Blackboard bold

Blackboard bold used on a blackboard

Blackboard boldis a style of writingboldsymbols on ablackboardby doubling certain strokes, commonly used in mathematicallectures,and the derived style oftypefaceused in printed mathematical texts. The style is most commonly used to represent thenumber sets$\mathbb {N}$(natural numbers),$\mathbb {Z}$(integers),$\mathbb {Q}$(rational numbers),$\mathbb {R}$(real numbers), and$\mathbb {C}$(complex numbers).

To imitate a bold typeface on atypewriter,a character can be typed over itself (calleddouble-striking);symbols thus produced are calleddouble-struck,and this name is sometimes adopted for blackboard bold symbols,for instance inUnicodeglyphnames.

Intypography,a typeface with characters that are not solid is calledinline,handtooled,oropen face.

History

Typewritten lecture notes byGunning(1966), showing "blackboard bold" styleRandCachieved by double-striking each letter with significant offset.

Typewritten lecture notes byNarasimhan(1966), with "blackboard bold" styleRandCachieved with an inline typewriter face.

Traditionally, various symbols were indicated byboldfacein print but on blackboards and inmanuscripts"by wavy underscoring, or enclosure in a circle, or even by wavy overscoring".

Most typewriters have no dedicated bold characters at all. To produce a bold effect on a typewriter, a character can bedouble-struckwith or without a small offset. By the mid 1960s, typewriter accessories such as the "Doublebold" could automatically double-strike every character while engaged.While this method makes a character bolder, and can effectively emphasize words or passages, in isolation a double-struck character is not always clearly different from its single-struck counterpart.

Blackboard bold originated from the attempt to write bold symbols on typewriters and blackboards that were legible but distinct, perhaps starting in the late 1950s in France, and then taking hold at thePrinceton Universitymathematics department in the early 1960s.Mathematical authors began typing faux-bold letters by double-striking them with a significant offset or over-striking them with the letterI,creating new symbols such as IR, IN, CC, orZZ; at the blackboard, lecturers began writing bold symbols with certain doubled strokes.The notation caught on: blackboard bold spread from classroom to classroom and is now used around the world.

A page fromLoomis&Sternberg(1968), showing an early example of "blackboard bold" styleRandCin a printed book.

The style made its way into print starting in the mid 1960s. Early examples includeRobert GunningandHugo Rossi'sAnalytic Functions of Several Complex Variables(1965)andLynn LoomisandShlomo Sternberg'sAdvanced Calculus(1968).Initial adoption was sporadic, however, and most publishers continued using boldface. In 1979,Wileyrecommended its authors avoid "double-backed shadow or outline letters, sometimes called blackboard bold", because they could not always be printed;in 1982, Wiley refused to include blackboard bold characters in mathematical books because the type was difficult and expensive to obtain.

Donald Knuthpreferred boldface to blackboard bold and so did not include blackboard bold in theComputer Moderntypeface that he created for theTeXmathematical typesetting system he first released in 1978.When Knuth's 1984The TeXbookneeded an example of blackboard bold for the index, he produced$\mathrm {I\!R}$using the lettersIandRwith a negative space between;in 1988 Robert Messer extended this to a full set of "poor person's blackboard bold" macros, overtyping each capital letter with carefully placedIcharacters or vertical lines.

Not all mathematical authors were satisfied with such workarounds. TheAmerican Mathematical Societycreated a simple chalk-style blackboard bold typeface in 1985 to go with theAMS-TeXpackage created byMichael Spivak,accessed using the\Bbbcommand (for "blackboard bold" ); in 1990, the AMS released an update with a new inline-style blackboard bold font intended to better matchTimes.Since then, a variety of other blackboard bold typefaces have been created, some following the style of traditional inline typefaces and others closer in form to letters drawn with chalk.

Unicodeincluded the most common blackboard bold letters among the "Letterlike Symbols"in version 1.0 (1991), inherited from theXerox Character Code Standard.Later versions of Unicode extended this set to all uppercase and lowercaseLatinletters and a variety of other symbols, among the "Mathematical Alphanumeric Symbols".

In professionally typeset books, publishers and authors have gradually adopted blackboard bold, and its use is now commonplace,but some still use ordinary bold symbols. Some authors use blackboard bold letters on the blackboard or in manuscripts, but prefer an ordinary bold typeface in print; for example,Jean-Pierre Serrehas used blackboard bold in lectures, but has consistently used ordinary bold for the same symbols in his published works.TheChicago Manual of Style's recommendation has evolved over time: In 1993, for the 14th edition, it advised that "blackboard bold should be confined to the classroom" (13.14); In 2003, for the 15th edition, it stated that "open-faced (blackboard) symbols are reserved for familiar systems of numbers" (14.12). The international standardISO 80000-2:2019 listsRas the symbol for the real numbers but notes "the symbolsIRand$\mathbb {R}$are also used ", and similarly forN,Z,Q,C,andP(prime numbers).

