article is divided into logical units such as an abstract, sections, theorems,
a bibliography, and so on. The logical units are typed separately. After all the
1Of course, markup languages have always dominated typographic work of high quality. On the
Internet, the most trendy communications on the World Wide Web are written in a markup language
called HTML (HyperText Markup Language).
2ASCII stands for American Standard Code for Information Interchange.

xxii
Introduction
units have been typed, LATEX organizes the placement and formatting of these
elements.
Notice line 4 of the source ﬁle of the sampart.tex sample article
\documentclass{amsart}
on page 364. Here the general design is speciﬁed by the amsart “document
class”, which is the AMS article document class. When submitting your article
to a journal that is equipped to handle LATEX articles (and the number of such
journals is increasing rapidly), only the name of the document class is replaced by
the editor to make the article conform to the design of the journal.
LATEX relieves you of tedious bookkeeping chores. Consider a completed article,
with theorems and equations numbered and properly cross-referenced. Upon ﬁ-
nal reading, some changes must be made—for example, section 4 has to be placed
after section 7, and a new theorem has to be inserted somewhere in the middle.
Such a minor change used to be a major headache! But with LATEX, it becomes
almost a pleasure to make such changes. LATEX automatically redoes all the num-
bering and cross-references.
Typing the same bibliographic references in article after article is a tedious chore.
With LATEX you may use BIBTEX, a program that helps you create and main-
tain bibliographic databases, so references need not be retyped for each article.
BIBTEX will select and format the needed references from the databases.
All the features of LATEX are made available by the LaTeX format, which you
should use to typeset the sample documents in this book.
AMS-LATEX
The AMS enhanced the capabilities of LATEX in three different areas. You decide
which of these are important to you.
1. Math enhancements. The ﬁrst area of improvement is a wide variety of tools
for typesetting math. AMS-LATEX provides
excellent tools to deal with multiline math formulas requiring special align-
ment. For instance, in the following formula, the equals sign (=) is verti-
cally aligned and so are the explanatory comments:
x = (x + y)(x + z)
(by distributivity)
= x + yz
(by Condition (M))
= yz

Introduction
xxiii
numerous constructs for typesetting math, exempliﬁed by the following
formula:



−x2,
if x < 0;
f (x) = α + x, if 0 ≤ x ≤ 1;

x2,
otherwise.
special spacing rules for dozens of formula types, for example
a ≡ b (mod Θ)
If the above formula is typed inline, it becomes: a ≡ b (mod Θ); the spac-
ing is automatically changed.
multiline “subscripts” as in
α2i,j
i<n
j<m
user-deﬁned symbols for typesetting math, such as
ˆ
∗
∗
Trunc f (x),
ˆ
A,
formulas numbered in a variety of ways:
– automatically,
– manually (by tagging),
– by groups, with a group number such as (2), and individual numbers
such as (2a), (2b), and so on.
the proof environment and three theorem styles; see the sampart.tex
sample article (pages 361–363) for examples.
2. Document classes. AMS-LATEX provides a number of document classes, in-
cluding the AMS article document class, amsart, which allows the input of
the title page information (author, address, e-mail, and so on) as separate
entities. As a result, a journal can typeset even the title page of an article
according to its own speciﬁcations without having to retype it.
Many users prefer the visual design of the amsart document class to the sim-
pler design of the classical LATEX article document class.
3. Fonts. There are hundreds of binary operations, binary relations, negated bi-
nary relations, bold symbols, arrows, extensible arrows, and so on, provided
by AMS-LATEX, which also makes available additional math alphabets such
as Blackboard bold, Euler Fraktur, Euler Script, and math bold italic. Here
are just a few examples:
,
,
,
,
A, p, E

xxiv
Introduction
We have barely scratched the surface of this truly powerful set of enhance-
ments.
What is in the book?
Part I (Chapter 1) will help you get started quickly with LATEX; if you read it
carefully, you’ll certainly be ready to start typing your ﬁrst article and tackle LATEX
in more depth.
Part I guides you through:
marking up text, which is quite easy;
marking up math, which is not so straightforward (four sections ease you into
mathematical typesetting: the ﬁrst discusses the basic building blocks; the sec-
ond shows how to build up a complicated formula in simple steps; the third is a
formula gallery; and the fourth deals with equations and multiline formulas);
the anatomy of an article;
how to set up an article template;
typing your ﬁrst article.
Part II introduces the two most basic skills in depth: typing text and typing
math.
Chapters 2 and 3 introduce text and displayed text. Chapter 2 is very im-
portant; when typing your LATEX document, you spend most of your time typing
text. The topics covered include special characters and accents, hyphenation, fonts,
and spacing. Chapter 3 covers displayed text including lists and tables, and for the
mathematician, proclamations (theorem-like structures) and proofs.
Chapters 4 and 5 discuss math and displayed math. Of course, typing math
is the heart of any mathematical typesetting system. Chapter 4 discusses this topic
in detail, including basic constructs, operators, delimiters, building new symbols,
fonts, and grouping of equations. Chapter 5 presents one of the major contribu-
tions of AMS-LATEX: aligned multiline formulas. This chapter also contains other
multiline formulas.
Part III discusses the parts of a LATEX document. In Chapter 6, you learn
about the structure of a LATEX document. The most important topics are section-
ing and cross-referencing. In Chapter 7, the standard LATEX document classes are
presented: article, report, book, and letter, along with a description of the
standard LATEX distribution. In Chapter 8, the AMS document classes are dis-
cussed. In particular, the title page information for the amsart document class
and a description of the standard AMS-LATEX distribution is presented.
Part IV (Chapter 9) introduces techniques to customize LATEX to speed up
typing source ﬁles and typesetting of documents. LATEX really speeds up with user-
deﬁned commands, user-deﬁned environments, and custom formats. You’ll learn
how parameters that effect the behavior of LATEX are stored in counters and length
commands, how to change them, and how to design custom lists.

Introduction
xxv
In Part V (Chapters 10 and 11), we’ll discuss two programs: BIBTEX and
MakeIndex that complement the standard LATEX distribution; they give a helping
hand in making large bibliographies and indices.
Appendices A and B will probably be needed quite often in your work: they
contain math symbol tables and text symbol tables.
Appendix C presents the AMS-LATEX sample article, sampart.tex, ﬁrst in
typeset form (pages 361–363), then in “mixed” form, showing the source ﬁle and
the typeset article together (pages 364–371). You can learn a lot about LATEX and
AMS-LATEX just by reading the source ﬁle a paragraph at a time and see how that
paragraph looks typeset. Then Appendix D rewrites this sample article utilizing
the user-deﬁned commands collected in lattice.sty of section 9.5.
Appendix E relates some historical background material on LATEX: how did
it develop and how does it work. Appendix F is a brief introduction to the use
of PostScript fonts in a LATEX document. Appendix G shows how you can obtain
LATEX and AMS-LATEX, and how you can keep them up-to-date through the In-
ternet. A work session is reproduced (in part) using “anonymous ftp” (ﬁle transfer
protocol).
Appendix H will help those who have worked with (Plain) TEX, LATEX ver-
sion 2.09, AMS-TEX, or AMS-LATEX version 1.1, programs from which the new
LATEX and AMS-LATEX developed. Some tips are given to smooth the transition
to the new LATEX and AMS-LATEX.
Finally, Appendix I points the way for further study. The most important
book for extending and customizing LATEX is The LATEX Companion, the work of
Michel Goossens, Frank Mittelbach, and Alexander Samarin [12].

xxvi
Introduction
Typographical conventions
To make this book easy to read, I use some very simple conventions on the use of
fonts.
Explanatory text is set in the Galliard font, as this text is.
This book is about typesetting math in LATEX. So often you are told to type
in some material and shown how it’ll look typeset.
I use this font, Computer Modern typewriter style, to show what
you have to type.
All characters have the same width so it’s
easy to distinguish it from the other fonts used in this book.
I use the same font for commands (\parbox), environments (align), documents
(sampart.tex), document classes (article), directories and folders (work), coun-
ters (tocdepth), and so on.
The names of packages (amsmath), extensions of LATEX, are printed in a sans
serif font, as traditional.
When I show you how something looks when typeset, I use this font, Com-
puter Modern roman, which you’ll most likely see when you use LATEX. This
looks suﬃciently diﬀerent from the other two fonts I use so that you should have
little diﬃculty recognizing typeset LATEX material. If the typeset material is a
separate paragraph (or paragraphs), I make it visually stand out even more by
adding the little corner symbols on the margin to oﬀset it.
When I give explanations in the text: “Compare iﬀ with iff, typed as iff and
if{f}, respectively.” I use the same fonts but since they are not visually set off, it
may be a little harder to see that iﬀ is in Computer Modern roman and iff is in
Computer Modern typewriter style.
Commands are introduced, as a rule, with examples:
\\[0.5in]
However, sometimes it’s necessary to more formally deﬁne the syntax of a com-
mand. For instance:
\\[length ]
where length is a placeholder: it represents the length you have to type in. I use
the Computer Modern typewriter style italic font for placeholders.

PART I
A short course
1

C H A P T E R
1
Typing
your ﬁrst article
In this chapter, you’ll start writing your ﬁrst article. All you have to do is to type
the (electronic) source ﬁle; LATEX does the rest.
In the next few sections, I’ll introduce you to the most important commands
for typesetting text and math by working through examples. Go to the latter parts
of this book for more detail.
√
The source ﬁle is made up of text, math (for instance,
5 ), and instructions
to LATEX. This is how you type the last sentence:
The source file is made up of \emph{text}, \emph{math} (for
instance, $\sqrt{5}$), and \emph{instructions} to \LaTeX.
In this sentence,
The source file is made up of \emph{text}, \emph{math} (for
instance,
is text,
$\sqrt{5}$
is math, and
3

4
Chapter 1
Typing your ﬁrst article
\emph{text }
is an instruction (a command). Commands, as a rule, start with a backslash \ and
are meant to instruct LATEX; this particular command, \emph, emphasizes text
given as its argument (between the braces). Another kind of instruction is called
an environment. For instance,
\begin{flushright}
and
\end{flushright}
bracket a flushright environment—what is typed inside this environment comes
out right justiﬁed (lined up against the right margin) in the printed form.
In practice, text, math, and instructions are intertwined. For example,
\emph{My first integral} $\int \zeta^{2}(x) \, dx$
which produces
My ﬁrst integral
ζ2(x) dx
is a mixture of all three. Nevertheless, to some extent I try to introduce the three
topics: typing text, typing math, and giving instructions to LATEX (commands and
environments) as if they were separate topics.
I introduce the basic features of LATEX by working with a number of sample
documents. If you wish to obtain these documents electronically, create a sub-
directory (folder) on your computer, say, ftp, and proceed to download all the
sample ﬁles as described in section G.6. Also create a subdirectory (folder) called
work. Whenever you want to use one of these documents, copy it from the ftp
subdirectory (folder) to the work subdirectory (folder), so that the original remains
unchanged; alternatively, type in the examples as shown in the book. In this book,
the ftp directory and the work directory will refer to the directories (folders) you hereby
create without further elaboration.
1.1
Typing a very short “article”
First we discuss how to use the keyboard in LATEX, and then type a very short “ar-
ticle” containing only text.
1.1.1
The keyboard
In LATEX, to type text, use the following keys:

1.1
Typing a very short “article”
5
a-z
A-Z
0-9
+
=
*
/
(
)
[
]
You may also use the punctuation marks
,
;
.
?
!
:
‘
’
-
and the spacebar, the tab key, and the return (or enter) key.
There are thirteen special keys (on most keyboards):
#
$
%
&
~
^ \ {
} @ " |
used mostly in LATEX instructions. There are special commands to type most of
these special characters (as well as composite characters, such as accented charac-
ters) if you need them in text. For instance, $ is typed as \$, is typed as \_, and
% is typed as \% (while ¨a is typed as \"{a}); however, @ is typed as @. See sections
2.4.4 and 2.4.6 and the tables of Appendix B for more detail.
Every other key is prohibited! (Unless special steps are taken; more about
this in section 2.1.) Do not use the computer’s modiﬁer keys, such as Alt, Ctrl,
Command, Option, to produce special characters. LATEX will either reject or mis-
understand them. When trying to typeset a source ﬁle that contains a prohibited
character, LATEX will display the error message:
! Text line contains an invalid character.
l.222 completely irreducible^^?
^^?
In this message l.222 means line 222 of your source ﬁle. You must edit this line.
The log ﬁle (see section 1.11.3) also contains this message.
1.1.2
Your ﬁrst note
We start our discussion on how to type a note in LATEX with a simple example.
Suppose you want to use LATEX to produce the following:
It is of some concern to me that the terminology used in multi-section math
courses is not uniform.
In several sections of the course on matrix theory, the term “hamiltonian-
reduced” is used. I, personally, would rather call these “hyper-simple”. I invite
others to comment on this problem.
Of special concern to me is the terminology in the course by Prof. Rudi
Hochschwabauer. Since his ﬁeld is new, there is no accepted terminology. It is
imperative that we arrive at a satisfactory solution.

6
Chapter 1
Typing your ﬁrst article
Create a new ﬁle in the work directory with the name note1.tex and type
the following (if you prefer not to type it, copy the ﬁle from the ftp directory; see
page 4):
% Sample file: note1.tex
% Typeset with LaTeX format
\documentclass{article}
\begin{document}
It is of some concern to me
that
the terminology used in
multi-section
math courses is not uniform.
In several sections of the course on
matrix theory, the
term
‘‘hamiltonian-reduced’’ is used.
I, personally, would rather call these ‘‘hyper-simple’’. I
invite others to comment on this
problem.
Of special concern to me is the terminology in the course
by Prof.~Rudi Hochschwabauer.
Since his field is new, there is
no accepted
terminology.
It is imperative
that we arrive at a satisfactory solution.
\end{document}
The ﬁrst two lines start with %; they are comments ignored by LATEX. (The %
character is very useful. If, for example, while typing the source ﬁle you want to
make a comment, but do not want that comment to appear in the typeset version,
start the line with %. The whole line will be ignored during typesetting. You can
also comment out a part of a line:
... % ...
The part of a line past the % character will be ignored.)
The line after the two comments names the “document class”, which speciﬁes
how the document will be formatted.
The text of the note is typed within the “document environment”, that is,
between the two lines
\begin{document}
and
\end{document}

1.1
Typing a very short “article”
7
Now typeset note1.tex; you should get the same typeset document as shown on
page 5.
As seen in the previous example, LATEX is somewhat different from most word
processors. It ignores the way you format the text, and follows only the formatting
instructions given by the markup commands. LATEX takes note of whether you put
a space in the text, but it ignores how many spaces are inserted. In LATEX, one or
more blank lines mark the end of a paragraph. Tabs are treated as spaces. Note that
you typed the left double quote as ‘‘ (two left single quotes) and the right double
quote as ’’ (two right single quotes). The left single quote key is not always easy
to ﬁnd; it usually hides in the upper left or upper right corner of the keyboard. The
symbol ˜ is called a “tie” and keeps Prof. and Rudi together.
1.1.3
Lines too wide
LATEX reads the text in the source ﬁle one line at a time and when the end of a para-
graph is reached, LATEX typesets it (see section E.2 for a more detailed discussion).
Most of the time, there is no need for corrective action. Occasionally, however,
LATEX gets into trouble splitting the paragraph into typeset lines. To illustrate this,
modify note1.tex: in the second sentence replace “term” by “strange term”,
and in the fourth sentence delete “Rudi ”. Save this modiﬁed ﬁle with the name
note1b.tex in the work directory. (You’ll ﬁnd note1b.tex in the ftp directory—
see page 4).
Typesetting note1b.tex, you get:
It is of some concern to me that the terminology used in multi-section math
courses is not uniform.
In several sections of the course on matrix theory, the strange term “hamiltonian-
reduced” is used. I, personally, would rather call these “hyper-simple”. I invite
others to comment on this problem.
Of special concern to me is the terminology in the course by Prof. Hochschwabauer.
Since his ﬁeld is new, there is no accepted terminology. It is imperative that we
arrive at a satisfactory solution.
The ﬁrst line of paragraph two is about 1/4 inch too wide. The ﬁrst line of
paragraph three is even wider. On your monitor, LATEX displays the message:
Overfull \hbox (15.38948pt too wide) in paragraph at lines 10--15
[]\OT1/cmr/m/n/10 In sev-eral sec-tions of the course on ma-trix
the-ory, the strange term ‘‘hamiltonian-
[]
Overfull \hbox (23.27834pt too wide) in paragraph at lines 16--22
[]\OT1/cmr/m/n/10 Of spe-cial con-cern to me is the ter-mi-nol-ogy

8
Chapter 1
Typing your ﬁrst article
in the course by Prof. Hochschwabauer.
[]
You’ll ﬁnd the same message in the log ﬁle note1b.log (see section 1.11.3).
The reference
Overfull \hbox (15.38948pt too wide) in paragraph at lines 10--15
is made to paragraph two (lines 10–15); the typeset version has a line (line number
unspeciﬁed within the typeset paragraph) which is 15.38948pt too wide. LATEX
uses points (pt) to measure distances; there are about 72 points to an inch. The
next two lines
[]\OT1/cmr/m/n/10 In sev-eral sec-tions of the course on ma-trix
the-ory, the strange term ‘‘hamiltonian-
identify the source of the problem: LATEX would not hyphenate
hamiltonian-reduced,
since it (automatically) hyphenates a hyphenated word only at the hyphen. You
may wonder what \OT1/cmr/m/n/10 signiﬁes. It says that the current font is the
Computer Modern roman font at size 10 points (see section 2.6.1).
The second reference
Overfull \hbox (23.27834pt too wide) in paragraph at lines 16--22
is made to paragraph three (lines 16–22). The problem is with the word
Hochschwabauer
which the hyphenation routine of LATEX can’t handle. (If you use a German hy-
phenation routine, it’ll have no difﬁculty hyphenating Hochschwabauer.)
If you encounter such a problem, try to reword the sentence or add an op-
tional hyphen \-, which encourages LATEX to hyphenate at this point if necessary.
For instance, rewrite Hochschwabauer as
Hoch\-schwabauer
and the second problem goes away.
Sometimes a small horizontal overﬂow is difﬁcult to spot. The draft docu-
ment class option is very useful in this case: it’ll paint an ugly slug on the margin to
mark an overfull line; see sections 7.1.2 and 8.4 for document class options. You
may invoke this option by changing the \documentclass line to
\documentclass[draft]{article}
You’ll ﬁnd this version of note1b.tex under the name noteslug.tex in the ftp
directory.

