The Blackb oard Bold Font Yossi Gil Department of Computer Science Technion|Israel Institute of Technology Technion City, Haifa 32000, Israel [email protected] 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 " 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 M A of the standard L T X2 distribution. " E 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. E 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.