The Blackb oard Bold Font

Yossi Gil

Department of Computer Science

Technion|Israel Institute of Technology

Technion City, Haifa 32000, Israel

yogi@cs.technion.ac.il

Septemb er 29, 2004

Blackboardbold also called double struck isatyp eface in which certain lines

of the symb ol usually vertical, or near-vertical lines are doubled. It is primarily

used for upp er case letters, which usually describ e sets of numb ers.

The name originated from the attempt to distinguish b old letters on black-

b oards by double striking them.

In this short do cument I try to demonstrate the common use of the black-

A

b oard b old font in mathematics and what you havetodoinLT X2 in order

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E

to generate these symb ols.

1 Accessing the blackb oard font

The blackb oard font is delivered with A S-LaTeX distribution, which is part

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A

of the standard L T X2 distribution.

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The following is one the shortest p ossible do cument that uses the blackb oard

b old font.

\documentclass{article}

\usepackage{amssymb}

\begin{document}

$\mathbb{A},\mathbb{B}, \mathbb{C}, \ldots$

\end{document}

The output is then

A ; B ; C ;:::

It would have b een sucient to write \usepackage{amsfonts}.However, the

amssymb package includes the amsfonts package. 1

2 The Symb ols

Only upp er case letters exist in the blackb oard b old font.

A B C D E F G

H I J K L M N

O P Q R S T U

V W XYZ

3 Using the font

Excessivechange of typ efaces, unrestrained creativityininventing new notation,

will deter the reader. It is even worse to overload existing and familiar notation

with idiosyncratic meaning. Here is a list of the common uses of the blackb oard

b old font in mathematics. If p ossible, try not to deviate from this list.

1. Numb er theory often uses the set N of natural numb ers whichmayor

may not include 0, and the ring Zof integers. It is also common to use P

to denote the set of primes. We can write

P  N  Z:

2. Fields in common use include Q the rational numb ers, R the real num-

b ers, and C the complex plane, which ob ey the relation

Z Q  R:

All the ab ove are in nite elds, also called elds of 0 characteristic.

3. An unsp eci ed eld is usually denoted by K. The notation F is usually

reserved to nite elds. A nite eld of order n is usually denoted as F .

n

However, since for all p 2 P

Zmo d p = F

p

p eople often use the notation Z for the nite eld of prime order p.

p

A nite eld is called a Galois eld and is completed de ned by its or-

k

der, whichmust b e a prime p ower. Therefore p eople often write GFp 

where p 2 P is the eld characteristic and k 2 N, k  1 instead of F k .

p

4. The eld obtained by an algebraic closure of the rational numb ers is writ-

Q or A .Wehave ten as either

Q=A C:

5. There are several extensions of the complex eld, none of which is a eld.

The quaternions H are a four dimensional set in whichmultiplication

is non-commutative. The o ctonions H are a nonasso ciative extension of 2

the quaternions. The sedenions S are a 16-dimensional extension of the

o ctonions. Thus,

C  H  O  S:

6. The unit disk in the complex plane is denoted by D .

7. The symbol B is used for denoting a ball, S is the sphere, while T often

n

denotes a torus. A sup erscript is used to denote the dimension. Thus, B

is an n-dimensional ball.

I like to de ne meaningful macro names for all mathematical symb ols I use.

Table 1 lists the macros provided by the bbbold package for this purp ose.

\FiniteField F \Algebraics A

\Field K \ComplexPlane C

\Booleans B \Quaternions H

\Primes P \Octonions O

\Naturals N \Sendenions S

\Integers Z \UnitDisk D

\Rationals Q \Ball B

\Reals R \Sphere S

\Algebraics A \Torus T

Table 1: Macros provided by the bbbold package.

4 Alternatives

There is an ever increasing need for mathematical notation. If you need a

variation on capital letters, you maywant to consider the following alternatives.

A

Calligraphic letters L T X o ers a calligraphic style for upp er case letters.

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This is obtained by the \mathcal command, as in $\mathcal{ABC}$ which

pro duces AB C .

Boldface letters To obtain characters mathematics, use the \mathbf com-

mand, whichworks for b oth upp er and lower case letters. e.g., the input

$\mathbf{a B}$ will pro duce aB.

Boldface italics letters Command \boldsymbol, made available by the amssymb

package, makes it p ossible to use the b oldface version of a symb ol, while

keeping its slanted app earance. Thus, typing

$\boldsymbol{A} \ne A \ne \mathbf{A}$ 3

you will see that

A 6= A 6= A:

Bold Greek If you, like others, do not think that the di erence b etween the

ab ove three version of \A" is sucient to let them denote di erententities,

you can still use \boldsymbol to generate a visually distinctiveversion of

Greek letters. Consider the di erence b etween and ,between
and
,

etc. It is even p ossible to generate b old face version of lower case Greek

letters, suchas 6= and 6= .

Sans Serif letters I like using these for prop er words. Thus,

$t_\mathsf{QuickSort}$

will pro duce

t :

QuickSort

Fraktur letters One of the nicest and relatively unknown options you have

is to invoke the \mathfrak command to to generate Fraktur typ eface,

available for b oth upp er case letters

ABCDEFGHIJKLMNOPQRSTUVWXYZ;

pro duced by \mathfrak{ABCDEFGHIJKLMNOPQRSTUVWXYZ} and lower case

letters

abcedefghijklmnopqrstuvwxyz

pro duced by \mathfrak{abcedefghijklmnopqrstuvwxyz}. 4