1.1
Typing a very short “article”
9
1.1.4
More text features
Next you’ll produce the following note in LATEX:
November 5, 1995
From the desk of George Gr¨
atzer
February 7–21 please use my temporary e-mail address:
George Gratzer@umanitoba.ca
Type in the following source ﬁle, save it as note2.tex in the work directory
(you’ll also ﬁnd note2.tex in the ftp directory):
% Sample file: note2.tex
% Typeset with LaTeX format
\documentclass{article}
\begin{document}
\begin{flushright}
\today
\end{flushright}
\textbf{From the desk of George Gr\"{a}tzer}\\[10pt]
February~7--21 \emph{please} use my temporary e-mail address:
\begin{center}
\texttt{George\_Gratzer@umanitoba.ca}
\end{center}
\end{document}
This note introduces several additional features of LATEX:
The \today command displays today’s date.
Use environments to right justify or center text. Use the \emph command to em-
phasize text; the text to be emphasized is surrounded by { and }. Use \textbf
for bold text; the text to be made bold is also surrounded by { and }. Simi-
larly, use \texttt for typewriter style text. \emph, \textbf, and \texttt
are examples of commands with arguments. Note that command names are case
sensitive; do not type \Textbf or \TEXTBF in lieu of \textbf.
LATEX commands (almost) always start with \ followed by the command name,
for instance, \textbf. The command name is terminated by the ﬁrst non-alpha-
betic character.

10
Chapter 1
Typing your ﬁrst article
Use double hyphens for number ranges (en-dash): 7--21 prints 7–21; use triple
hyphens (---) for the “em-dash” punctuation mark—such as the one in this sen-
tence.
If you want to create additional space between lines (as in the last note under
the line From the desk . . . ), use the command \\[10pt] with an appropriate
amount of vertical space. (\\ is the newline command—see section 2.7.1; the
variant used in the above example is the newline with additional vertical space.)
The distance may be given in points, centimeters (cm), or inches (in). (72.27
points make an inch.)
There are special rules for accented characters and some European characters. For
instance, ¨
a is typed as \"{a}. Accents are explained in section 2.4.6 (see also the
tables in Appendix B).
You’ll seldom need to know more than this about typing text. For more detail,
however, see Chapters 2 and 3. All text symbols are organized into tables in Ap-
pendix B.
1.2
Typing math
Now you can start mixing text with math formulas.
1.2.1
The keyboard
In addition to the regular text keys (section 1.1.1), three more keys are needed to
type math:
| < >
(| is the shifted \ key on many keyboards.)
1.2.2
A note with math
You’ll begin typesetting math with the following note:
In ﬁrst year Calculus, we deﬁne intervals such as (u, v) and (u, ∞). Such an
interval is a neighborhood of a if a is in the interval. Students should realize that
∞ is only a symbol, not a number. This is important since we soon introduce
concepts such as limx→∞ f(x).
When we introduce the derivative
f (x) − f (a)
lim
,
x→a
x − a
we assume that the function is deﬁned and continuous in a neighborhood of a.

1.2
Typing math
11
To create the source ﬁle for this mixed math and text note, create a new doc-
ument with an editor. Name it math.tex, place it in the work directory, and type
in the following source ﬁle—or copy math.tex from the ftp directory:
% Sample file: math.tex
% Typeset with LaTeX format
\documentclass{article}
\begin{document}
In first year Calculus, we
define intervals
such as
$(u, v)$ and $(u, \infty)$.
Such an interval is a
\emph{neighborhood} of
$a$
if
$a$ is in the interval.
Students should
realize that
$\infty$ is only a
symbol, not a number.
This is important since
we soon introduce concepts
such as $\lim_{x \to \infty} f(x)$.
When we introduce the derivative
\[
\lim_{x \to a} \frac{f(x) - f(a)}{x - a},
\]
we assume that the function is defined and continuous
in a neighborhood of
$a$.
\end{document}
This note introduces the basic techniques of typesetting math with LATEX:
There are two kinds of math formulas and environments: inline and displayed.
Inline math environments open and close with $.
Displayed math environments open with \[ and close with \].
LATEX ignores the spaces you insert in math environments with two exceptions:
spaces that delimit commands (see section 2.3.1) and spaces in the argument of
commands that temporarily revert into text mode. (\mbox is such a command;
see section 4.5.) Thus spacing in math is important only for the readability of
the source ﬁle. To summarize:
Rule
Spacing in text and math
In text mode, many spaces equal one space, while in math mode, the spaces are
ignored.
The same formula may be typeset differently depending on which math environ-
ment it’s in. The expression x → a is typed as a subscript to lim in the inline

12
Chapter 1
Typing your ﬁrst article
formula limx→a f(x), typed as $\lim_{x \to a} f(x)$, but it’s placed below
lim in the displayed version:
lim f (x)
x→a
typed as
\[
\lim_{x \to a} f(x)
\]
A math symbol is invoked by a command. Examples: the command for ∞ is
\infty and the command for → is \to. The math symbols are organized into
tables in Appendix A.
To access most of the symbols listed in Appendix A by name, use the amssymb
package; in other words, the article should start with
\documentclass{article}
\usepackage{amssymb}
The amssymb package loads the amsfonts package, which contains the commands
for using the AMSFonts (see section 4.14.2).
Some commands such as \sqrt need arguments enclosed in { and }. To type-
√
set
5, type $\sqrt{5}$, where \sqrt is the command and 5 is the argument.
Some commands need more than one argument. To get
3 + x
5
type
\[
\frac{3+x}{5}
\]
\frac is the command, 3+x and 5 are the arguments.
There are many mistakes you can make, even in such a simple note. You’ll
now introduce mistakes in math.tex, by inserting and deleting % signs to make the
mistakes visible to LATEX one at a time. Recall that lines starting with % are ignored
by LATEX. Type the following source ﬁle, and save it under the name mathb.tex
in the work directory (or copy over the ﬁle mathb.tex from the ftp directory).
% Sample file: mathb.tex
% Typeset with LaTeX format
\documentclass{article}

1.2
Typing math
13
\begin{document}
In first year Calculus, we
define intervals
such as
%$(u, v)$ and $(u, \infty)$.
Such an interval is a
$(u, v)$ and
(u, \infty)$.
Such an interval is a
{\emph{neighborhood} of $a$
if $a$ is in the interval.
Students should
realize that
$\infty$ is only a
symbol, not a number.
This is important since
we soon introduce concepts
such as $\lim_{x \to \infty} f(x)$.
%such as $\lim_{x \to \infty f(x)$.
When we introduce the derivative
\[
\lim_{x \to a} \frac{f(x) - f(a)}{x - a}
%\lim_{x \to a} \frac{f(x) - f(a)
x - a}
\]
we assume that the function is defined and continuous
in a neighborhood of
$a$.
\end{document}
Exercise 1
Note that in line 8, the second $ is missing. When you typeset the
mathb.tex ﬁle, LATEX sends the error message:
! Missing $ inserted.
<inserted text>
$
l.8 ..., v)$ and
(u, \infty
)$.
Such an interval is a
?
Since you omitted $, LATEX reads (u, \infty) as text; but the \infty command
instructs LATEX to typeset a math symbol, which can only be done in math mode.
So LATEX offers to put a $ in front of \infty. LATEX suggests a cure, but in this
example it comes too late. Math mode should start just prior to (u.
Exercise 2
In the mathb.tex ﬁle, delete % at the beginning of line 7 and insert
a % at the beginning of line 8 (this eliminates the previous error); delete % at the
beginning of line 15 and insert a % at the beginning of line 14 (this introduces a
new error: the closing brace of the subscript is missing). Save the changes, and
typeset the note. You get the error message:
! Missing } inserted.
<inserted text>
}

14
Chapter 1
Typing your ﬁrst article
l.15 ...im_{x \to \infty f(x)$
.
?
LATEX is telling you that a closing brace } is missing, but it’s not sure where. LATEX
noticed that the subscript started with { and it reached the end of the math formula
before ﬁnding }. You must look in the formula for a { that is not closed, and close
it with }.
Exercise 3
Delete % at the beginning of line 14 and insert a % at the beginning
of line 15, which removes the last error, and delete % at the beginning of line 20
and insert a % at the beginning of line 19 (introducing the ﬁnal error: deleting the
closing brace of the ﬁrst argument of \frac). Save and typeset the ﬁle. You get
the error message:
! LaTeX Error: Bad math environment delimiter.
l.21 \]
There is a bad math environment delimiter in line 21, namely, \]. So the reference
to
! Bad math environment delimiter.
is to the displayed formula. Since the environment delimiter is correct, something
must have gone wrong with the displayed formula. This is what happened: LATEX
was trying to typeset
\lim_{x \to a} \frac{f(x) - f(a)
x - a}
but \frac needs two arguments. LATEX found f(x) - f(a) x - a as the ﬁrst
argument. While looking for the second, it found \], which is obviously an error
(it was looking for a { ).
1.2.3
Building blocks of a formula
A formula is built up from various types of components. We group them as follows:
Arithmetic
Subscripts and superscripts
Accents
Binomial coefﬁcients
Congruences
Delimiters
Operators
Ellipses
Integrals

1.2
Typing math
15
Matrices
Roots
Sums and products
Text
Some of the commands in the following examples are deﬁned in the amsmath pack-
age; in other words, to typeset these examples with the article document class,
the article should start with
\documentclass{article}
\usepackage{amssymb,amsmath}
Arithmetic The arithmetic operations a + b, a − b, −a, a/b, ab are typed as
expected:
$a + b$, $a - b$, $-a$, $a / b$, $a b$
If you wish to use · or × for multiplication, as in a · b or a × b, use \cdot or
\times, respectively. The expressions a · b and a × b are typed as follows:
$a \cdot b$
$a \times b$
Displayed fractions, such as
1 + 2x
x + y + xy
are typed with \frac:
\[
\frac{1 + 2x}{x + y + xy}
\]
The \frac command is seldom used inline.
Subscripts and superscripts Subscripts are typed with (underscore) and super-
scripts with ^ (caret). Remember to enclose the subscripts and superscripts
with { and }. To get a1, type the following characters:
Go into inline math mode:
$
type the letter a:
a
subscript command:
_
bracket the subscripted 1:
{1}
exit inline math mode:
$
that is, type $a_{1}$. Omitting the braces in this example causes no harm;
however, to get a10, you must type $a_{10}$. Indeed, $a_10$ prints a10.
Further examples: ai , a2, ai1 are typed as
1

16
Chapter 1
Typing your ﬁrst article
$a_{i_{1}}$, $a^{2}$, $a^{i_{1}}$
Accents The four most often used math accents are:
¯
a
typed as
$\bar{a}$
ˆ
a
typed as
$\hat{a}$
˜
a
typed as
$\tilde{a}$
a
typed as
$\vec{a}$
Binomial coefﬁcients The amsmath package provides the \binom command for
binomial coefﬁcients. For example,
a
is typed inline as
b+c
$\binom{a}{b + c}$
whereas the displayed version
a
n2−1
2
b + c
n + 1
is typed as
\[
\binom{a}{b + c} \binom{\frac{n^{2} - 1}{2}}{n + 1}
\]
Congruences The two most important forms are:
a ≡ v (mod θ)
typed as
$a \equiv v \pmod{\theta}$
a ≡ v (θ)
typed as
$a \equiv v \pod{\theta}$
The second form requires the amsmath package.
Delimiters These are parenthesis-like symbols that vertically expand to enclose a
formula. For example: (a + b)2, which is typed as $(a + b)^{2}$, and
2
1 + x
2 + y2
which is typed as
\[
\left( \frac{1 + x}{2 + y^{2}} \right)^{2}
\]

1.2
Typing math
17
contain such delimiters. The \left( and \right) commands tell LATEX to
size the parentheses correctly (relative to the size of the symbols inside the
parentheses). Two further examples:
a + b ,
A2
2
would be typed as:
\[
\left| \frac{a + b}{2} \right|,
\quad \left\| A^{2} \right\|
\]
where \quad is a spacing command (see section 4.11 and Appendix A).
Operators To typeset the sine function sin x, type: $\sin x$. Note that $sin x$
prints: sinx, where the typeface of sin is wrong, as is the spacing.
LATEX calls \sin an operator; there are a number of operators listed in sec-
tion 4.7.1 and Appendix A. Some are just like \sin; others produce a more
complex display:
lim f (x) = 0
x→0
which is typed as
\[
\lim_{x \to 0} f(x) = 0
\]
Ellipses The ellipsis ( . . . ) in math sometimes needs to be printed as low dots
and sometimes as (vertically) centered dots. Print low dots with the \ldots
command as in F (x1, x2, . . . , xn), typed as
$F(x_{1}, x_{2}, \ldots , x_{n})$
Print centered dots with the \cdots command as in x1 +x2 +· · ·+xn, typed
as
$x_{1} + x_{2} + \cdots + x_{n}$
If you use the amsmath package, there is a good chance that the command
\dots will print the ellipsis as desired.
Integrals The command for an integral is \int; the lower limit is a subscript and
π
the upper limit is a superscript. Example:
sin x dx = 2 is typed as
0
$\int_{0}^{\pi} \sin x \, dx = 2$
\, is a spacing command (see section 4.11 and Appendix A).

18
Chapter 1
Typing your ﬁrst article
Matrices The amsmath package provides you with a matrix environment:
a + b + c
uv
x − y
27
a + b
u + v
z
134
which is typed as follows:
\[
\begin{matrix}
a + b + c & uv
& x - y & 27\\
a + b
& u + v & z
& 134
\end{matrix}
\]
The matrix elements are separated by &; the rows are separated by \\. The ba-
sic form gives no parentheses; for parentheses, use the pmatrix environment;
for brackets, the bmatrix environment; for vertical lines (determinants, for
example), the vmatrix environment; for double vertical lines, the Vmatrix
environment. For example,
a + b + c
uv
30
7
A =
a + b
u + v
3
17
is typed as follows:
\[
\mathbf{A} =
\begin{pmatrix}
a + b + c & uv\\
a + b & u + v
\end{pmatrix}
\begin{pmatrix}
30 & 7\\
3 & 17
\end{pmatrix}
\]
√
Roots \sqrt produces the square root; for instance,
5 is typed as
$\sqrt{5}$
√
and
a + 2b is typed as
$\sqrt{a + 2b}$

1.2
Typing math
19
√
The nth root, n 5, is done with two arguments:
$\sqrt[n]{5}$
Note that the ﬁrst argument is in brackets [ ]; it’s an optional argument (see
section 2.3).
Sums and products The command for sum is \sum and for product is \prod. The
examples
n
n
x2
x2
i
i
i=1
i=1
are typed as
\[
\sum_{i=1}^{n} x_{i}^{2} \qquad \prod_{i=1}^{n} x_{i}^{2}
\]
\qquad is a spacing command; it separates the two formulas (see section 4.11
and Appendix A).
Sums and products are examples of large operators; all of them are listed in
section 4.8 and Appendix A. They display in a different style (and size) when
used in an inline formula:
n
x2
n
x2.
i=1
i
i=1
i
Text Place text in a formula with an \mbox command. For instance,
a = b
by assumption
is typed as
\[
a = b \mbox{\qquad by assumption}
\]
Note the space command \qquad in the argument of \mbox. You could also
have typed
\[
a = b \qquad \mbox{by assumption}
\]
because \qquad works in text as well as in math.
If you use the amsmath package, then the \text command is available in lieu
of the \mbox command. It works just like the \mbox command except that
it automatically changes the size of its argument as required, as in apower,
typed as

20
Chapter 1
Typing your ﬁrst article
$a^{ \text{power} }$
If you do not want to use the large amsmath package, the tiny amstext package
also provides the \text command (see section 8.5).
1.2.4
Building a formula step-by-step
It is simple to build up complicated formulas from the components described in
section 1.2.3. Take the formula
[ n ]
3
2
xi2
µ(i) 2 (i2 − 1)
i,i+1
i+3
3 ρ(i) − 2 + 3 ρ(i) − 1
i=1
3
for instance. You should build this up in several steps. Create a new ﬁle in the work
directory. Call it formula.tex and type in the lines:
% File: formula.tex
% Typeset with LaTeX format
\documentclass{article}
\usepackage{amssymb,amsmath}
\begin{document}
\end{document}
and save it. At present, the ﬁle has an empty document environment.1 Type each
part of the formula as an inline or displayed formula so that you can typeset the
document and check for errors.
Step 1
Let’s start with n :
2
$\left[ \frac{n}{2} \right]$
Type this into formula.tex and test it by typesetting the document.
Step 2
Now you can do the sum:
[ n ]
2
i=1
For the superscript, you can cut and paste the formula created in Step 1 (without
the dollar signs), to get
\[
\sum_{i = 1}^{ \left[ \frac{n}{2} \right] }
\]
1The quickest way to create this ﬁle is to open mathb.tex, save it under the new name formula.tex,
and delete the lines in the document environment. Then add the line
\usepackage{amssymb,amsmath}