Encoding

Blackboard bold variants, from top to bottom: "poor person's blackboard bold", AMSFonts mathbb based onTimes,doublestroke package based onComputer Modern,STIX Twoinspired byMonotypeSpecial Alphabets 4

TeX,the standard typesetting system for mathematical texts, does not contain direct support for blackboard bold symbols, but theAmerican Mathematical Societydistributes the popularAMSFontscollection, loaded from theamssymbpackage, which includes a blackboard bold typeface for uppercase Latin letters accessed using\mathbb(e.g.\mathbb{R}produces$\mathbb {R}$).

InUnicode,a few of the more common blackboard bold characters (ℂ, ℍ, ℕ, ℙ, ℚ, ℝ, and ℤ) are encoded in theBasic Multilingual Plane(BMP) in theLetterlike Symbols(2100–214F) area, named DOUBLE-STRUCK CAPITAL C etc. The rest, however, are encoded outside the BMP, inMathematical Alphanumeric Symbols(1D400–1D7FF), specifically from 1D538–1D550 (uppercase, excluding those encoded in the BMP), 1D552–1D56B (lowercase) and 1D7D8–1D7E1 (digits). Blackboard bold Arabic letters are encoded inArabic Mathematical Alphabetic Symbols(1EE00–1EEFF), specifically 1EEA1–1EEBB.

Usage

"𝕏" redirects here. For the social network, seeTwitter.

The following table shows all available Unicode blackboard bold characters.

The first column shows the letter as typically rendered by theLaTeXmarkup system. The second column shows the Unicode code point. The third column shows the Unicode symbol itself (which will only display correctly on browsers that support Unicode and have access to a suitable typeface). The fourth column describes some typical usage in mathematical texts.Some of the symbols (particularly$\mathbb {C},\mathbb {Q},\mathbb {R}$and$\mathbb {Z}$) are nearly universal in their interpretation,while others are more varied in use.

