Blackboard bold
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Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or nearvertical lines) are doubled. The symbols usually denote number sets. One way of producing Blackboard bold is to doublestrike a character with a small offset on a typewriter.^{[1]} Thus they are referred to as double struck.
Origin
In some texts these symbols are simply shown in bold type: blackboard bold in fact originated from the attempt to write bold letters on blackboards in a way that clearly differentiated them from nonbold letters i.e. by using the edge rather than point of the chalk. It then made its way back in print form as a separate style from ordinary bold,^{[1]} possibly starting with the original 1965 edition of Gunning and Rossi's textbook on complex analysis.^{[2]}
It is sometimes erroneously claimed^{[1]}Template:Failed verification that Bourbaki introduced the blackboard bold notation, but whereas individual members of the Bourbaki group may have popularized doublestriking bold characters on the blackboard, their printed books use ordinary bold.^{[3]}
Rejection
Some mathematicians, therefore, do not recognize blackboard bold as a separate style from bold: JeanPierre Serre, for example, has publicly inveighed against the use of "blackboard bold" anywhere other than on a blackboard,{{ safesubst:#invoke:Unsubstdate=__DATE__ $B= {{#invoke:Category handlermain}}{{#invoke:Category handlermain}}^{[citation needed]} }}. He uses doublestruck letters when writing bold on the blackboard,^{[4]} whereas his published works consistently use ordinary bold for the same symbols.^{[5]} Donald Knuth also advises against the use of blackboard bold in print.^{[6]}Template:Not in citation
The Chicago Manual of Style in 1993 (14th edition) advises: "blackboard bold should be confined to the classroom" (13.14) whereas in 2003 (15th edition) it states that "openfaced (blackboard) symbols are reserved for familiar systems of numbers" (14.12).
Encoding
TeX, the standard typesetting system for mathematical texts, does not contain direct support for blackboard bold symbols, but the addon AMS Fonts package (amsfonts
) by the American Mathematical Society provides this facility; a blackboard bold R is written as \mathbb{R}
. The amssymb
package loads amsfonts
.
In Unicode, a few of the more common blackboard bold characters (C, H, N, P, Q, R and Z) are encoded in the Basic Multilingual Plane (BMP) in the Letterlike Symbols (2100β214F) area, named DOUBLESTRUCK CAPITAL C etc. The rest, however, are encoded outside the BMP, from U+1D538
to U+1D550
(uppercase, excluding those encoded in the BMP), U+1D552
to U+1D56B
(lowercase) and U+1D7D8
to U+1D7E1
(digits). Being outside the BMP, these are relatively new and not widely supported.
Usage
The following table shows all available Unicode blackboard bold characters.
The symbols are nearly universal in their interpretation, unlike their normallytypeset counterparts, which are used for many different purposes.
The first column shows the letter as typically rendered by the ubiquitous LaTeX markup system. The second column shows the Unicode codepoint. The third column shows the symbol itself (which will only display correctly on browsers that support Unicode and have access to a suitable font). The fourth column describes known typical (but not universal) usage in mathematical texts.
LaTeX  Unicode (Hex)  Symbol  Mathematics usage 

U+1D538

πΈ  Represents affine space or the ring of adeles. Sometimes represents the algebraic numbers, the algebraic closure of Q (β or β, although Q is often used instead). It may also represent the algebraic integers, an important subring of the algebraic numbers.  
U+1D552

π  
U+1D539

πΉ  Sometimes represents a ball, a boolean domain, or the Brauer group of a field.  
U+1D553

π  
U+2102

β  Represents the complex numbers.  
U+1D554

π  
U+1D53B

π»  Represents the unit (open) disk in the complex plane (for example as a model of the Hyperbolic plane), or the decimal fractions (see number).  
U+1D555

π  
U+2145

β  
U+2146

β  May represent the differential symbol.  
U+1D53C

πΌ  Represents the expected value of a random variable, or Euclidean space, or a field in a tower of fields.  
U+1D556

π  
U+2147

β  Sometimes used for the mathematical constant e.  
U+1D53D

π½  Represents a field. Often used for finite fields, with a subscript to indicate the order. Also represents a Hirzebruch surface or a free group, with a subset to indicate the number of generators (or generating set, if infinite).  
U+1D557