1.2
Typing math
21
Step 3
Next, do the two formulas in the binomial:
i + 3
xi2
i,i+1
3
Type them as separate formulas in formula.tex:
\[
x_{i, i + 1}^{i^{2}} \qquad \left[ \frac{i + 3}{3} \right]
\]
Step 4
Now it’s easy to do the binomial. Type the following formula by cutting
and pasting the previous formulas:
\[
\binom{ x_{i,i + 1}^{i^{2}} }{ \left[ \frac{i + 3}{3} \right] }
\]
which prints:
xi2
i,i+1
i+3
3
3
Step 5
Next type the formula under the square root µ(i) 2 (i2 − 1) as
$\mu(i)^{ \frac{3}{2} } (i^{2} - 1)$
3
and then the square root
µ(i) 2 (i2 − 1) as
$\sqrt{ \mu(i)^{ \frac{3}{2} } (i^{2} - 1) }$
Step 6
The two cube roots, 3 ρ(i) − 2 and 3 ρ(i) − 1, are easy to type:
$\sqrt[3]{ \rho(i) - 2 }$
$\sqrt[3]{ \rho(i) - 1 }$
Step 7
So now get the fraction:
3
µ(i) 2 (i2 − 1)
3 ρ(i) − 2 + 3 ρ(i) − 1
typed, cut, and pasted as
\[
\frac{ \sqrt{ \mu(i)^{ \frac{3}{2}} (i^{2} -1) } }
{ \sqrt[3]{\rho(i) - 2} + \sqrt[3]{\rho(i) - 1} }
\]

22
Chapter 1
Typing your ﬁrst article
Step 8
Finally, get the formula
[ n ]
3
2
xi2
µ(i) 2 (i2 − 1)
i,i+1
i+3
3 ρ(i) − 2 + 3 ρ(i) − 1
i=1
3
by cutting and pasting the pieces together, leaving only one pair of displayed math
delimiters:
\[
\sum_{i = 1}^{ \left[ \frac{n}{2} \right] }
\binom{ x_{i, i + 1}^{i^{2}} }
{ \left[ \frac{i + 3}{3} \right] }
\frac{ \sqrt{ \mu(i)^{ \frac{3}{2}} (i^{2} - 1) } }
{ \sqrt[3]{\rho(i) - 2} + \sqrt[3]{\rho(i) - 1} }
\]
Notice the use of
spacing to help distinguish the braces (note that some editors help you balance
the braces);
separate lines for the various pieces.
Keep the source ﬁle readable. Of course, this is for your beneﬁt, since LATEX does
not care. It would also accept
\[\sum_{i=1}^{\left[\frac{n}{2}\right]}\binom{x_{i,i+1}^{i^{2}}}
{\left[\frac{i+3}{3}\right]}\frac{\sqrt{\mu(i)^{\frac{3}
{2}}(i^{2}-1)}}{\sqrt[3]{\rho(i)-2}+\sqrt[3]{\rho(i)-1}}\]
Problems arise with this haphazard style when you make a mistake. Try to ﬁnd the
error in the next version:
\[\sum_{i=1}^{\left[\frac{n}{2}\right]}\binom{x_{i,i+1}^{i^{2}}}
{\left[\frac{i+3}{3}\right]}\frac{\sqrt{\mu(i)^{\frac{3}
{2}}}(i^{2}-1)}}{\sqrt[3]{\rho(i)-2}+\sqrt[3]{\rho(i)-1}}\]
(Answer: \frac{3}{2} should be followed by }} and not by }}}.)
1.3
Formula gallery
In this section, I present the formula gallery (gallery.tex in the ftp directory),
a collection of formulas—some simple, some complex—that illustrate the power
of LATEX and AMS-LATEX. Most of the commands in these examples have not yet
been discussed, but comparing the source formula with the typeset version should
answer most of your questions. Occasionally, I’ll give you a helping hand with
some comments.

1.3
Formula gallery
23
Many of these formulas are from text books and research articles. The last
six are reproduced from the document testart.tex that was distributed by the
AMS with AMS-LATEX version 1.1. Some of these examples require the amssymb
and amsmath packages. So make sure to include the line
\usepackage{amssymb,amsmath}
following the documentclass line of any article using such constructs. The pack-
ages (if any) required for each formula shall be indicated.
Formula 1
A set-valued function:
x → { c ∈ C | c ≤ x }
\[
x \mapsto \{\, c \in C \mid c \leq x \,\}
\]
Note that both | and \mid print |. Use | for absolute value signs. In this formula,
\mid is used because it provides extra spacing (see section 4.6.4). To equalize the
spacing around c ∈ C and c ≤ x, a thin space was added inside each brace (see
section 4.11). The same technique is used in a number of other formulas below.
Formula 2
The \left| and \right| commands print the vertical bars | whose
size adjusts to the size of the formula. The \mathfrak command provides access
to the Fraktur math alphabet (which requires the amsfonts or the eufrak package):
( Ij | j ∈ J ) < m
typed as
\[
\left| \bigcup (\, I_{j} \mid j \in J \,) \right|
< \mathfrak{m}
\]
Formula 3
Note that you need spacing both before and after the text fragment
“for some” in the following example. The argument of \mbox is typeset in text
mode, so a single space is recognized.
A = { x ∈ X | x ∈ Xi
for some i ∈ I }
\[
A = \{\, x \in X \mid x \in X_{i}
\mbox{\quad for some } i \in I \,\}
\]

24
Chapter 1
Typing your ﬁrst article
Formula 4
Space to show the logical structure:
a1, a2 ≤ a , a
iff
a
or
a
and a
1
2
1 < a1
1 = a1
2 ≤ a2
\[
\langle a_{1}, a_{2} \rangle \leq \langle a’_{1}, a’_{2}\rangle
\qquad \mbox{if{f}} \qquad a_{1} < a’_{1} \quad
\mbox{or}
\quad a_{1} = a’_{1} \mbox{ and } a_{2} \leq a’_{2}
\]
Note that in if{f} (in the argument of \mbox) the second f is in braces to avoid
the use of the ligature—the merging of the two f ’s (see section 2.4.5).
Formula 5
Here are some examples of Greek letters:
Γu = { γ | γ < 2χ, Bα
u , Bγ ⊆ u }
\[
\Gamma_{u’} = \{\, \gamma \mid \gamma < 2\chi,
\ B_{\alpha} \nsubseteq u’, \ B_{\gamma} \subseteq u’ \,\}
\]
See Appendix A for a complete listing of Greek letters. The \nsubseteq command
requires the amssymb package.
Formula 6
\mathbb gives the Blackboard bold math alphabet (available only in
uppercase):
A = B2 × Z
\[
A = B^{2} \times \mathbb{Z}
\]
Blackboard bold requires the amsfonts package.
Formula 7
The \left( and \right) commands tell LATEX to size the paren-
theses correctly (relative to the size of the symbols in the parentheses).
c
( si | i ∈ I )
=
( sc | i ∈ I )
i
\[
\left( \bigvee (\, s_{i} \mid i \in I \,) \right)^{c} =
\bigwedge (\, s_{i}^{c} \mid i \in I \,)
\]
Notice how the superscript is placed right on top of the subscript in sc.
i

1.3
Formula gallery
25
Formula 8
y ∨
( [Bγ] | γ ∈ Γ ) ≡ z ∨
( [Bγ] | γ ∈ Γ ) (mod Φx)
\[
y \vee \bigvee (\, [B_{\gamma}] \mid \gamma
\in \Gamma \,) \equiv z \vee \bigvee (\, [B_{\gamma}]
\mid \gamma \in \Gamma \,) \pmod{ \Phi^{x} }
\]
Formula 9
Use \nolimits so that the “limit” of the large operator is displayed
as a subscript:
f (x) =
( xj | j ∈ Ii ) | i < ℵα
m
m
\[
f(\mathbf{x}) = \bigvee\nolimits_{\!\mathfrak{m}}
\left(\,
\bigwedge\nolimits_{\mathfrak{m}}
(\, x_{j} \mid j \in I_{i} \,) \mid i < \aleph_{\alpha}
\,\right)
\]
The \mathfrak command requires the amsfonts or the eufrak package. A negative
space (\!) was inserted to bring m a little closer to
(see section 4.11).
Formula 10
The \left. command gives a blank left delimiter.
F (x)|b = F (b) − F (a)
a
\[
\left. F(x) \right|_{a}^{b} = F(b) - F(a)
\]
Formula 11
1
2
u + v ∼ w ∼ z
α
\[
u \underset{\alpha}{+} v \overset{1}{\thicksim} w
\overset{2}{\thicksim} z
\]
The \underset and \overset commands require the amsmath package.

26
Chapter 1
Typing your ﬁrst article
Formula 12
In this formula, \mbox would not work properly, so we use \text.
def
f (x) = x2 − 1
\[
f(x) \overset{ \text{def} }{=} x^{2} - 1
\]
This formula requires the amsmath package.
Formula 13
n
a + b + · · · + z
\[
\overbrace{a + b + \cdots + z}^{n}
\]
Formula 14
a + b + c
uv
= 7
a + b
c + d
\[
\begin{vmatrix}
a + b + c & uv\\
a + b & c + d
\end{vmatrix}
= 7
\]
a + b + c
uv
= 7
a + b
c + d
\[
\begin{Vmatrix}
a + b + c & uv\\
a + b & c + d
\end{Vmatrix}
= 7
\]
The vmatrix and Vmatrix environments require the amsmath package.

1.3
Formula gallery
27
Formula 15
The \mathbf{N} command makes a bold N. (\textbf{N} would
use a different font, namely, N.)
bij ˆ
yj =
b(λ) ˆ
y
ij
j + (bii − λi)ˆ
yi ˆ
y
j∈N
j∈N
\[
\sum_{j \in \mathbf{N}} b_{ij} \hat{y}_{j} =
\sum_{j \in \mathbf{N}} b^{(\lambda)}_{ij} \hat{y}_{j} +
(b_{ii} - \lambda_{i}) \hat{y}_{i} \hat{y}
\]
Formula 16
To produce the formula:
n
1
ˆ
x
ˆ
j
Hc =
kij det K(i|i)
2
j=1
try
\[
\left( \prod^n_{\, j = 1} \hat x_{j} \right) H_{c} =
\frac{1}{2} \hat k_{ij} \det \hat{ \mathbf{K} }(i|i)
\]
However, this produces:


n

1
ˆ
x 
ˆ
j
Hc =
kij det ˆ
K(i|i)
2
j=1
Correct the overly large parentheses by using the \biggl and \biggr commands
in place of \left( and \right), respectively (see section 4.6.2). Adjust the small
hat over K by using \widehat:
\[
\biggl( \prod^n_{\, j = 1} \hat x_{j} \biggr) H_{c} =
\frac{1}{2} \hat{k}_{ij} \det \widehat{ \mathbf{K} }(i|i)
\]
Formula 17
In this formula, use \overline{I} to get I (the variant \bar{I},
which prints ¯
I , is less pleasing to me):
det K(t = 1, t1, . . . , tn) =
(−1)|I|
ti
(Dj + λjtj) det A(λ)(I|I) = 0
I∈n
i∈I
j∈I

28
Chapter 1
Typing your ﬁrst article
\[
\det \mathbf{K} (t = 1, t_{1}, \dots, t_{n}) =
\sum_{I \in \mathbf{n} }(-1)^{|I|}
\prod_{i \in I} t_{i}
\prod_{j \in I} (D_{j} + \lambda_{j} t_{j})
\det \mathbf{A}^{(\lambda)} (\overline{I} | \overline{I}) = 0
\]
Formula 18
Note that \| provides the
math symbol in this formula:
H(z + v) − H(z + v ) − BH(z)(v − v )
lim
= 0
(v,v )→(0,0)
v − v
\[
\lim_{(v, v’) \to (0, 0)}
\frac{H(z + v) - H(z + v’) - BH(z)(v - v’)}
{\| v - v’ \|} = 0
\]
Formula 19
This formula uses the calligraphic math alphabet:
|∂u|2Φ0(z)eα|z|2 ≥ c4α
|u|2Φ0eα|z|2 + c5δ−2
|u|2Φ0eα|z|2
D
D
A
\[
\int_{\mathcal{D}} | \overline{\partial u} |^{2}
\Phi_{0}(z) e^{\alpha |z|^2} \geq
c_{4} \alpha \int_{\mathcal{D}} |u|^{2} \Phi_{0}
e^{\alpha |z|^{2}} + c_{5} \delta^{-2} \int_{A}
|u|^{2} \Phi_{0} e^{\alpha |z|^{2}}
\]
Formula 20
The \hdotsfor command places dots spanning multiple columns
in a matrix.
The \dfrac command is the displayed variant of \frac (see section 4.4.1).


ϕ · Xn,1

(x + ε2)2
· · · (x + εn−1)n−1 (x + εn)n
 ϕ

1 × ε1


ϕ · X
ϕ · X

n,1
n,2
· · ·

(x + εn−1)n−1 (x + εn)n
A = 
 ϕ2 × ε1
ϕ2 × ε2
 + In


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


ϕ · X

n,1
ϕ · Xn,2
· · ·
ϕ · Xn,n−1
ϕ · Xn,n
ϕn × ε1
ϕn × ε2
ϕn × εn−1
ϕn × εn
\[
\mathbf{A} =
\begin{pmatrix}

1.4
Typing equations and aligned formulas
29
\dfrac{\varphi \cdot X_{n, 1}}
{\varphi_{1} \times \varepsilon_{1}}
& (x + \varepsilon_{2})^{2} & \cdots
& (x + \varepsilon_{n - 1})^{n - 1}
& (x + \varepsilon_{n})^{n}\\
\dfrac{\varphi \cdot X_{n, 1}}
{\varphi_{2} \times \varepsilon_{1}}
& \dfrac{\varphi \cdot X_{n, 2}}
{\varphi_{2} \times \varepsilon_{2}}
& \cdots & (x + \varepsilon_{n - 1})^{n - 1}
& (x + \varepsilon_{n})^{n}\\
\hdotsfor{5}\\
\dfrac{\varphi \cdot X_{n, 1}}
{\varphi_{n} \times \varepsilon_{1}}
& \dfrac{\varphi \cdot X_{n, 2}}
{\varphi_{n} \times \varepsilon_{2}}
& \cdots & \dfrac{\varphi \cdot X_{n, n - 1}}
{\varphi_{n} \times \varepsilon_{n - 1}}
& \dfrac{\varphi\cdot X_{n, n}}
{\varphi_{n} \times \varepsilon_{n}}
\end{pmatrix}
+ \mathbf{I}_{n}
\]
This formula requires the amsmath and the amssymb packages. I’ll show in sec-
tion 9.1.2 how to write this formula so that it’s short and more readable.
1.4
Typing equations and aligned formulas
1.4.1
Equations
The equation environment creates a displayed math formula and automatically
generates a number. The equation
π
(1)
sin x dx = 2
0
is typed as
\begin{equation} \label{E:firstInt}
\int_{0}^{\pi} \sin x \, dx = 2
\end{equation}
Of course, the number generated depends on how many equations precede the
given one.

30
Chapter 1
Typing your ﬁrst article
To refer to this formula without having to remember a (changeable) num-
ber, assign a name to the equation in the argument of a \label command; I’ll
call the name of the equation a label. In this section, let’s call the ﬁrst equation
“firstInt” (ﬁrst integral). I use the convention that the label of an equation starts
with “E:”.
The number of this formula is referenced with the \ref command. For ex-
ample, to get the reference “see (1)”, type
see~(\ref{E:firstInt})
Alternatively, with the amsmath package, you can use the \eqref command. For
instance,
see~\eqref{E:firstInt}
also produces “see (1)”.
An advantage of this cross-referencing system is that if a new equation is intro-
duced, or the existing equations are rearranged, the numbering will automatically
be adjusted to reﬂect these changes.
Rule
Typeset twice
For renumbering to work, you have to typeset the source ﬁle twice.
See sections 6.3.2 and E.2.4. LATEX will send a warning if you forget.
At the end of the typesetting, LATEX stores the labels in the aux ﬁle (see sec-
tion 1.11.3). For every label, it stores the number the label is associated with and
also the page number on which the label occurs in the typeset version.
An equation will be numbered whether or not there is a \label command
attached to it. Of course, if there is no \label command, the number generated
by LATEX for the equation can’t be referenced automatically.
The system described here is called symbolic referencing. The argument of
\label is the “symbol” for the number, and \ref provides the referencing. LATEX
uses the same mechanism for all numberings it automatically generates: number-
ing of section titles, equations, theorems, lemmas, and bibliographic references—
except that for bibliographic references the commands are \bibitem and \cite,
respectively (see section 1.7.4).
With the amsmath package, equations can also be tagged by attaching a name
to the formula with the \tag command; the tag replaces the number.
Example:
π
(Int)
sin x dx = 2
0
is typed as

1.4
Typing equations and aligned formulas
31
\begin{equation}
\int_{0}^{\pi} \sin x \, dx = 2 \tag{Int}
\end{equation}
Tags (of the type discussed here) are absolute; this equation is always referred
to as (Int). Equation numbers, on the other hand, are relative; they change as
equations are added, deleted, or rearranged.
1.4.2
Aligned formulas
LATEX, with the help of the amsmath package, has many ways to typeset multiline
formulas. Right now, you’ll be introduced to three constructs: simple align, an-
notated align, and cases; see Chapter 5 for a discussion of many others.
The align math environment is used for simple and annotated align. Each
line in this environment is an equation, which LATEX automatically numbers.
Simple align
Simple align is used to align two or more formulas. To obtain the formulas
r2 = s2 + t2
(2)
2u + 1 = v + wα
(3)
y + z
(4)
x = √s + 2u
type (using \\ as a line separator)
\begin{align}
r^{2}
&= s^{2} + t^{2}
\label{E:eqn1}\\
2u + 1 &= v + w^{\alpha}
\label{E:eqn2}\\
x
&= \frac{y + z}{\sqrt{s + 2u}}
\label{E:eqn3}
\end{align}
(These equations are numbered (2), (3), and (4) because they are preceded by one
numbered equation earlier in this section.)
The align environment can also be used to break a long formula into two.
Since numbering both lines is undesirable, you may prevent the numbering of the
second line with the \notag command.
f (x) + g(x)
1 + f (x)g(x)
(5)
h(x) =
+ √
dx
1 + f 2(x)
1 − sin x
1 + f (x)
=
dx − 2 tan−1(x − 2)
1 + g(x)
This formula may be typed as