LaTeX	Unicode code point(hex)	Unicode symbol	Mathematics usage
$\mathbb {A}$	U+1D538	𝔸	Representsaffine space,$\mathbb {A} ^{n}$,or thering of adeles.Occasionally represents thealgebraic numbers,thealgebraic closureof$\mathbb {Q}$(more commonly written${\overline {\mathbb {Q} }}$orQ), or thealgebraic integers,an important subring of the algebraic numbers.
$\mathbb {B}$	U+1D539	𝔹	Sometimes represents aball,aboolean domain,or theBrauer groupof a field.
$\mathbb {C}$	U+2102	ℂ	Represents thesetofcomplex numbers.
$\mathbb {D}$	U+1D53B	𝔻	Represents theunit diskin thecomplex plane,for example as theconformal disk modelof thehyperbolic plane.By generalisation$\mathbb {D} ^{n}$may mean then-dimensionalball.Occasionally$\mathbb {D}$may mean the decimal fractions (seenumber),split-complex numbers,ordomain of discourse.
$\mathbb {E}$	U+1D53C	𝔼	Represents theexpected valueof arandom variable,orEuclidean space,or afieldin atower of fields,or theEudoxus reals.
$\mathbb {F}$	U+1D53D	𝔽	Represents afield.Often used forfinite fields,with a subscript to indicate the order.Also represents aHirzebruch surfaceor afree group,with a subscript to indicate the number of generators (or generating set, if infinite).
$\mathbb {G}$	U+1D53E	𝔾	Represents aGrassmannianor agroup,especially analgebraic group.
$\mathbb {H}$	U+210D	ℍ	Represents thequaternions(the H stands forHamilton),or theupper half-plane,orhyperbolic space,orhyperhomologyof a complex.
$\mathbb {I}$	U+1D540	𝕀	The closedunit intervalor theidealofpolynomialsvanishing on asubset.Occasionally theidentity mappingon analgebraic structure,or anindicator function.The set ofimaginary numbers(i.e., the set of all real multiples of the imaginary unit).
$\mathbb {J}$	U+1D541	𝕁	Sometimes represents theirrational numbers,$\mathbb {R} \smallsetminus \mathbb {Q}$.
$\mathbb {K}$	U+1D542	𝕂	Represents afield.This is derived from theGermanwordKörper,which is German for field (literally, 'body'; in French the term iscorps). May also be used to denote acompact space.
$\mathbb {L}$	U+1D543	𝕃	Represents the Lefschetz motive. SeeMotive (algebraic geometry).
$\mathbb {M}$	U+1D544	𝕄	Sometimes represents themonster group.The set of allm-by-nmatricesis sometimes denoted$\mathbb {M} (m,n)$.Ingeometric algebra,represents the motor group of rigid motions. Infunctional programmingandformal semantics,denotes the type constructor for amonad.
$\mathbb {N}$	U+2115	ℕ	Represents the set ofnatural numbers.May or may not includezero.
$\mathbb {O}$	U+1D546	𝕆	Represents theoctonions.
$\mathbb {P}$	U+2119	ℙ	Representsprojective space,theprobabilityof an event,theprime numbers,apower set,the positive reals, theirrational numbers,or aforcingposet.
$\mathbb {Q}$	U+211A	ℚ	Represents the set ofrational numbers.(The Q stands forquotient.)
$\mathbb {R}$	U+211D	ℝ	Represents the set ofreal numbers.
$\mathbb {S}$	U+1D54A	𝕊	Represents asphere,or thesphere spectrum,or occasionally thesedenions.
$\mathbb {T}$	U+1D54B	𝕋	Represents thecircle group,particularly theunit circlein the complex plane (and$\mathbb {T} ^{n}$then-dimensionaltorus), or aHecke algebra(Hecke denoted his operators asTnor$\mathbb {T} _{n}$), or thetropical semiring,ortwistor space.
$\mathbb {U}$	U+1D54C	𝕌
$\mathbb {V}$	U+1D54D	𝕍	Represents avector spaceor anaffine varietygenerated by a set of polynomials, or in probability theory and statistics thevariance.
$\mathbb {W}$	U+1D54E	𝕎	Represents thewhole numbers(here in the sense of non-negative integers), which also are represented by$\mathbb {N} _{0}$.
$\mathbb {X}$	U+1D54F	𝕏	Occasionally used to denote an arbitrarymetric space.
$\mathbb {Y}$	U+1D550	𝕐
$\mathbb {Z}$	U+2124	ℤ	Represents the set ofintegers.(The Z is forZahlen,German for 'numbers', andzählen,German for 'to count'.) When it has a positive integer subscript, it can mean thefinite cyclic groupof that size.
	U+1D552	𝕒
	U+1D553	𝕓
	U+1D554	𝕔
	U+1D555	𝕕
	U+1D556	𝕖
	U+1D557	𝕗
	U+1D558	𝕘
	U+1D559	𝕙
	U+1D55A	𝕚
	U+1D55B	𝕛
$\mathbb {k}$	U+1D55C	𝕜	Represents afield.
	U+1D55D	𝕝
	U+1D55E	𝕞
	U+1D55F	𝕟
	U+1D560	𝕠
	U+1D561	𝕡
	U+1D562	𝕢
	U+1D563	𝕣
	U+1D564	𝕤
	U+1D565	𝕥
	U+1D566	𝕦
	U+1D567	𝕧
	U+1D568	𝕨
	U+1D569	𝕩
	U+1D56A	𝕪
	U+1D56B	𝕫
	U+2145	ⅅ
	U+2146	ⅆ
	U+2147	ⅇ
	U+2148	ⅈ
	U+2149	ⅉ
	U+213E	ℾ
	U+213D	ℽ
	U+213F	ℿ
	U+213C	ℼ
	U+2140	⅀
	U+1D7D8	𝟘	In algebra of logical propositions, it represents a contradiction or falsity.
	U+1D7D9	𝟙	Inset theory,thetop elementof aforcingposet,or occasionally the identity matrix in amatrix ring.Also used for theindicator functionand theunit step function,and for theidentity operatororidentity matrix.Ingeometric algebra,represents the unit antiscalar, the identity element under the geometric antiproduct. In algebra of logical propositions, it represents a tautology.
	U+1D7DA	𝟚	Incategory theory,the interval category.
	U+1D7DB	𝟛
	U+1D7DC	𝟜
	U+1D7DD	𝟝
	U+1D7DE	𝟞
	U+1D7DF	𝟟
	U+1D7E0	𝟠
	U+1D7E1	𝟡
	U+1EEA1	𞺡	Arabic Mathematical Double-Struck Beh (based on ب)
	U+1EEA2	𞺢
	U+1EEA3	𞺣
	U+1EEA5	𞺥
	U+1EEA6	𞺦
	U+1EEA7	𞺧
	U+1EEA8	𞺨
	U+1EEA9	𞺩
	U+1EEAB	𞺫
	U+1EEAC	𞺬
	U+1EEAD	𞺭
	U+1EEAE	𞺮
	U+1EEAF	𞺯
	U+1EEB0	𞺰
	U+1EEB1	𞺱
	U+1EEB2	𞺲
	U+1EEB3	𞺳
	U+1EEB4	𞺴
	U+1EEB5	𞺵
	U+1EEB6	𞺶
	U+1EEB7	𞺷
	U+1EEB8	𞺸
	U+1EEB9	𞺹
	U+1EEBA	𞺺
	U+1EEBB	𞺻

In addition, a blackboard-boldμ_n(not found in Unicode oramsmathLaTeX) is sometimes used bynumber theoristsandalgebraic geometersto designate thegroup schemeofn-throots of unity.

Note: Only uppercase Roman letters are given LaTeX renderings because Wikipedia's implementation uses theAMSFontsblackboard bold typeface, which does not support other characters.

See also

• Latin letters used in mathematics, science, and engineering
• Mathematical Alpha numeric symbols
• Set notation