π  
U+1D53E

πΎ  Represents a Grassmannian or a group, especially an algebraic group.  
U+1D558

π  
U+210D

β  Represents the quaternions (the H stands for Hamilton), or the upper halfplane, or hyperbolic space, or hyperhomology of a complex.  
U+1D559

π  
U+1D540

π  Occasionally used to denote the identity mapping on an algebraic structure, or the set of imaginary numbers (i.e., the set of all real multiples of the imaginary unit).  
U+1D55A

π  
U+2148

β  Occasionally used for the imaginary unit.  
U+1D541

π  Sometimes represents the irrational numbers, R\Q (β\β).  
U+1D55B

π  
U+2149

β  
U+1D542

π  Represents a field, typically a scalar field. This is derived from the German word KΓΆrper, which is German for field (literally, "body"; cf. the French term corps). May also be used to denote a compact space.  
U+1D55C

π  
U+1D543

π  Represents the Lefschetz motive. See motives.  
U+1D55D

π  
U+1D544

π  Represents the monster group. The set of all mbyn matrices is denoted π(m, n).  
U+1D55E

π  
U+2115

β  Represents the natural numbers. May or may not include zero.  
U+1D55F

π  
U+1D546

π  Represents the octonions.  
U+1D560

π  
U+2119

β  Represents projective space, the probability of an event, the prime numbers, a power set, the positive reals, the irrational numbers, or a forcing partially ordered set (poset).  
U+1D561

π‘  
U+211A

β  Represents the rational numbers. (The Q stands for quotient.)  
U+1D562

π’  
U+211D

β  Represents the real numbers.  
U+1D563

π£  
U+1D54A

π  Represents the sedenions, or a sphere.  
U+1D564

π€  
U+1D54B

π  Represents a torus, or the circle group, or a Hecke algebra (Hecke denoted his operators as T_{n} or π_{β}), or the Tropical semiring, or twistor space.  
U+1D565

π₯  
U+1D54C

π  
U+1D566

π¦  
U+1D54D

π  Represents a vector space.  
U+1D567

π§  
U+1D54E

π  Represents the whole numbers (here in the sense of nonnegative integers), which also are represented by β_{0}.  
U+1D568

π¨  
U+1D54F

π  Occasionally used to denote an arbitrary metric space.  
U+1D569

π©  
U+1D550

π  
U+1D56A

πͺ  
U+2124

β€  Represents the integers. (The Z is for Zahlen, which is German for "numbers".)  
U+1D56B

π«  
U+213E

βΎ  
U+213D

β½  
U+213F

βΏ  
U+213C

βΌ  
U+2140

β  
U+1D7D8

π  
U+1D7D9

π  Often represents, in set theory, the top element of a forcing partially ordered set (poset), or occasionally for the identity matrix in a matrix ring. Also used for the indicator function.  
U+1D7DA

π  
U+1D7DB

π  
U+1D7DC

π  
U+1D7DD

π  
U+1D7DE

π  
U+1D7DF

π  
U+1D7E0

π  
U+1D7E1

π‘ 
In addition, a blackboardbold Greek letter mu (not found in Unicode) is sometimes used by number theorists and algebraic geometers (with a subscript n) to designate the group (or more specifically group scheme) of nth roots of unity.^{[7]}
See also
References
 β ^{1.0} ^{1.1} ^{1.2} Google Groups
 β {{#invoke:citation/CS1citation CitationClass=book }}
 β E.g., {{#invoke:citation/CS1citation CitationClass=book }}
 β "Writing Mathematics Badly" video talk (part 3/3), starting at 7′08″
 β E.g., {{#invoke:citation/CS1citation CitationClass=book }}
 β Krantz, S., Handbook of Typography for the Mathematical Sciences, Chapman & Hall/CRC, Boca Raton, Florida, 2001, p. 35.
 β {{#invoke:citation/CS1citation CitationClass=book }}
External links
 http://www.w3.org/TR/MathML2/doublestruck.html shows blackboard bold symbols together with their Unicode encodings. Encodings in the Basic Multilingual Plane are highlighted in yellow.