32
Chapter 1
Typing your ﬁrst article
\begin{align} \label{E:longInt}
h(x) &= \int
\left(
\frac{ f(x) + g(x) }
{ 1+ f^{2}(x) }
+ \frac{ 1+ f(x)g(x) }
{ \sqrt{1 - \sin x} }
\right) \, dx\\
&= \int \frac{ 1 + f(x) }
{ 1 + g(x) }
\, dx - 2 \tan^{-1}(x-2) \notag
\end{align}
See the split subsidiary math environment in section 5.5.2 for a better way to
split a long formula into (two or more) aligned parts, and on how to center the
formula number (5) between the two lines.
The rules are easy for simple align:
Rule
Simple align
Separate the lines with \\.
In each line, indicate the alignment point with &.
Place a \notag in each line that you do not wish numbered.
Place a \label in each numbered line you may want to reference with \ref or
\eqref.
Annotated align
Annotated align will align the formulas and the annotation (explanatory text) sep-
arately:
(6)
x = x ∧ (y ∨ z)
(by distributivity)
= (x ∧ y) ∨ (x ∧ z)
(by condition (M))
= y ∨ z.
This is typed as:
\begin{align} \label{E:DoAlign}
x &= x \wedge (y \vee z)
& &\text{(by distributivity)}\\
&= (x \wedge y) \vee (x \wedge z)
& &\text{(by condition (M))} \notag\\
&= y \vee z. \notag
\end{align}

1.5
The anatomy of an article
33
The rules for annotated align are similar to the rules of simple align. In each
line, in addition to the alignment point (marked by &), there is also a mark for the
start of the annotation: & &.
The align environment does much more than simple and annotated aligns
(see section 5.4).
Cases
The cases construct is a subsidiary math environment; it must be used in a dis-
played math environment or in an equation environment (see section 5.5). Here
is a typical example:



−x2,
if x < 0;
f (x) = α + x, if 0 ≤ x ≤ 1;

x2,
otherwise.
which may be typed as follows:
\[
f(x)=
\begin{cases}
-x^{2},
&\text{if $x < 0$;}\\
\alpha + x,
&\text{if $0 \leq x \leq 1$;}\\
x^{2},
&\text{otherwise.}
\end{cases}
\]
The rules for cases are simple:
Rule
cases
Separate the lines with \\.
In each line, indicate the alignment point for the annotation with &.
1.5
The anatomy of an article
The sampart.tex sample article (typeset on pages 361–363) uses the AMS article
document class, amsart. In this introductory chapter, I want to start off with the
popular article document class of LATEX, which is easier to use. So we’ll use a
simpliﬁed and shortened sample article, intrart.tex (in the ftp directory). Type
it in as we discuss the parts of an article.
The preamble of an article is the initial part of the source ﬁle up to the line
\begin{document}

34
Chapter 1
Typing your ﬁrst article
See Figure 1.1. The preamble contains instructions for the entire article, for in-
stance, the
\documentclass
command.
Here is the preamble of the introductory sample article:
% Introductory sample article: intrart.tex
% Typeset with LaTeX format
\documentclass{article}
\usepackage{amssymb,amsmath}
\newtheorem{theorem}{Theorem}
\newtheorem{definition}{Definition}
\newtheorem{notation}{Notation}
The preamble names the document class, article, and then names the LATEX en-
hancements, or packages, used by the article. This article loads two packages:
The amssymb package provides the names of all the math symbols in Appen-
dix A and the amsmath package provides many of the math constructs used.
\documentclass{...}
\usepackage{...}
preamble
...
\begin{document}
\title{...}
\author{...}
top matter
\date{...}
\maketitle
\begin{abstract}
...
abstract
\end{abstract}
body
\section{...}
\section{...}
\begin{thebibliography}{9}
...
bibliography
\end{thebibliography}
\end{document}
Figure 1.1: A schematic view of an article

1.5
The anatomy of an article
35
A proclamation is a theorem, deﬁnition, corollary, note, and so on. In the
preamble, three proclamations are deﬁned. For instance,
\newtheorem{theorem}{Theorem}
deﬁnes the theorem environment, which you can use in the body of your article
(see section 1.7.3). LATEX will automatically number and visually format the the-
orems.
The article proper, called the body of the article, is contained in the document
environment, that is, between the lines
\begin{document}
and
\end{document}
as illustrated in Figure 1.1. The body of the article is also logically split up into
several parts; we’ll discuss these in detail in section 6.1.
The body of the article starts with the top matter, which contains the “title
page” information. It follows the line:
\begin{document}
and concludes with the line
\maketitle
Here is the top matter of the introductory sample article:
\title{A construction of complete-simple\\
distributive lattices}
\author{George~A. Menuhin\thanks{Research supported
by the NSF under grant number~23466.}\\
Computer Science Department\\
University of Winnebago\\
Winnebago, Minnesota 23714\\
\texttt{menuhin@ccw.uwinnebago.edu}}
\date{March 15, 1995}
\maketitle
The body continues with an (optional) abstract, contained in an abstract
environment:
\begin{abstract}
In this note we prove that there exist \emph{complete-simple
distributive lattices}, that is, complete distributive
lattices in which there are only two complete congruences.
\end{abstract}

36
Chapter 1
Typing your ﬁrst article
And here is the rest of the body of the introductory sample article:
\section{Introduction} \label{S:intro}
In this note we prove the following result:
\begin{theorem}
There exists an infinite complete distributive lattice $K$
with only the two trivial complete congruence relations.
\end{theorem}
\section{The $\Pi^{*}$ construction} \label{S:P*}
The following construction is crucial in our proof of our Theorem:
\begin{definition} \label{D:P*}
Let $D_{i}$, $i \in I$, be complete distributive
lattices satisfying condition~\textup{(J)}.
Their
$\Pi^{*}$ product is defined as follows:
\[
\Pi^{*} ( D_{i} \mid i \in I ) =
\Pi ( D_{i}^{-} \mid i \in I ) + 1;
\]
that is, $\Pi^{*} ( D_{i} \mid i \in I )$ is
$\Pi ( D_{i}^{-} \mid i \in I )$ with a new unit element.
\end{definition}
\begin{notation}
If $i \in I$ and $d \in D_{i}^{-}$, then
\[
\langle \dots, 0, \dots, \overset{i}{d}, \dots, 0,
\dots \rangle
\]
is the element of $\Pi^{*} ( D_{i} \mid i \in I )$ whose
$i$th component is $d$ and all the other components
are $0$.
\end{notation}
See also Ernest~T. Moynahan~\cite{eM57a}.
Next we verify the following result:
\begin{theorem} \label{T:P*}
Let $D_{i}$, $i \in I$, be complete distributive
lattices satisfying condition~\textup{(J)}.
Let $\Theta$
be a complete congruence relation on

1.5
The anatomy of an article
37
$\Pi^{*} ( D_{i} \mid i \in I )$.
If there exists an $i \in I$ and a $d \in D_{i}$ with
$d < 1_{i}$ such that for all $d \leq c < 1_{i}$,
\begin{equation} \label{E:cong1}
\langle \dots, 0, \dots,\overset{i}{d},
\dots, 0, \dots \rangle \equiv \langle \dots, 0, \dots,
\overset{i}{c}, \dots, 0, \dots \rangle \pmod{\Theta},
\end{equation}
then $\Theta = \iota$.
\end{theorem}
\emph{Proof.} Since
\begin{equation} \label{E:cong2}
\langle \dots, 0, \dots, \overset{i}{d}, \dots, 0,
\dots \rangle \equiv \langle \dots, 0, \dots,
\overset{i}{c}, \dots, 0, \dots \rangle \pmod{\Theta},
\end{equation}
and $\Theta$ is a complete congruence relation, it follows
from condition~(C) that
\begin{align} \label{E:cong}
& \langle \dots, \overset{i}{d}, \dots, 0,
\dots \rangle \equiv\\
&\qquad \qquad \quad \bigvee ( \langle \dots, 0, \dots,
\overset{i}{c}, \dots, 0, \dots \rangle \mid d \leq c < 1 )
\equiv 1 \pmod{\Theta}. \notag
\end{align}
Let $j \in I$, $j \neq i$, and let $a \in D_{j}^{-}$.
Meeting both sides of the congruence \eqref{E:cong2} with
$\langle \dots, 0, \dots, \overset{j}{a}, \dots, 0,
\dots \rangle$, we obtain
\begin{align} \label{E:comp}
0 = & \langle \dots, 0, \dots, \overset{i}{d}, \dots, 0, \dots
\rangle \wedge \langle \dots, 0, \dots, \overset{j}{a},
\dots, 0, \dots \rangle \equiv\\
&\langle \dots, 0, \dots, \overset{j}{a}, \dots, 0, \dots
\rangle \pmod{\Theta}, \notag
\end{align}
Using the completeness of $\Theta$ and \eqref{E:comp},
we get:
\[

38
Chapter 1
Typing your ﬁrst article
0 \equiv \bigvee ( \langle \dots, 0, \dots, \overset{j}{a},
\dots, 0,\dots\rangle \mid a \in D_{j}^{-} ) = 1 \pmod{\Theta},
\]
hence $\Theta = \iota$.
\begin{thebibliography}{9}
\bibitem{sF90}
Soo-Key Foo, \emph{Lattice constructions}, Ph.D. thesis,
University of Winnebago, Winnebago MN, December 1990.
\bibitem{gM68}
George~A. Menuhin, \emph{Universal Algebra}, D.~van Nostrand,
Princeton-Toronto-London-Mel\-bourne, 1968.
\bibitem{eM57}
Ernest~T. Moynahan, \emph{On a problem of M.~H. Stone}, Acta
Math. Acad. Sci. Hungar. \textbf{8} (1957), 455--460.
\bibitem{eM57a}
Ernest~T. Moynahan, \emph{Ideals and congruence relations in
lattices. II}, Magyar Tud. Akad. Mat. Fiz. Oszt. K\"{o}zl.
\textbf{9} (1957), 417--434.
\end{thebibliography}
At the end of the body, the bibliography is typed between the lines
\begin{thebibliography}{9}
\end{thebibliography}
The argument “9” of the thebibliography environment tells LATEX to make room
for single digit numbering, since in this article there are fewer than 10 articles. In
the typeset article, the bibliography is entitled “References”.
Observe that we refer to condition (J) in the deﬁnition as \textup{(J)}. We
do this so that if the text of the deﬁnition is emphasized (as it is), then (J) should
still be typeset as (J) and not as (J); see section 2.6.4 for the \textup command.
1.5.1
The typeset article
Here is the typeset introductory sample article (note that the equation numbers are
on the right, the default in the article document class; elsewhere in this book you
ﬁnd the AMS default, equations on the left—see sections 7.1.2 and 8.4 on how to
change the default).

1.5
The anatomy of an article
39
A construction of complete-simple
distributive lattices
George A. Menuhin∗
Computer Science Department
Winnebago, Minnesota 23714
menuhin@ccw.uwinnebago.edu
March 15, 1995
Abstract
In this note we prove that there exist complete-simple distributive lat-
tices, that is, complete distributive lattices in which there are only two
complete congruences.
1
Introduction
In this note we prove the following result:
Theorem 1 There exists an inﬁnite complete distributive lattice K with only
the two trivial complete congruence relations.
2
The Π∗ construction
The following construction is crucial in our proof of our Theorem:
Deﬁnition 1 Let Di, i ∈ I, be complete distributive lattices satisfying condi-
tion (J). Their Π∗ product is deﬁned as follows:
Π∗(Di | i ∈ I) = Π(D− | i ∈ I) + 1;
i
that is, Π∗(Di | i ∈ I) is Π(D− | i ∈ I) with a new unit element.
i
Notation 1 If i ∈ I and d ∈ D−, then
i
i
. . . , 0, . . . , d, . . . , 0, . . .
is the element of Π∗(Di | i ∈ I) whose ith component is d and all the other
components are 0.
∗Research supported by the NSF under grant number 23466.
1

40
Chapter 1
Typing your ﬁrst article
See also Ernest T. Moynahan [4].
Next we verify the following result:
Theorem 2 Let Di, i ∈ I, be complete distributive lattices satisfying condi-
tion (J). Let Θ be a complete congruence relation on Π∗(Di | i ∈ I). If there
exists an i ∈ I and a d ∈ Di with d < 1i such that for all d ≤ c < 1i,
i
i
. . . , 0, . . . , d, . . . , 0, . . . ≡ . . . , 0, . . . , c, . . . , 0, . . .
(mod Θ),
(1)
then Θ = ι.
Proof. Since
i
i
. . . , 0, . . . , d, . . . , 0, . . . ≡ . . . , 0, . . . , c, . . . , 0, . . .
(mod Θ),
(2)
and Θ is a complete congruence relation, it follows from condition (C) that
i
. . . , d, . . . , 0, . . . ≡
(3)
i
( . . . , 0, . . . , c, . . . , 0, . . . | d ≤ c < 1) ≡ 1 (mod Θ).
Let j ∈ I, j = i, and let a ∈ D−. Meeting both sides of the congruence (2)
j
j
with . . . , 0, . . . , a, . . . , 0, . . . , we obtain
i
j
0 = . . . , 0, . . . , d, . . . , 0, . . . ∧ . . . , 0, . . . , a, . . . , 0, . . . ≡
(4)
j
. . . , 0, . . . , a, . . . , 0, . . .
(mod Θ),
Using the completeness of Θ and (4), we get:
j
0 ≡
( . . . , 0, . . . , a, . . . , 0, . . . | a ∈ D−) = 1 (mod Θ),
j
hence Θ = ι.
References
[1] Soo-Key Foo, Lattice Constructions, Ph.D. thesis, University of Winnebago,
Winnebago, MN, December 1990.
[2] George A. Menuhin, Universal Algebra, D. van Nostrand, Princeton-
Toronto-London-Melbourne, 1968.
[3] Ernest T. Moynahan, On a problem of M. H. Stone, Acta Math. Acad. Sci.
Hungar. 8 (1957), 455–460.
[4] Ernest T. Moynahan, Ideals and congruence relations in lattices. II, Magyar
Tud. Akad. Mat. Fiz. Oszt. K¨
ozl. 9 (1957), 417–434.
2

1.6
Article templates
41
1.6
Article templates
Before you start writing your ﬁrst article, I suggest you create two article templates
for your own use.
There are two templates for articles written in the article document class in
this book: article.tpl for articles with one author and article2.tpl for articles
with two authors.2 You can ﬁnd these in the ftp directory (see page 4). So copy
article.tpl into the work directory or type it in as follows:
% Sample file: article.tpl
% Typeset with LaTeX format
\documentclass{article}
\usepackage{amsmath,amssymb}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{corollary}{Corollary}
\newtheorem{notation}{Notation}
\begin{document}
\title{%
titleline1\\
titleline2}
\author{name\thanks{support}\\
addressline1\\
addressline2\\
addressline3}
\date{date}
\maketitle
\begin{abstract}
abstract
\end{abstract}
\begin{thebibliography}{99}
\end{thebibliography}
\end{document}
2In section 8.3, we discuss a template ﬁle, amsart.tpl, for the AMS document class amsart.

42
Chapter 1
Typing your ﬁrst article
Now copy article2.tpl into the work directory, or type it in. It is identical
to article.tpl except for the argument of the \author command:
\author{name1\thanks{support1}\\
address1line1\\
address1line2\\
address1line3
\and
name2\thanks{support2}\\
address2line1\\
address2line2\\
address2line3}
Note the \and command; it separates the two authors.
Now let’s customize the template ﬁles. Open article.tpl and save it un-
der a name of your choosing; I saved it under the name ggart.tpl (in the ftp
directory—see page 4). In this personalized template ﬁle, I edit the top matter:
\title{titleline1\\
titleline2}
\author{G. Gr\"{a}tzer\thanks{Research supported by the
NSERC of Canada.}\\
University of Manitoba\\
Department of Mathematics\\
Winnipeg, Man. R3T 2N2\\
Canada}
\date{date}
I did not edit the \title lines because they change from article to article. There
is also a personalized ggart2.tpl for two authors.
1.7
Your ﬁrst article
Your ﬁrst article will be typeset using the article document class. To start, open
the personalized article template created in section 1.6, and save it under the name
of your ﬁrst article. The name must be one word (no spaces) ending with .tex.
1.7.1
Editing the top matter
Edit the top matter to contain the article information (title, date, and so on). Here
are some simple rules to follow:

1.7
Your ﬁrst article
43
Rule
Top matter for the article document class
1. If the title is only one line long, then there is no \\ in the argument of the
\title command; otherwise, separate the lines of the title with \\. There is
no \\ at the end of the last line.
2. Separate the lines of the address with \\. There is no \\ at the end of the last
line.
3. \thanks places a footnote at the bottom of the ﬁrst page. If it is not needed,
delete it.
4. Multiple authors are separated by \and. There is only one \author command,
and it contains all the information (name, address, support) about all the au-
thors.
5. The \title command is the only compulsory command. The others are op-
tional.
6. If there is no \date command, the current date will be shown. If you do not
want a date, type the form \date{}; if you want a speciﬁc date, say February
21, 1995, write
\date{February 21, 1995}
1.7.2
Sectioning
An article, as a rule, is divided into sections. To start the section entitled “Intro-
duction”, type
\section{Introduction} \label{S:intro}
Introduction is the title of the section, S:intro is the label. I use the convention
that “S:” starts the label for a section. The number of the section is automatically
assigned by LATEX, and you can refer to this section number by \ref{S:intro},
as in
In section~\ref{S:intro}, we introduce ...
(the tilde ˜ is an unbreakable space, it keeps the word “section” and the section
number together—see section 2.4.3).
For instance, the section title of this section was typed as follows:
\section{Typing your first article} \label{S:FirstArticle}
A reference to this section is made by typing

44
Chapter 1
Typing your ﬁrst article
\ref{S:FirstArticle}
Sections have subsections, and subsections have subsubsections, followed by
paragraphs and subparagraphs. The corresponding commands are
\subsection
\subsubsection
\paragraph
\subparagraph
1.7.3
Invoking proclamations
In the preamble of article.tpl, you typed the theorem, lemma, proposition, def-
inition, corollary, and notation proclamations. Each of these proclamations deﬁnes
an environment. For example, type a theorem in a theorem environment; the body
of the theorem (that is, the part of the source ﬁle that produces the theorem) is be-
tween the two lines:
\begin{theorem} \label{T:xxx}
and
\end{theorem}
where T:xxx is the label for the theorem. Of course, xxx should be somewhat
descriptive of the contents of the theorem. The theorem number is automatically
assigned by LATEX, and it can be referenced by \ref{T:xxx} as in
it follows from Theorem~\ref{T:xxx}
(the tilde ˜ keeps the word “Theorem” and the theorem number together—see
section 2.4.3). I use the convention that the label for a theorem starts with “T:”.
1.7.4
Inserting references
Finally, we discuss the bibliography. Below are typical entries for the most often
used types of references: an article in a journal, a book, an article in a conference
proceedings, an article (chapter) in a book, a Ph.D. thesis, and a technical report
(see inbibl.tpl in the ftp directory).
\bibitem{eM57}
Ernest~T. Moynahan, \emph{On a problem of M.~H. Stone},
Acta Math. Acad. Sci. Hungar. \textbf{8} (1957), 455--460.
\bibitem{gM68}
George~A. Menuhin, \emph{Universal Algebra}, D.~van Nostrand,
Princeton-Toronto-London-Melbourne, 1968.
\bibitem{pK69}

1.7
Your ﬁrst article
45
Peter~A. Konig, \emph{Composition of functions}, Proceedings
of the Conference on Universal Algebra (Kingston, 1969).
\bibitem{hA70}
Henry~H. Albert, \emph{Free torsoids}, Current Trends in
Lattice Theory, D.~van Nostrand, 1970.
\bibitem{sF90}
Soo-Key Foo, \emph{Lattice constructions}, Ph.D. thesis,
University of Winnebago, 1990.
\bibitem{gF86}
Grant~H. Foster, \emph{Computational complexity in lattice
theory}, Tech. report, Carnegie Mellon University, 1986.
I use the convention that the label for the \bibitem consists of the initials of the au-
thor and the year of publication: a publication by Andrew B. Reich in 1987 would
have the label aR87 (the second publication would be aR87a). For joint publica-
tions, the label consists of the initials of the authors and the year of publication;
for instance, a publication by John Bradford and Andrew B. Reich in 1987 would
have the label BR87. Of course, you can use any label you choose (subject to the
rule in section 6.4.2).
Suppose you want to include as the ﬁfth item in the bibliography the following
article:
John Bradford and Andrew B. Reich, Duplexes in posets, Proc. Amer. Math.
Soc. 112 (1987), 115–125.
Modeling it after Moynahan’s article, type it as:
\bibitem{BR87}
John~Bradford and Andrew~B. Reich, \emph{Duplexes in posets},
Proc. Amer. Math. Soc. \textbf{112} (1987), 115--125.
A reference to this article is made with \cite{BR87}, for instance:
this result was ﬁrst published in [5]
typed as
this result was first published in~\cite{BR87}
Note that you have to arrange the references in the thebibliography envi-
ronment in the order you wish to see them. LATEX only takes care of the numbering
and the citations in the text.

46
Chapter 1
Typing your ﬁrst article
Tip
The thebibliography environment properly handles periods. You do not have
to mark periods for abbreviations (in the form .\ —as discussed in section 2.2.2)
in the name of a journal, so
Acta Math. Acad. Sci. Hungar.
is correct.
1.8
LATEX error messages
You’ll probably make a number of mistakes in your ﬁrst article. The mistakes come
in various forms:
Typographical errors, which LATEX will blindly typeset. View the typeset version,
ﬁnd the errors, and correct the source ﬁle.
Errors in mathematical formulas or in the formatting of the text.
Errors in your instructions—commands and environments—to LATEX.
Let’s look at some examples by introducing a number of errors in the source
ﬁle of the intrart.tex introductory sample article and see what error messages
occur.
Example 1
Go to line 21 (you do not have to count lines, since most editors
have a “go to line” command) and remove the closing brace so it reads:
\begin{abstract
Upon typesetting intrart.tex, LATEX informs you of a mistake:
Runaway argument?
{abstract
! Paragraph ended before \end was complete.
<to be read again>
\par
l.26
Line 26 of the ﬁle is the line after \end{abstract}. From the error message, you
can tell that something is wrong with the abstract environment.
Example 2
Now correct line 21, go to line 25, change it from
\end{abstract}
to
\end{abstrac}
and typeset again. LATEX will inform you:

1.8
LATEX error messages
47
! LaTeX Error: \begin{abstract} on input line 21
ended by \end{abstrac}.
l.25 \end{abstrac}
Pressing return, LATEX will recover from this error.
Example 3
Instead of correcting the error in line 25, comment it out:
% \end{abstrac}
and introduce an additional error in line 67. This line presently reads:
lattices satisfying condition~\textup{(J)}.
Let $\Theta$
Change \Theta to \Teta:
lattices satisfying condition~\textup{(J)}.
Let $\Teta$
Typesetting the article now, the message is:
! Undefined control sequence.
l.67 ...xtup{(J)}.
Let $\Teta
$
and pressing return gives the message:
! LaTeX Error: \begin{abstract} on input line 21 ended
by \end{document}.
l.131 \end{document}
These two mistakes are easy to identify. \Teta is a typo for \Theta. Observe
how LATEX tries to match
\begin{abstract}
with
\end{document}
Undo the two changes (lines 25 and 67).
Example 4
In line 73, change
\langle \dots, 0, \dots,\overset{i}{d},
to
\langle \dots, 0, \dots,\overset{i}{d,

48
Chapter 1
Typing your ﬁrst article
This results in the message:
Runaway argument?
\def \\{\@amsmath@err {\Invalid@@ \\}\@eha } \label {E\ETC.
! Paragraph ended before \equation was complete.
<to be read again>
\par
l.79
Line 79 is the blank line following \end{theorem}. LATEX skipped over the de-
fective construct \overset and the incomplete equation environment, indicat-
ing the error past the end of the theorem environment. The error message indi-
cates that the error may have been caused by the new paragraph (\par). Of course,
there can be no new paragraph in either the second argument of \overset or the
displayed formula. The solution does not come easily except by isolating the last
paragraph and investigating it.
Error messages from LATEX are not always as helpful as one would like, but
there is always some information to be gleaned from them. As a rule, the error
message should at least inform you of the line number (or paragraph or formula)
where the error was caught. Try to identify the structure that caused the error:
a command, an environment, or so forth. Keep in mind that it could be quite far
from the line where LATEX indicated the error. Try reading the section of this book
that describes that command or environment; it should help in correcting the error.
The next best defense is to isolate your problem. Create a current.tex ﬁle
that is the same as the present article, except that there is only one paragraph in the
document environment. When this paragraph is typeset correctly, cut and paste it
into your source ﬁle. If there is only one paragraph in the document, the error is
easier to ﬁnd. If the error is of the type as in the last example, split the paragraph
into smaller paragraphs. See also section 2.5 on how to use the comment environ-
ment for ﬁnding errors.
1.9
Logical and visual design
This book attempts to show how to typeset an article, not how to write it. Never-
theless, it seems appropriate to point out some approaches to article design.
The typeset version of our intrart.tex introductory sample article (pp. 39–
40) looks impressive. (For another example of a typeset article, see sampart.tex
on pp. 361–363.) To produce an article like this, you have to realize that there are
two aspects of article design: the visual and the logical. Let’s borrow an example
from the sample article to illustrate this: a theorem. You tell LATEX to typeset a
theorem and number it. Here is how you type the theorem:
\begin{theorem} \label{T:P*}

1.9
Logical and visual design
49
Let $D_{i}$, $i \in I$, be complete distributive
lattices satisfying condition~\textup{(2)}.
Let $\Theta$
be a complete congruence relation on
$\Pi^{*} ( D_{i} \mid i \in I )$.
If there exists an $i \in I$ and a $d \in D_{i}$ with
$d < 1_{i}$ such that for all $d \leq c < 1_{i}$,
\begin{equation} \label{E:cong1}
\langle \dots, 0, \dots,\overset{i}{d},
\dots, 0, \dots \rangle \equiv \langle \dots, 0, \dots,
\overset{i}{c}, \dots, 0, \dots \rangle \pmod{\Theta},
\end{equation}
then $\Theta = \iota$.
\end{theorem}
You ﬁnd the typeset form on page 40.
The logical design is the theorem itself, which is placed in the theorem envi-
ronment. For the visual design, LATEX makes literally hundreds of decisions: the
vertical space before and after the theorem; the bold Theorem heading and its
numbering; the vertical space before and after the equation, and its numbering;
the spacing of all the math symbols (inline and displayed formulas are spaced dif-
ferently); the text of the theorem to be emphasized; and so on.
The decisions were made by professional designers, whose expertise is hidden
in TEX itself, in LATEX, in the document class, and in the packages. Could you
have typeset this theorem yourself? Probably not. Aesthetic decisions are difﬁcult
for lay people to make. But even if you could have guessed the correct spacing, you
would have faced the problem of consistency (guaranteeing that the next theorem
will look the same), and just as importantly, you would have spent a great deal
of time and energy on the visual design of the theorem, as opposed to the logical
design. The idea is to concentrate on the logical design and let LATEX take care of
the visual design.
This approach has the advantage that by changing the document class (or its
options; see sections 7.1.2 and 8.4), the visual design can be changed. If you code
the visual design into the article (“hard coding” it, as a programmer would say),
it’s very difﬁcult to change.
LATEX uses four major tools to separate the logical and visual designs of an
article:
Commands Information is given to LATEX as arguments of commands; it’s up to
LATEX to process the information. For instance, the title page information
(especially in the amsart document class) is given in this form; the organiza-
tion of the title page is completely up to the document class and its options.
A more subtle example is the use of a command for distinguishing a term or

50
Chapter 1
Typing your ﬁrst article
notation. For instance, you may want to use an \env command for environ-
ment names. You may deﬁne \env as follows (\newcommand is explained in
section 9.1.1):
\newcommand{\env}[1]{\texttt{#1}}
which typesets all environment names in typewriter style (see section 2.6.2).
Logically, you have decided that an environment name should be marked up.
Visually, you may change the decision any time. By changing the deﬁnition
to
\newcommand{\env}[1]{\textbf{#1}}
all environment names will be typeset in bold (see section 2.6.7).
The following more mathematical example is taken from sampart2.tex (see
Appendix D and the ftp directory). This article deﬁnes the construct D 2
with the command
\newcommand{\Ds}{D^{\langle 2 \rangle}}
If a referee (or coauthor) suggests a different notation, changing this one line
will carry out the change throughout the whole article.
Environments Important logical structures are placed in environments. For in-
stance, you can give a list as an environment by saying that this is a list and
these are the items (see section 3.1). Again, exactly how the list is typeset
is up to LATEX; you can even switch from one list type to another by just
changing the name of the environment.
Proclamations These deﬁne numbered environments. If the amsthm package is
used, you can further specify which one of three styles to use for typeset-
ting; at any time you can change the style or the numbering scheme in the
preamble (see the typeset sampart.tex on pages 361–363 for examples of
proclamations printed in the three styles).
Cross-referencing Since theorems and sections are logical units, they can be freely
moved around. This gives tremendous freedom in reorganizing the source
ﬁle to improve the logical design.
You write articles to communicate. The closer you get to a separation of logi-
cal and visual design, the more you are able to concentrate on communicating your
ideas. Of course, you can never quite reach this ideal. For instance, a “line too
wide” warning (see sections 1.1.3 and 2.7.1) is a problem of visual design. When
the journal changes the document class, unless the new document class retains the
same fonts and line width, new “line too wide” problems arise. However, LATEX
is successful well over 95% of the time in solving visual design problems without
your intervention. This is getting fairly close to the ideal.

1.10
A brief overview
51
package
package
package
package
package
document class
LATEX
AMS
T
CM
EX
Fonts
Fonts
Figure 1.2: The structure of LATEX
1.10
A brief overview
Having ﬁnished the short course, maybe it’s time to pause and get a brief overview
of how LATEX works. As I pointed out in the Introduction, at the core of LATEX
is a programming language called TEX, providing many typesetting instructions.
Along with TEX comes a set of fonts called Computer Modern (CM). The CM fonts
and the TEX programming language form the foundation of a typical TEX system.
TEX is expandable, that is, additional commands can be deﬁned in terms of
more basic ones. One of the best known expansions of TEX is LATEX; it introduces
the idea of a logical unit that you read about in section 1.9.
Visual layout in LATEX is determined by the document class; for example, you
now have some familiarity with the article document class. Expansions of LATEX
are called packages; you have already come across the amssymb and amsmath pack-
ages.
The structure of LATEX is illustrated in Figure 1.2. This ﬁgure suggests that,
in order to work with a LATEX document, you ﬁrst have to install TEX and the
CM fonts, then LATEX, and ﬁnally specify the document class and the necessary
packages. The AMSFonts font set is useful but it’s not absolutely necessary.

52
Chapter 1
Typing your ﬁrst article
Figure 1.2 illustrates my view of TEX and LATEX: it is the foundation on which
many useful packages—extensions of LATEX—are built. It is essential that you un-
derstand the packages that make your work easier. An important example of this is
the central focus of this book: typesetting math in LATEX. When typesetting math,
invoke the amsmath package. In Part I, you invoke the amsmath package directly; in
later parts of this book, I point out when a described feature needs the amsmath (or
some other) package. The AMS document classes automatically load the amsmath
and amsfonts packages.
1.11
Using LATEX
Figure 1.3 illustrates the steps taken to produce a typeset document. As illustrated
in Figure 1.3, you open the source ﬁle or create a new one using an editor; call the
source ﬁle myart.tex. Once the document is ready, typeset it with TEX using the
LaTeX format. This step produces three ﬁles:
myart.dvi, the typeset article in machine readable format;
myart.aux, the auxiliary ﬁle; it is used by LATEX for internal “book keeping”,
including cross-referencing;
myart.log, the log ﬁle; LATEX records the typesetting session in the log ﬁle, in-
cluding the warnings and the errors.
Use a video driver to display the typeset article, myart.dvi, on the monitor, and
a printer driver to print the typeset article, myart.dvi on a printer.
It should be emphasized that of the four programs used, only one (TEX) is the
same for all computers and all implementations. If you use TEX in an “integrated
environment”, then all four programs appear as one.
1.11.1
AMS-LATEX revisited
Now that you understand the structure of LATEX, we can again discuss AMS-LATEX,
a set of enhancements to LATEX by the AMS. As outlined in the Introduction,
the AMS enhancements to LATEX fall into three groups: the AMS math enhance-
ments, the document classes, and the AMSFonts. Each consists of several packages.
An AMS document class automatically invokes the following AMS packages
(see section 8.5 for a more detailed discussion and for the package interdependency
diagram, Figure 8.3):
amsmath, the main AMS math package;
amsthm, proclamations with style and the proof environment;
amsopn, operator names;
amstext, the \text command;
amsfonts, commands for math alphabets;
amsbsy, bold symbol commands.

1.11
Using LATEX
53
printed article
create
editor
print
printer driver
edit
TEX
myart.tex
myart.dvi
the source file
the typeset version
typeset
LaTeX format
view
video driver
myart.aux
myart.log
the aux file
the log file
view the typeset
version on screen
Figure 1.3: Using LATEX

54
Chapter 1
Typing your ﬁrst article
They do not automatically input the amssymb package, which provides the
math symbol names. You can additionally input this and other AMS-LATEX or
LATEX packages as needed.
When we discuss a feature of LATEX that requires a package, I point this out in
the text. I do not always point out, however, the interdependencies of the docu-
ment classes and of the packages. For instance, the \text command (section 2.9)
is provided by the amstext package, which is loaded automatically by the amsmath
package, which in turn is loaded automatically by each of the AMS-LATEX docu-
ment classes. These interdependencies are discussed in section 8.5.
1.11.2
Interactive LATEX
As a rule, LATEX typesets an article non-interactively. Occasionally, you may want
to use LATEX interactively, that is, give LATEX an instruction and ask it to carry it
out. If LATEX can’t carry out your instructions, it displays a prompt:
The ** prompt means that LATEX wants to know the name of a source ﬁle to
typeset. Probably, you misspelled a name, or you are in the wrong directory.
The ? prompt asks “What should I do about the error I found?” Press return
to continue; most of the time LATEX recovers from the error, and completes the
typesetting. If LATEX can’t recover from the error at the ? prompt, press X to
exit. Typing H instead may yield useful advice.
The * signiﬁes interactive mode: LATEX is waiting for an instruction. To get to
such a prompt, comment out the line
\end{document}
(by inserting % as the ﬁrst character of the line) in the source ﬁle and typeset.
Interactive instructions (such as \show—see section 9.1.6) may be given at the
* prompt. Typing
\end{document}
at the * prompt exits LATEX.
1.11.3
Files
A number of ﬁles are created when a document called, say, myart.tex is typeset.
When the typesetting takes place, a number of messages appear on the monitor.
These are stored in the log ﬁle, myart.log. The typeset document is written in
the myart.dvi ﬁle. LATEX also writes one or more auxiliary ﬁles, as necessary. The
most important one is myart.aux, the aux ﬁle (see section E.2.4).

1.11
Using LATEX
55
1.11.4
Versions
All components of LATEX interact. Since all of them have many versions, make
sure they are up-to-date and compatible. While writing this book, I used LATEX 2ε
(LATEX version 2e), issued on December 1, 1994. You can check the version num-
bers and dates by reading the ﬁrst few lines of the ﬁles in an editor or by checking
the dates shown on the ﬁle list discussed below.
LATEX is updated every six months; in-between updates, the ltpatch.ltx
document is posted periodically on the CTAN (see Appendix G). Get this ﬁle and
place it in your TEX input directory. When you rebuild your formats, ltpatch.ltx
will patch LATEX.
When you typeset a LATEX document, LATEX introduces itself in the log ﬁle
with a line such as
LaTeX2e <1994/12/01> patch level 3
giving you the release date and patch level. If you use a new feature of LATEX that
was introduced recently, place in the preamble of your document the command
\NeedsTeXFormat{LaTeX2e}[1994/12/01]
where the date is the release date of the version you must use.
As of this writing, AMS-LATEX is at version 1.2 and the AMSFonts font set
is at version 2.2. See Appendix G on how to get updated versions of AMS-LATEX
and the AMSFonts.
BIBTEX is at version 0.99 (version 1.0 is expected soon). In this book, I use
the amsplain.bst bibliographic style ﬁle (version 1.2a).
If you include the \listfiles command in the preamble of your document,
the log ﬁle will contain a detailed listing of all the ﬁles used in the typesetting of
your document.
Here are a few lines from such a listing:
*File List*
book.cls
1994/12/09 v1.2x Standard LaTeX document class
leqno.clo
1994/12/09 v1.2x Standard LaTeX option
bk10.clo
1994/12/09 v1.2x Standard LaTeX file (size option)
amsmath.sty
1995/02/23 v1.2b AMS math features
Ueus57.fd
1994/10/17 v2.2d AMS font definitions
latexsym.sty
1994/09/25 v2.1f Standard LaTeX package
xspace.sty
1994/11/15 v1.03 Space after command names (DPC)
Ulasy.fd
1994/09/25 v2.1f LaTeX symbol font definitions
***********

56
Chapter 1
Typing your ﬁrst article
1.12
What’s next?
Having read thus far, you probably know enough about LATEX to write your ﬁrst
article. The best way to learn LATEX is by experimentation. Later, you may want
to read Parts II–V.
If you look at the source ﬁles of the sample articles, your ﬁrst impression may
be how very verbose LATEX is. In actual practice, LATEX is fairly easy to type. There
are two basic tools to make typing LATEX more efﬁcient.
Firstly, you should have a good editor. For instance, you should be able to
train your editor so that a single keystroke produces the text:
\begin{theorem} \label{T:}
\end{theorem}
with the cursor in the position following “:” (where you type the label).
Secondly, customizing LATEX will make repetitious structures such as
\begin{equation}
\langle \dots, 0, \dots, \overset{i}{d}, \dots, 0,
\dots \rangle \equiv \langle
\dots, 0, \dots, \overset{i}{c}, \dots, 0, \dots
\rangle \pmod{\Theta},
\end{equation}
which prints
i
i
(3.1)
. . . , 0, . . . , d, . . . , 0, . . . ≡ . . . , 0, . . . , c, . . . , 0, . . .
(mod Θ),
(see page 369) become much shorter and (with practice) more readable. Utiliz-
ing the user-deﬁned commands \con (for congruence), \vct (for vector), and \gQ
(for Greek theta), in sampart2.tex (in the ftp directory and in Appendix C), this
formula becomes
\begin{equation}
\con \vct{i}{d}=\vct{i}{c}(\gQ),
\end{equation}
which is about as long as the typeset formula itself.
The topic of user-deﬁned commands is taken up in Part IV.
Finally, custom formats (section 9.7) substantially speed up the typesetting of
an average document.

APPENDIX
A
Math symbol tables
A.1
Hebrew letters
Type:
Print:
Type:
Print:
\aleph
ℵ
\beth
\daleth
\gimel
All symbols but \aleph need the amssymb package.
345

346
Appendix A
A.2
Greek characters
Type:
Print:
Type:
Print:
Type:
Print:
\alpha
α
\beta
β
\gamma
γ
\digamma
\delta
δ
\epsilon
\varepsilon
ε
\zeta
ζ
\eta
η
\theta
θ
\vartheta
ϑ
\iota
ι
\kappa
κ
\varkappa
κ
\lambda
λ
\mu
µ
\nu
ν
\xi
ξ
\pi
π
\varpi
\rho
ρ
\varrho
\sigma
σ
\varsigma
ς
\tau
τ
\upsilon
υ
\phi
φ
\varphi
ϕ
\chi
χ
\psi
ψ
\omega
ω
\digamma and \varkappa require the amssymb package.
Type:
Print:
Type:
Print:
\Gamma
Γ
\varGamma
Γ
\Delta
∆
\varDelta
∆
\Theta
Θ
\varTheta
Θ
\Lambda
Λ
\varLambda
Λ
\Xi
Ξ
\varXi
Ξ
\Pi
Π
\varPi
Π
\Sigma
Σ
\varSigma
Σ
\Upsilon
Υ
\varUpsilon
Υ
\Phi
Φ
\varPhi
Φ
\Psi
Ψ
\varPsi
Ψ
\Omega
Ω
\varOmega
Ω
All symbols whose name begins with var need the amsmath package.

Math symbol tables
347
A.3
LATEX binary relations
Type:
Print:
Type:
Print:
\in
∈
\ni
\leq
≤
\geq
≥
\ll
\gg
\prec
\succ
\preceq
\succeq
\sim
∼
\cong
∼
=
\simeq
\approx
≈
.
\equiv
≡
\doteq
=
\subset
⊂
\supset
⊃
\subseteq
⊆
\supseteq
⊇
\sqsubseteq
\sqsupseteq
\smile
\frown
\perp
⊥
\models
|=
\mid
|
\parallel
\vdash
\dashv
\propto
∝
\asymp
\bowtie
\sqsubset
`
\sqsupset
a
\Join
I
The latter three symbols need the latexsym package.

348
Appendix A
A.4
AMS binary relations
Type:
Print:
Type:
Print:
\leqslant
\geqslant
\eqslantless
\eqslantgtr
\lesssim
\gtrsim
\lessapprox
\gtrapprox
\approxeq
\lessdot
\gtrdot
\lll
\ggg
\lessgtr
\gtrless
\lesseqgtr
\gtreqless
\lesseqqgtr
\gtreqqless
\doteqdot
\eqcirc
\circeq
\fallingdotseq
\risingdotseq
\triangleq
\backsim
\thicksim
∼
\backsimeq
\thickapprox
≈
\preccurlyeq
\succcurlyeq
\curlyeqprec
\curlyeqsucc
\precsim
\succsim
\precapprox
\succapprox
\subseteqq
\supseteqq
\Subset
\Supset
\vartriangleleft
\vartriangleright
\trianglelefteq
\trianglerighteq
\vDash
\Vdash
\Vvdash
\smallsmile
\smallfrown
\shortmid
\shortparallel
\bumpeq
\Bumpeq
\between
\pitchfork
\varpropto
∝
\backepsilon
\blacktriangleleft
\blacktriangleright
\therefore
∴
\because
All symbols require the amssymb package.

Math symbol tables
349
A.5
AMS negated binary relations
Type:
Print:
Type:
Print:
\ne
=
\notin
/
∈
\nless
\ngtr
\nleq
\ngeq
\nleqslant
\ngeqslant
\nleqq
\ngeqq
\lneq
\gneq
\lneqq
\gneqq
\lvertneqq
\gvertneqq
\lnsim
\gnsim
\lnapprox
\gnapprox
\nprec
\nsucc
\npreceq
\nsucceq
\precneqq
\succneqq
\precnsim
\succnsim
\precnapprox
\succnapprox
\nsim
\ncong
\nshortmid
\nshortparallel
\nmid
\nparallel
\nvdash
\nvDash
\nVdash
\nVDash
\ntriangleleft
\ntriangleright
\ntrianglelefteq
\ntrianglerighteq
\nsubseteq
\nsupseteq
\nsubseteqq
\nsupseteqq
\subsetneq
\supsetneq
\varsubsetneq
\varsupsetneq
\subsetneqq
\supsetneqq
\varsubsetneqq
\varsupsetneqq
All symbols but \ne require the amssymb package.

350
Appendix A
A.6
Binary operations
Type:
Print:
Type:
Print:
\pm
±
\mp
\times
×
\cdot
·
\circ
◦
\bigcirc
\div
÷
\diamond
\ast
∗
\star
\cap
∩
\cup
∪
\sqcap
\sqcup
\wedge
∧
\vee
∨
\triangleleft
\triangleright
\bigtriangleup
\bigtriangledown
\oplus
⊕
\ominus
\otimes
⊗
\oslash
\odot
\bullet
•
\dagger
†
\ddagger
‡
\setminus
\
\uplus
\wr
\amalg
\lhd
¡
\rhd
£
\unlhd
¢
\unrhd
¤
\dotplus
\centerdot
\ltimes
\rtimes
\leftthreetimes
\rightthreetimes
\circleddash
\smallsetminus
\barwedge
\doublebarwedge
\curlywedge
\curlyvee
\veebar
\intercal
\Cap
\Cup
\circledast
\circledcirc
\boxminus
\boxtimes
\boxdot
\boxplus
\divideontimes
\And
&
This table is divided into four parts. The ﬁrst part contains the binary operations
in LATEX. The second part requires the latexsym package. The third part contains
the AMS additions; they require the amssymb package. The symbol \And requires
the amsmath package.

Math symbol tables
351
A.7
Arrows
Type:
Print:
Type:
Print:
\leftarrow
←
\rightarrow or \to
→
\longleftarrow
←−
\longrightarrow
−→
\Leftarrow
⇐
\Rightarrow
⇒
\Longleftarrow
⇐=
\Longrightarrow
=⇒
\leftrightarrow
↔
\longleftrightarrow
←→
\Leftrightarrow
⇔
\Longleftrightarrow
⇐⇒
\uparrow
↑
\downarrow
↓
\Uparrow
⇑
\Downarrow
⇓
\updownarrow
\Updownarrow
\nearrow
\searrow
\swarrow
\nwarrow
\mapsto
→
\longmapsto
−→
\hookleftarrow
←
\hookrightarrow
→
\leftharpoonup
\rightharpoonup
\leftharpoondown
\rightharpoondown
\rightleftharpoons
\leadsto
Y
\leftleftarrows
\rightrightarrows
\leftrightarrows
\rightleftarrows
\Lleftarrow
\Rrightarrow
\twoheadleftarrow
\twoheadrightarrow
\leftarrowtail
\rightarrowtail
\looparrowleft
\looparrowright
\upuparrows
\downdownarrows
\upharpoonleft
\upharpoonright
\downharpoonleft
\downharpoonright
\leftrightsquigarrow
\rightsquigarrow
\multimap
\nleftarrow
\nrightarrow
\nLeftarrow
\nRightarrow
\nleftrightarrow
\nLeftrightarrow
This table is divided into three parts. The top part contains the symbols provided
by LATEX; the last command, \leadsto, requires the latexsym package. The middle
table contains the AMS arrows; they all require the amssymb package. The bottom
table lists the negated arrow symbols; they also require amssymb.

352
Appendix A
A.8
Miscellaneous symbols
Type:
Print:
Type:
Print:
\hbar
\ell
\imath
ı
\jmath

\wp
℘
\Re
\Im
\partial
∂
\infty
∞
\prime
\emptyset
∅
\backslash
\
\forall
∀
\exists
∃
\smallint
∫
\triangle
√
\surd
\Vert
\top
\bot
⊥
\P
¶
\S
§
\dag
†
\ddag
‡
\flat
\natural
\sharp
\angle
∠
\clubsuit
♣
\diamondsuit
♦
\heartsuit
♥
\spadesuit
♠
\neg
¬
\Box
P
\Diamond
Q
\mho
H
\hslash
\complement
\backprime
\vartriangle
\Bbbk
k
\varnothing
∅
\diagup
\diagdown
\blacktriangle
\blacktriangledown
\triangledown
\Game
\square
\blacksquare
\lozenge
♦
\blacklozenge
\measuredangle
\sphericalangle
\circledS
\bigstar
\Finv
\eth
ð
\nexists
This table is divided into two parts. The top part contains the symbols provided
by LATEX; the last three commands require the latexsym package. The bottom table
lists symbols from the AMS; they all require the amssymb package.

Math symbol tables
353
A.9
Math spacing commands
Short form:
Full form:
Size:
Short form:
Full form:
\,
\thinspace
\!
\negthinspace
\:
\medspace
\negmedspace
\;
\thickspace
\negthickspace
\quad
\qquad
The \medspace, \thickspace, \negmedspace, and \negthickspace
commands require the amsmath package.
A.10
Delimiters
Name:
Type:
Print:
Name:
Type:
Print:
Left paren
(
(
Right paren
)
)
Left bracket
[
[
Right bracket
]
]
Left brace
\{
{
Right brace
\}
}
Reverse slash
\backslash
\
Forward slash
/
/
Left angle
\langle
Right angle
\rangle
Vertical line
|
|
Double vert. line
\|
Left ﬂoor
\lfloor
Right ﬂoor
\rfloor
Left ceiling
\lceil
Right ceiling
\rceil
Upper left corner
\ulcorner
Upper right corner
\urcorner
Lower left corner
\llcorner
Lower right corner
\lrcorner
The corners require the amsmath package.
Name:
Type:
Print:
Upward arrow
\uparrow
↑
Double upward arrow
\Uparrow
⇑
Downward arrow
\downarrow
↓
Double downward arrow
\Downarrow
⇓
Up-and-down arrow
\updownarrow
Double up-and-down arrow
\Updownarrow

354
Appendix A
A.11
Operators
\arccos
\arcsin
\arctan
\arg
\cos
\cosh
\cot
\coth
\csc
\dim
\exp
\hom
\ker
\lg
\ln
\log
\sec
\sin
\sinh
\tan
\tanh
\varliminf
\varlimsup
\varinjlim
\varprojlim
The \var commands require the amsmath package.
\det
\gcd
\inf
\injlim
\lim
\liminf
\limsup
\max
\min
\projlim
\Pr
\sup
The \injlim and \projlim commands require the amsmath package.
Type:
Inline
Displayed
Type:
Inline
Displayed
n
n
n
n
\prod_{i=1}^{n}
\coprod_{i=1}^{n}
i=1
i=1
i=1
i=1
n
n
n
n
\bigcap_{i=1}^{n}
\bigcup_{i=1}^{n}
i=1
i=1
i=1
i=1
n
n
n
n
\bigwedge_{i=1}^{n}
\bigvee_{i=1}^{n}
i=1
i=1
i=1
i=1
n
n
n
n
\bigsqcup_{i=1}^{n}
\biguplus_{i=1}^{n}
i=1
i=1
i=1
i=1
n
n
n
n
\bigotimes_{i=1}^{n}
\bigoplus_{i=1}^{n}
i=1
i=1
i=1
i=1
n
n
n
n
\bigodot_{i=1}^{n}
\sum_{i=1}^{n}
i=1
i=1
i=1
i=1

Math symbol tables
355
A.12
Math accents
\hat{a}
ˆ
a
\Hat{a}
ˆ
a
\widehat{a}
a
a\sphat
a
\tilde{a}
˜
a
\Tilde{a}
˜
a
\widetilde{a}
a
a\sptilde
a∼
\acute{a}
´
a
\Acute{a}
´
a
\bar{a}
¯
a
\Bar{a}
¯
a
\breve{a}
˘
a
\Breve{a}
˘
a
a\spbreve
a˘
\check{a}
ˇ
a
\Check{a}
ˇ
a
a\spcheck
a∨
.
\dot{a}
˙a
\Dot{a}
˙a
a\spdot
a
..
\ddot{a}
¨
a
\Ddot{a}
¨
a
a\spddot
a
...
...
\dddot{a}
a
a\spdddot
a
....
\ddddot{a}
a
\grave{a}
`
a
\Grave{a}
`
a
\imath
ı
\vec{a}
a
\Vec{a}
a
\jmath

The \dddot and \ddddot commands and all the capitalized commands require the
amsmath package; the commands in the fourth column require the amsxtra package.
A.13
Math font commands
Type:
Print:
\mathbf{A}
A
\mathit{A}
A
\mathsf{A}
A
\mathrm{A}
A
\mathtt{A}
A
\mathnormal{A}
A
\mathbb{A}
A
\mathfrak{A}
A
\mathcal{A}
A
\boldsymbol{\alpha}
α
The \mathbb, \mathfrak, and \mathcal commands require the amsfonts package.
The \boldsymbol command requires the amsbsy package.

APPENDIX
B
Text symbol tables
B.1
Special text characters
Type:
Print:
Type:
Print:
Type:
Print:
\#
#
\$
$
\%
%
\&
&
\~{}
˜
\_
\^{}
ˆ
\{
{
\}
}
$|$
|
@
@
$*$
∗
$\backslash$
\
356

Text symbol tables
357
B.2
Text accents
Type:
Print:
Type:
Print:
Type:
Print:
\‘{o}
`
o
\’{o}
´
o
\"{o}
¨
o
\H{o}
˝
o
\^{o}
ˆ
o
\~{o}
˜
o
\v{o}
ˇ
o
\u{o}
˘
o
\={o}
¯
o
\b{o}
o
\.{o}
˙o
\d{o}
o.
¯
\c{o}
¸
o
\r{o}
˚
o
\t{oo}
oo
\i
ı
\j

B.3
Some European characters
Type:
Print:
Type:
Print:
Type:
Print:
\aa
˚
a
\AA
˚
A
\ae
æ
\AE
Æ
\o
ø
\O
Ø
\oe
œ
\OE
Œ
\l
l
\L
L
\ss
ß
\SS
SS
?‘
¿
!‘
¡
B.4
Extra text symbols
Type:
Print:
\dag
†
\ddag
‡
\S
§
\P
¶
\copyright
c
\pounds
£
\textbullet
•
\textvisiblespace
\textcircled{a}
a
\textperiodcentered
·

358
Appendix B
B.5
Text spacing commands
Short form:
Full form:
Size:
Short form:
Full form:
\,
\thinspace
\!
\negthinspace
\:
\medspace
\negmedspace
\;
\thickspace
\negthickspace
\quad
\qquad
The \medspace, \thickspace, \negmedspace, and \negthickspace
commands require the amsmath package.
B.6
Text font commands
command
command
switch to
with argument
declaration
\textnormal{...}
{\normalfont ...}
document font family
\textrm{...}
{\rmfamily ...}
roman font family
\textsf{...}
{\sffamily ...}
sans serif font family
\texttt{...}
{\ttfamily ...}
typewriter style font family
\textup{...}
{\upshape ...}
upright shape
\textit{...}
{\itshape ...}
italic shape
\textsl{...}
{\slshape ...}
slanted shape
\textsc{...}
{\scshape ...}
small capitals
\emph{...}
{\em ...}
emphasis
\textbf{...}
{\bfseries ...}
bold (extended)
\textmd{...}
{\mdseries ...}
normal weight and width

Text symbol tables
359
B.7
Text font size changes
\tiny
sample text
\scriptsize
sample text
\footnotesize
sample text
\small
sample text
\normalsize
sample text
\large
sample text
\Large
sample text
\LARGE
sample text
\huge
sample text
\Huge
sample text
B.8
AMS text font size changes
\Tiny
sample text
\tiny
sample text
\SMALL or \scriptsize
sample text
\Small or \footnotesize
sample text
\small
sample text
\normalsize
sample text
\large
sample text
\Large
sample text
\LARGE
sample text
\huge
sample text
\Huge
sample text

Afterword
This book is based on my earlier book Math into TEX: A simple introduction to
AMS-LATEX [13]. Although the topic changed considerably, I borrowed a fair
amount of material from that book.
So it may be appropriate to begin by thanking here those who helped me with
the earlier book. Harry Lakser was extremely generous with his time; Michael Doob
and Craig Platt assisted me with TEX and UNIX; and David Kelly and Arthur Ger-
hard read and commented on an early version of that manuscript. Michael Downes,
Frank Mittelbach, and Ralph Freese read various drafts. Richard Ribstein read the
third and fourth drafts very conscientiously.
The ﬁrst draft of this new book was read for the publisher by
David Carlisle (of the LATEX3 team)
Michael J. Downes (the project leader of the AMS team)
Fernando Q. Gouvˆea (Colby College)
Frank Mittelbach (the project leader of the LATEX3 team)
Tobias Oetiker (De Montfort University)
Nico A. F. M. Poppelier (Elsevier Science Publishers)
Together they produced a huge tutorial on LATEX for my beneﬁt. I hope that I
succeeded in passing on to you some of what I learned from them.
On February 8, 1995, a short announcement was posted on the Internet (in
the comp.text.tex newsgroup) asking for volunteers to read the ﬁrst draft. The
response was overwhelming.
I received reports from the following volunteers:
Jeff Adler (University of Chicago, Chicago, IL, USA)
Helmer Aslaksen (National University of Singapore, Singapore, Republic of Sin-
gapore)
Andrew Caird (University of Michigan Center for Parallel Computing, Ann Ar-
bor, MI, USA)
416

Afterword
417
Michael Carley (Trinity College, Dublin, Ireland)
Miroslav Dont (Czech Technical University, Prague, The Czech Republic)
Simon P. Eveson (University of York, Heslington, York, England)
Weiqi Gao (St. Louis, MO, USA)
Suleyman Guleyupoglu (Concurrent Technologies Corporation, Johnstown, PA,
USA)
Peter Gruter (Laboratoir Kastler Brossel, Paris, France)
Chris F.W. Hendriks (National Aerospace Laboratory, Amsterdam, The Nether-
lands)
Mark Higgins (Global Seismology, British Geological Survey, Edinburgh, Scot-
land)
Zhihui Huang (University of Michigan, Ann Arbor, MI, USA)
David M. Jones (Information and Computation, MIT Laboratory for Computer
Science, Cambridge, MA, USA)
Alexis Kotte and John van der Koijk (University Hospital Utrecht, Utrecht, The
Netherlands)
Donal Lyons (Trinity College, Dublin, Ireland)
Michael Lykke (Roskild, Denmark)
Steve Niu (University of Toronto, Toronto, ON, Canada)
Piet van Oostrum (Utrecht University, Utrecht, The Netherlands)
Denis Roegel (CRIN, Nancy, France)
Kevin Ruland (Washington University, St. Louis, MO, USA)
Thomas R. Scavo (Syracuse, NY, USA)
Peter Schmitt (University of Vienna, Vienna, Austria)
Nandor Sieben (Arizona State University, Tempe, AZ, USA)
Paul Thompson (Case Western Reserve University, Cleveland, OH, USA)
Ronald M. Tol (University of Groningen, Groningen, The Netherlands)
Ernst U. Wallenborn (Federal Institute of Technology, Zurich, Switzerland)
Doug Webb (Knoxville, TN, USA)
There were many volunteers, ranging in expertise from users who wanted to
learn more about LATEX, to experts in charge of large LATEX installations, to inter-
nationally known experts whose names are known to many LATEX users; and rang-
ing in background from graduate students, to professional mathematicians, com-
puter scientists, chemical engineers, psychiatrists, and consultants. I would like to
thank them all for their enthusiastic reports. They shared their learning and their
teaching experiences. This has become a much better book for their contributions.
I also received carefully crafted reports on Chapter 10 from Oren Patashnik
(Stanford University), the author of BIBTEX.
Based on these reports (ranging in size from two pages to over thirty pages),
the manuscript has been rewritten, most of it has been reorganized, and sections
have been added or deleted. I felt that as a result of the major changes proba-
bly many new errors have been introduced. So the second draft was again sent to

418
Afterword
three readers; Jeff Adler, Simon P. Eveson, and David M. Jones sent me 27 pages
of reports, conﬁrming my suspicions. My deepest appreciation to these three indi-
viduals for their excellent repeat performance.
In the meanwhile, Merry Obrecht Sawdey undertook the visual design of the
book. She also lent a helping hand in the ﬁnal typesetting of the book.
The manuscript then was sent to the technical editor, Thomas R. Scavo, who
ﬂooded it with red ink; it would be hard to overstate the importance of his work.
The ﬁnal version of the book was checked again by him.
Last but not least, I want to thank Edwin Beschler, who believed in the project
from the very beginning.

Index
#, 5
. . .
(ellipsis), 17
\# (#), 356
/ (math delimiter), 353
$, 5
/ (slash), 5, 15
$ (inline math delimiter), 11
: (colon), 5
\$ ($), 356
\: (medium space, also \medspace),
%, 5, 6
353, 358
commenting out, 6
; (semicolon), 5
\% (%), 5, 356
\; (thick space, also \thickspace),
&, 5
353, 358
alignment point, 32, 33
<, 10
\& (&), 356
=, 5
\’ ( ´ accent), 357
\= ( ¯ accent), 357
(, 5
>, 10
( (math delimiter), 353
" (double quote), 5, 7
), 5
\" ( ¨ accent), 357
) (math delimiter), 353
? (question mark), 5
*, 5
?‘ (¿), 357
$*$ (∗), 356
! (exclamation mark), 5
@, 5
!‘ (¡), 357
+, 5, 15
\! (negative thin space, also
, (comma), 5
\negthinspace), 353, 358
\, (thin space, also \thinspace), 353,
[, 5
358
[ (math delimiter), 353
- (dash, hyphen, minus), 15
\[ (start displayed math), 11
\- (optional hyphen), 8
\, 5, 9
-- (number ranges, en-dash), 10
start of command, 9
--- (em-dash), 10
\ (math delimiter), 353
. (period), 5
{, 5
\. ( ˙ accent), 357
{ (math delimiter), 353
419

420
Index
\{ ({), 356
Alt key, 5
}, 5
\amalg ( ), 347, 350
} (math delimiter), 353
AMS document class, xxiii
\} (}), 356
amsart (document class), xxiii, 33, 41
\\
amsbsy (package), 54
row/line separator, 18, 31–33
AMSFonts, 51, 52
], 5
amsfonts (package), xxiii, 12, 51, 54
] (math delimiter), 353
AMS-LATEX, xx, xxii–xxiv, xxv, 23,
\] (end displayed math), 11
52–54
ˆ, 5
amsmath (package), 15–20, 23, 30–33,
\^ ( ˆ accent), 357
35, 41, 50, 51, 54, 346, 350,
(underscore), 5, 15
353–355, 358
\ (
accent), 357
amsopn (package), 54
‘ (left quote), 5
amssymb (package), 12, 15, 20, 23,
’ (right quote), 5
35, 41, 54, 346–352
|, 5, 10
AMS-TEX, xxv
$|$ (|), 356
amstext (package), 19, 54
| (math delimiter), 28, 353
amsthm (package), 50, 54
\| ( math delimiter), 353
amsxtra (package), 355
\~ ( ˜ accent), 357
\And ( & ), 350
˜, 5
\and, 42, 43
\angle (∠), 352
\AA (˚
A), 357
annotated align, 31, 32
\aa (˚
a), 357
\approx (≈), 347
abstract environment, 36
\approxeq, 348
abstract in article, 36
\arccos (arccos operator), 354
accent
\arcsin (arcsin operator), 354
European, 10, 357
\arctan (arctan operator), 354
math, 16, 355
\arg (arg operator), 354
accented character, 10, 357
argument of command, 9, 11, 12, 14
\Acute (´ math accent), 355
optional, 19
\acute (´ math accent), 355
arithmetic operation, 15
Jeff Adler, 416, 418
arrow, 351
\AE (Æ), 357
article, 34–46
\ae (˚
a), 357
abstract, 36
\aleph (ℵ), 345
bibliography, 38–40
align, 31
body, 35
annotated, 31, 32
design, 48
simple, 31
preamble, 34–35
align (math environment), 31, 32
sample
aligned multiline formula, 31
intrart, xx, 34–40, 46, 48
alignment point, 32, 33
sampart, xx–xxiii, 33, 49, 50
\alpha (α), 346

Index
421
sampart2, 50, 56
\bigtriangledown (
), 350
sectioning, 43
\bigtriangleup (
), 350
template, 41–42
\biguplus (
large operator), 354
top matter, 35, 36, 41, 42, 43
\bigvee (
large operator), 24, 25,
article (document class), 34, 41, 42,
354
49, 51
\bigwedge (
large operator), 25, 354
article.tpl, 41
binary operation, 350
article2.tpl, 41
binary relation, 348
Helmer Aslaksen, 416
negated, 349
\ast (∗), 350
\binom, 16, 21
\asymp ( ), 347
binomial, 16
aux ﬁle, 30, 55
Blackboard bold math alphabet, xxiii,
24
\b ( accent), 357
¯
\blacklozenge ( ), 352
\backepsilon ( ), 348
\blacksquare ( ), 352
\backprime ( ), 352
\blacktriangle ( ), 352
\backsim ( ), 348
\blacktriangledown ( ), 352
\backsimeq ( ), 348
\blacktriangleleft ( ), 348
\backslash (\), 352
\blacktriangleright ( ), 348
$\backslash$ (\), 356
blank delimiter, 25
\backslash (math delimiter), 353
blank line
\Bar (¯ math accent), 355
marking end of paragraph, 7
\bar (¯ math accent), 16, 27, 355
bmatrix (subsidiary math environment),
\barwedge ( ), 350
18
\Bbbk (k), 352
body of article, 35
\because ( ), 348
\bot (⊥), 352
Edwin Beschler, 418
\bowtie ( ), 347
\beta (β), 346
\Box (P), 352
\beth ( ), 345
\boxdot ( ), 350
\between ( ), 348
\boxminus ( ), 350
\bfseries, 358
\boxplus ( ), 350
\bibitem, 30, 38, 41, 45
brace
bibliography, 38–40
closing, 14
\bigcap (
large operator), 354
\Breve (˘ math accent), 355
\bigcirc (
), 350
\breve (˘ math accent), 355
\bigcup (
large operator), 23, 354
\bullet (•), 350
\biggl, 27
\Bumpeq ( ), 348
\biggr, 27
\bumpeq ( ), 348
\bigodot (
large operator), 354
\bigoplus (
large operator), 354
\c ( ¸ accent), 357
\bigotimes (
large operator), 354
Andrew Caird, 416
\bigsqcup (
large operator), 354
calligraphic alphabet, 28
\bigstar (
), 352
\Cap ( ), 350

422
Index
\cap (∩), 350
\copyright (), 357
Michael Carley, 417
\cos (cos operator), 354
David Carlisle, xviii, 416
\cosh (cosh operator), 354
cases (subsidiary math environment),
\cot (cot operator), 354
33
\coth (coth operator), 354
\cdot (·), 15, 350
cross-referencing, 30
\cdots (· · · ), 17
\csc (csc operator), 354
center, 9
Ctrl key, 5
\centerdot ( ), 350
\Cup ( ), 350
centimeter, 10
\cup (∪), 350
character, 4
\curlyeqprec ( ), 348
accented, 10, 357
\curlyeqsucc ( ), 348
European, 10, 357
\curlyvee ( ), 350
invalid, 5
\curlywedge ( ), 350
math, 10
custom format, 57
prohibited, 5
customized
special, 5, 356
preamble of article, 41–42
tab, 5, 7
top matter of article, 41–42
\Check (ˇ math accent), 355
\check (ˇ math accent), 355
\d ( . accent), 357
\chi (χ), 346
\dag († math symbol), 352
\circ (◦), 350
\dag († text symbol), 357
\circeq ( ), 348
\dagger (†), 350
\circledast ( ), 350
\daleth ( ), 345
dash, 10
\circledcirc ( ), 350
em-dash, 10
\circleddash ( ), 350
en-dash, 10
\circledS (
), 352
\cite, 30, 45
\dashv ( ), 347
\clubsuit (♣), 352
\ddag (‡ math symbol), 352
cm (dimensional unit), 10
\ddag (‡ text symbol), 357
CM fonts, 51
\ddagger (‡), 350
....
command
\ddddot (
math accent), 355
...
argument, 9, 11, 12, 14
\dddot (
math accent), 355
start with \, 9
\Ddot (¨ math accent), 355
Command key, 5
\ddot (¨ math accent), 355
commenting out, 6
delimiter, 16, 353
blank, 25
\complement ( ), 352
computer, xxi
\Delta (∆), 346
Computer Modern (CM) fonts, 51
\delta (δ), 346
\cong (∼
=), 347
\det (det operator), 354
congruence, 16
\dfrac, 28
\coprod (
large operator), 354
\diagdown (
), 352
\diagup (
), 352

Index
423
\Diamond (Q), 352
ellipsis ( . . . ), 17
\diamond ( ), 350
\em, 358
\diamondsuit (♦), 352
em-dash (—), 10
\digamma ( ), 346
\emph, 358
digit key, 4
emphasized text, 9
\dim (dim operator), 354
\emptyset (∅), 352
displayed
en-dash (–), 10
math, 11
enter (return) key, 5
\div (÷), 350
environment, 4, 9
\divideontimes ( ), 350
abstract, 36
document class, xxii–xxiv, 6, 15, 34,
document, 9, 20
35, 42, 49, 51
thebibliography, 38
AMS, xxiii
\epsilon ( ), 346
amsart, xxiii, 33, 41
\eqcirc ( ), 348
article, 34, 41, 42, 49, 51
\eqref, 30
option
\eqslantgtr ( ), 348
draft, 8
\eqslantless ( ), 348
document (environment), 9, 20
equation, 29–31
\documentclass, 12, 15
labelled, 30
Miroslav Dont, 417
referenced, 30
Michael Doob, 416
tagged, 31
\Dot ( ˙ math accent), 355
equation (math environment), 29–
\dot ( ˙ math accent), 355
31
.
\doteq (=), 347
\equiv (≡), 16, 347
\doteqdot ( ), 348
error message, 6, 13, 46–48
dotless i (ı), 357
\errorcontextlines, 6
dotless j (), 357
\eta (η), 346
\dotplus ( ), 350
\eth (ð), 352
\dots (· · · or . . . ), 17
eufrak (package), 23
double quote, 5, 7
Euler Script, xxiii
\doublebarwedge ( ), 350
European accent, 10, 357
\Downarrow ( ⇓ math delimiter), 353
European character, 10, 357
\Downarrow (⇓), 351
\EuScript, 355
\downarrow ( ↓ math delimiter), 353
euscript (package), 355
\downarrow (↓), 351
Simon P. Eveson, 417, 418
\downdownarrows ( ), 351
exclamation mark, 5
Michael J. Downes, xviii, 416
\exists (∃), 352
\downharpoonleft ( ), 351
\exp (exp operator), 354
\downharpoonright ( ), 351
draft (document class option), 8
\fallingdotseq ( ), 348
ﬁle
editor, 46, 51, 56
aux, 30, 55
\ell ( ), 352
log, 54, 55

424
Index
source, 3, 6, 7, 9, 11, 22, 30,
\gtrapprox ( ), 348
44, 46, 48, 51, 54
\gtrdot ( ), 348
\Finv ( ), 352
\gtreqless ( ), 348
\flat ( ), 352
\gtreqqless ( ), 348
flushleft (text environment), 4
\gtrless ( ), 348
flushright (text environment), 4, 9
\gtrsim ( ), 348
font
Suleyman Guleyupoglu, 417
Computer Modern (CM), 51
\gvertneqq ( ), 349
\footnotesize, 359
\forall (∀), 352
\H ( ˝ accent), 357
format
\Hat (ˆ math accent), 355
custom, 57
\hat (ˆ math accent), 16, 355
LaTeX, xxii, 6, 20
\hbar ( ), 352
plain, 51
\hdotsfor, 28
\frac, 15
\heartsuit (♥), 352
fraction, 15
Hebrew letters, 345
Fraktur math alphabet, xxiii, 23
Peter Hendriks, 417
Ralph Freese, 416
Mark Higgins, 417
\frown (
), 347
\hom (hom operator), 354
ftp
\hookleftarrow (← ), 351
directory, 4
\hookrightarrow ( →), 351
horizontal space
gallery.tex, 22
math, 353
\Game ( ), 352
text, 358
\Gamma (Γ), 346
\hslash ( ), 352
\gamma (γ), 346
HTML (markup language), xxi
Weiqi Gao, 417
Zhihui Huang, 417
\gcd (gcd operator), 354
\Huge, 359
\geq (≥), 347
\huge, 359
\geqslant ( ), 348
hyphen, 5, 10
Arthur Gerhard, 416
optional, 8
\gg (
), 347
\ggg (
), 348
\i (ı), 357
\gimel ( ), 345
\Im ( ), 352
\gnapprox ( ), 349
\imath (ı), 352, 355
\gneq ( ), 349
\in (∈), 347
\gneqq ( ), 349
in (dimensional unit), 10
\gnsim ( ), 349
inch, 10
Fernando Q. Gouvˆea, 416
\inf (inf operator), 354
\Grave (` math accent), 355
\infty (∞), 12, 352
\grave (` math accent), 355
\injlim (inj lim operator), 354
Greek alphabet, 346
inline
Peter Gruter, 417
math environment, 11

Index
425
math formula, 11
\Lambda (Λ), 346
instruction
\lambda (λ), 346
to LATEX, 9, 51
\langle ( math delimiter), 24, 353
\int ( ), 4, 17, 28
\LARGE, 359
integral, 17
\Large, 359
interactive LATEX, 54
\large, 359
\intercal ( ), 350
large operator, 19, 354
intrart (sample article), xx, 34–40,
LATEX, 51
46, 48
format, xxii, 6, 20
invalid character, 5
interactive mode, 54
\iota (ι), 346
LATEX 2ε, xviii
\itshape, 358
latexsym (package), 347, 350–352
\lceil (
math delimiter), 353
\j (), 357
\ldots (. . . ), 17
\jmath (), 352, 355
\leadsto (Y), 351
\Join (I), 347
\left, 17, 23, 24, 27
David M. Jones, xviii, 417, 418
left single quote, 5, 7
\left( ( ( math delimiter), 16, 24,
\kappa (κ), 346
32
David Kelly, 416
\left. (blank math delimiter), 25
\ker (ker operator), 354
\Leftarrow (⇐), 351
key, 4, 10
\leftarrow (←), 351
Alt, 5
\leftarrowtail (
), 351
Command, 5
\leftharpoondown (
), 351
Ctrl, 5
\leftharpoonup (
), 351
digit, 4
\leftleftarrows (
), 351
enter (return), 5
\Leftrightarrow (⇔), 351
letter, 4
\leftrightarrow (↔), 351
Option, 5
\leftrightarrows (
), 351
prohibited, 5
\leftrightsquigarrow (
), 351
return, 5
\leftthreetimes ( ), 350
space, 5
\leq (≤), 347
special, 5, 356
\leqslant ( ), 348
tab, 5, 7
\lessapprox ( ), 348
Donald E. Knuth, xviii, xx, 416
\lessdot ( ), 348
John van der Koijk, 417
\lesseqgtr ( ), 348
Alexis Kotte, 417
\lesseqqgtr ( ), 348
\L (L), 357
\lessgtr ( ), 348
\l (l), 357
\lesssim ( ), 348
label, 30, 43, 44
letter key, 4
\label, 30, 43, 44
\lfloor (
math delimiter), 353
Harry Lakser, 416
\lg (lg operator), 354

426
Index
\lhd (¡), 350
inline, 11
\lim (lim operator), 11, 17, 354
symbol, 346–355
\liminf (lim inf operator), 354
math environment
\limsup (lim sup operator), 354
align, 31, 32
line in text
annotated align, 31, 32
too wide, 7
equation, 29–31
line separator (\\), 18, 31, 32
inline, 11
line too wide, 7
simple align, 31
\listfiles, 55
subsidiary, 33
\ll (
), 347
math font
\Lleftarrow (
), 351
AMSFonts, xxiii, 51, 52
\lll (
), 348
Blackboard bold, xxiii, 24
\ln (ln operator), 354
calligraphic, 28
\lnapprox ( ), 349
Euler Script, xxiii
\lneq ( ), 349
Fraktur, xxiii, 23
\lneqq ( ), 349
math.tex (sample ﬁle), 11, 12
\lnsim ( ), 349
mathb.tex (sample ﬁle), 12, 13
\log (log operator), 354
\mathbb, 24, 355
log ﬁle, 54, 55
\mathbf, 355
logical unit, 49
\mathcal, 355
\Longleftarrow (⇐=), 351
\mathfrak, 355
\longleftarrow (←−), 351
\mathit, 355
\Longleftrightarrow (⇐⇒), 351
\mathrm, 355
\longleftrightarrow (←→), 351
\mathsf, 355
\longmapsto (−→), 351
\mathtt, 355
\Longrightarrow (=⇒), 351
matrix (subsidiary math environment),
\longrightarrow (−→), 351
18
\looparrowleft (
), 351
\max (max operator), 354
\looparrowright (
), 351
\mbox, 19
\lozenge (♦), 352
\mdseries, 358
\ltimes ( ), 350
\measuredangle ( ), 352
\lvertneqq ( ), 349
\medspace (medium space, also \:),
Michael Lykke, 417
353, 358
Donal Lyons, 417
\mho (H), 352
\mid (|), 23, 347
\maketitle, 35
\min (min operator), 354
\mapsto (→), 23, 351
minus, 15
markup language, xx
Frank Mittelbach, xviii, 416
HTML, xxi
\models (|=), 347
math, 10–33
\mp ( ), 350
accent, 16, 355
\mu (µ), 346
character, 10
multiline formula, 31
displayed, 11

Index
427
annotated aligned, 32
\nRightarrow (
), 351
simple align, 31
\nrightarrow (
), 351
\multimap (
), 351
\nshortmid ( ), 349
multiplication, 15
\nshortparallel ( ), 349
\nsim ( ), 349
\natural ( ), 352
\nsubseteq ( ), 349
\ncong ( ), 349
\nsubseteqq ( ), 349
\ne (=), 349
\nsucc ( ), 349
\nearrow (
), 351
\nsucceq ( ), 349
\neg (¬), 352
\nsupseteq ( ), 349
negated binary relation, 349
\nsupseteqq ( ), 349
\negmedspace (negative medium space),
nth root, 19
353, 358
\ntriangleleft ( ), 349
\negthickspace (negative thick space),
\ntrianglelefteq ( ), 349
353, 358
\ntriangleright ( ), 349
\negthinspace (negative thin space,
\ntrianglerighteq ( ), 349
also \!), 353, 358
\nu (ν), 346
\nexists ( ), 352
number range, 10
\ngeq ( ), 349
\nVDash ( ), 349
\ngeqq ( ), 349
\nVdash ( ), 349
\ngeqslant ( ), 349
\nvDash ( ), 349
\ngtr ( ), 349
\nvdash ( ), 349
\ni ( ), 347
\nwarrow (
), 351
Steve Niu, 417
\nLeftarrow (
), 351
\O (Ø), 357
\nleftarrow (
), 351
\o (ø), 357
\nLeftrightarrow (
), 351
\odot ( ), 350
\nleftrightarrow (
), 351
\OE (Œ), 357
\nleq ( ), 349
\oe (œ), 357
\nleqq ( ), 349
Tobias Oetiker, 416
\nleqslant ( ), 349
\Omega (Ω), 346
\nless ( ), 349
\omega (ω), 346
\nmid ( ), 349
\ominus ( ), 350
\normalsize, 359
Piet van Oostrum, 417
\notag, 32
operation
note1.tex (sample ﬁle), 6, 7
arithmetic, 15
note1b.tex (sample ﬁle), 7, 8
operator, 17, 354
note2.tex (sample ﬁle), 9
large, 19, 354
noteslug.tex (sample ﬁle), 8
\oplus (⊕), 350
\notin ( /
∈), 349
Option key, 5
\nparallel ( ), 349
optional
\nprec ( ), 349
argument, 19
\npreceq ( ), 349
hyphen, 8

428
Index
\oslash ( ), 350
pmatrix (subsidiary math environment),
\otimes (⊗), 350
18, 28
\overbrace, 26
\pmod, 16
Overfull \hbox, 7, 8
\pod, 16
overline, 27, 28
point (font size), 8
\overline, 27, 28
Nico A. F. M. Poppelier, 416
\overset, 25, 26, 38, 48
\pounds (£ text symbol), 357
\Pr (Pr operator), 354
\P (¶ math symbol), 352
preamble of article, 34–35
\P (¶ text symbol), 357
\prec ( ), 347
packages
\precapprox ( ), 348
amsbsy, 54
\preccurlyeq ( ), 348
amsfonts, xxiii, 12, 51, 54
\preceq ( ), 347
amsmath, 15–20, 23, 30–33, 35,
\precnapprox ( ), 349
41, 50, 51, 54, 346, 350,
\precneqq ( ), 349
353–355, 358
\precnsim ( ), 349
amsopn, 54
\precsim ( ), 348
amssymb, 12, 15, 20, 23, 35, 41,
\prime ( ), 352
54, 346–352
proclamation, 35
amstext, 19, 54
invoking, 44
amsthm, 50, 54
\prod (
large operator), 19, 354
amsxtra, 355
product, 19
eufrak, 23
prohibited character/key, 5
euscript, 355
\projlim (proj lim operator), 354
latexsym, 347, 350–352
prompt, 54
paragraph
\propto (∝), 347
end of, 7
\Psi (Ψ), 346
\parallel ( ), 347
\psi (ψ), 346
parentheses, 5
pt (dimensional unit), 8
\partial (∂), 28, 352
punctuation mark, 5
Oren Patashnik, 417
period, 5
\qquad (space command), 19, 353,
\perp (⊥), 347
358
personal computer, xxi
\quad (space command), 17, 353, 358
\Phi (Φ), 346
question mark, 5
\phi (φ), 346
quotation mark, 5, 7
\Pi (Π), 346
double quote, 5, 7
\pi (π), 346
single quote, 5, 7
\pitchfork ( ), 348
plain.fmt, 51
\r (˚ accent), 357
Craig Platt, 416
\rangle ( math delimiter), 24, 353
\pm (±), 350
\rceil (
math delimiter), 353
\Re ( ), 352

Index
429
\ref, 30, 31, 44
Rainer Sch¨
opf, xviii
reference in article, 38, 44
Peter Schmitt, 417
references, 38
\scriptsize, 359
referencing
\scshape, 358
bibliographic item, 45
\searrow (
), 351
label, 30, 43, 44
\sec (sec operator), 354
return key, 5
section, 44
\rfloor (
math delimiter), 353
\section, 44
\rhd (£), 350
sectioning of article, 43
\rho (ρ), 346
paragraph, 44
Richard Ribstein, 416
section, 44
\right, 17, 23, 24, 27
subparagraph, 44
right single quote, 5, 7
subsection, 44
\right) ( ) math delimiter), 16, 24,
subsubsection, 44
32
\setminus (\), 350
\Rightarrow (⇒), 351
\sffamily, 358
\rightarrow (→, also \to), 12, 351
\sharp ( ), 352
\rightarrowtail (
), 351
\shortmid ( ), 348
\rightharpoondown (
), 351
\shortparallel ( ), 348
\rightharpoonup (
), 351
\show, 54
\rightleftarrows (
), 351
Nandor Sieben, 417
\rightleftharpoons (
), 351
\Sigma (Σ), 346
\rightrightarrows (
), 351
\sigma (σ), 346
\rightsquigarrow (
), 351
\sim (∼), 347
\rightthreetimes ( ), 350
\simeq ( ), 347
\risingdotseq ( ), 348
simple align, 31
\rmfamily, 358
\sin (sin operator), 17, 354
Denis Roegel, 417
sine, 17, 354
root, 12, 19
single quote, 5, 7
nth, 19
\sinh (sinh operator), 354
square, 12, 19
\slshape, 358
row/line separator (\\), 18, 31, 32
\SMALL, 359
\Rrightarrow (
), 351
\Small, 359
\rtimes ( ), 350
\small, 359
Kevin Ruland, 417
\smallfrown ( ), 348
\smallint (∫ ), 352
\S (§ math symbol), 352
\smallsetminus ( ), 350
\S (§ text symbol), 357
\smallsmile ( ), 348
sampart (sample article), xx–xxiii, 33,
\smile (
), 347
49, 50
source ﬁle, xx, 3, 5–7, 9, 11, 22, 30,
sampart2 (sample article), 50, 56
44, 46, 48, 51, 54
Merry Obrecht Sawdey, 418
space
Thomas R. Scavo, 417, 418

430
Index
key, 5
\succ ( ), 347
rules
\succapprox ( ), 348
in text, 7
\succcurlyeq ( ), 348
vertical, 10
\succeq ( ), 347
spacebar, 5
\succnapprox ( ), 349
\spadesuit (♠), 352
\succneqq ( ), 349
\spbreve (˘ math accent), 355
\succnsim ( ), 349
\spcheck ( ∨
math accent), 355
\succsim ( ), 348
\spdddot ( ...
math accent), 355
sum, 19, 20, 27
\spddot ( ..
math accent), 355
\sum (
large operator), 19, 20, 27,
\spdot ( .
math accent), 355
354
special key/character, 5, 356
\sup (sup operator), 354
\sphat (
math accent), 355
superscript, 15
\sphericalangle ( ), 352
\Supset ( ), 348
Michael Spivak, xviii
\supset (⊃), 347
\sptilde ( ∼ math accent), 355
\supseteq (⊇), 347
\sqcap ( ), 350
\supseteqq ( ), 348
\sqcup ( ), 350
\supsetneq ( ), 349
√
\sqrt (
), 12, 19
\supsetneqq ( ), 349
√
\sqsubset (`), 347
\surd ( ), 352
\sqsubseteq ( ), 347
\swarrow (
), 351
\sqsupset (a), 347
symbolic referencing, 30
\sqsupseteq ( ), 347
\square ( ), 352
\t (
accent), 357
square root, 12, 19
tab character/key, 5, 7
\SS (SS), 357
\tag, 30
tagging equations, 31
\ss (ß), 357
\star ( ), 350
\tan (tan operator), 354
subscript, 15
\tanh (tanh operator), 354
\Subset ( ), 348
\tau (τ ), 346
\subset (⊂), 347
testart.tex, 23
T
\subseteq (⊆), 347
EX, xx, xxi, 9, 51
T
\subseteqq ( ), 348
EX log, 54, 55
text, 3, 4
\subsetneq ( ), 349
emphasized, 9
\subsetneqq ( ), 349
subsidiary math environment, 33
in math, 19
bmatrix, 18
\text, 19, 23, 54
text environment, 9
cases, 33
ﬂush right, 4, 9
matrix, 18
text symbol, 356–357
pmatrix, 18, 28
Vmatrix, 18, 26
\textbf, 9, 358
vmatrix, 18, 26
\textbullet (• text symbol), 357
\textcircled (
text symbol), 357

Index
431
\textit, 358
\trianglerighteq ( ), 348
\textmd, 358
\ttfamily, 358
\textperiodcentered (· text symbol),
\twoheadleftarrow (
), 351
357
\twoheadrightarrow (
), 351
\textrm, 358
\textsc, 358
\u ( ˘ accent), 357
\textsf, 358
\underset, 25
\textsl, 358
\unlhd (¢), 350
\textstyle, 25
\unrhd (¤), 350
\texttt, 358
\Uparrow ( ⇑ math delimiter), 353
\textup, 358
\Uparrow (⇑), 351
\textvisiblespace ( text symbol),
\uparrow ( ↑ math delimiter), 353
357
\uparrow (↑), 351
thebibliography (text environment),
\Updownarrow (
math delimiter), 353
38
\Updownarrow ( ), 351
\therefore (∴), 348
\updownarrow ( math delimiter), 353
\Theta (Θ), 346
\updownarrow ( ), 351
\theta (θ), 346
\upharpoonleft ( ), 351
\thickapprox (≈), 348
\upharpoonright ( ), 351
\thicksim (∼), 348
\uplus ( ), 350
\thickspace (thick space, also \;),
\upshape, 358
353, 358
\Upsilon (Υ), 346
\thinspace (thin space, also \,), 4,
\upsilon (υ), 346
353, 358
\upuparrows ( ), 351
Paul Thompson, 417
\usepackage, 12, 15, 20
\Tilde (˜ math accent), 355
\v ( ˇ accent), 357
\tilde (˜ math accent), 16, 355
\varDelta (∆), 346
\times (×), 15, 350
\varepsilon (ε), 346
\Tiny, 359
\varGamma (Γ ), 346
\tiny, 359
\varinjlim (lim
\title, 42, 43
−→ operator)), 354
\varkappa (κ), 346
title page, 35
\varLambda (Λ), 346, 348
\to (→, also \rightarrow), 12, 351
\varliminf (lim operator)), 354
Ronald M. Tol, 417
\varlimsup (lim operator)), 354
\top ( ), 352
\varnothing (∅), 352
top matter, 35, 36, 41–43
\varOmega (Ω), 346
\triangle (
), 352
\varPhi (Φ), 346
\triangledown ( ), 352
\varphi (ϕ), 28, 346
\triangleleft ( ), 350
\varPi (Π), 346
\trianglelefteq ( ), 348
\varpi ( ), 346
\triangleq ( ), 348
\varprojlim (lim
\triangleright ( ), 350
←− operator)), 354
\varpropto (∝), 348

432
Index
\varPsi (Ψ ), 346
\xi (ξ), 346
\varrho ( ), 346
\xvarsupsetneqqx ( ), 349
\varSigma (Σ), 346
\varsigma (ς), 346
\zeta (ζ), 4, 346
\varsubsetneq ( ), 349
\varsubsetneqq ( ), 349
\varsupsetneq ( ), 349
\varTheta (Θ), 346
\vartheta (ϑ), 346
\vartriangle ( ), 352
\vartriangleleft ( ), 348
\vartriangleright ( ), 348
\varUpsilon (Υ ), 346
\varXi (Ξ), 346
\Vdash ( ), 348
\vDash ( ), 348
\vdash ( ), 347
\Vec (
math accent), 355
\vec (
math accent), 16, 355
\vee (∨), 25, 350
\veebar ( ), 350
version number, 55
\Vert ( math symbol), 352
vertical space, 10
Vmatrix (subsidiary math environment),
18, 26
vmatrix (subsidiary math environment),
18, 26
\vspace, 10
\Vvdash (
), 348
Ernst U. Wallenborn, 417
Doug Webb, 417
\wedge (∧), 350
\widehat (
math accent), 27, 355
\widetilde (
math accent), 355
work directory, 4, 6, 7, 9, 11, 12, 20,
41, 42
World Wide Web, xxi
\wp (℘), 352
\wr ( ), 350
\Xi (Ξ), 346