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

To imitate a bold typeface on atypewriter, a character can be typed over itself (calleddouble-striking);^[1] symbols thus produced are calleddouble-struck, and this name is sometimes adopted for blackboard bold symbols,^[2] for instance inUnicodeglyph names.

Intypography, a typeface with characters that are not solid is calledinline,handtooled, oropen face.^[3]

Traditionally, various symbols were indicated byboldface in print but on blackboards and inmanuscripts "by wavy underscoring, or enclosure in a circle, or even by wavy overscoring".^[6]

Most typewriters have no dedicated bold characters at all. To produce a bold effect on a typewriter, a character can bedouble-struck with or without a small offset. By the mid 1960s, typewriter accessories such as the "Doublebold" could automatically double-strike every character while engaged.^[7] 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.^[8][9]

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 University mathematics department in the early 1960s.^[8][10] 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 asIR,IN,CC,orZZ;at the blackboard, lecturers began writing bold symbols with certain doubled strokes.^[8][10] The notation caught on: blackboard bold spread from classroom to classroom and is now used around the world.^[8]

The style made its way into print starting in the mid 1960s. Early examples includeRobert Gunning andHugo Rossi'sAnalytic Functions of Several Complex Variables (1965)^[12][10] andLynn Loomis andShlomo Sternberg'sAdvanced Calculus (1968).^[11] Initial adoption was sporadic, however, and most publishers continued using boldface. In 1979,Wiley recommended its authors avoid "double-backed shadow or outline letters, sometimes called blackboard bold", because they could not always be printed;^[13] in 1982, Wiley refused to include blackboard bold characters in mathematical books because the type was difficult and expensive to obtain.^[14]

Donald Knuth preferred boldface to blackboard bold and so did not include blackboard bold in theComputer Modern typeface that he created for theTeX mathematical typesetting system he first released in 1978.^[14] When Knuth's 1984The TeXbook needed an example of blackboard bold for the index, he produced${\displaystyle \mathrm{I}\mathrm{R}}$ using the lettersI andR with a negative space between;^[15] in 1988 Robert Messer extended this to a full set of "poor man's blackboard bold" macros, overtyping each capital letter with carefully placedI characters or vertical lines.^[16]

Not all mathematical authors were satisfied with such workarounds. TheAmerican Mathematical Society created a simple chalk-style blackboard bold typeface in 1985 to go with theAMS-TeX package created byMichael Spivak, accessed using the\Bbb command (for "blackboard bold"); in 1990, the AMS released an update with a new inline-style blackboard bold font intended to better matchTimes.^[17] 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.^[18]

Unicode included 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 lowercaseLatin letters and a variety of other symbols, among the "Mathematical Alphanumeric Symbols".^[19]

In professionally typeset books, publishers and authors have gradually adopted blackboard bold, and its use is now commonplace,^[14] 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 Serre has used blackboard bold in lectures, but has consistently used ordinary bold for the same symbols in his published works.^[20] 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 listsR as the symbol for the real numbers but notes "the symbolsIR and${\displaystyle \mathbb{R}}$ are also used", and similarly forN,Z,Q,C, andP (prime numbers).^[21]

TeX, the standard typesetting system for mathematical texts, does not contain direct support for blackboard bold symbols, but theAmerican Mathematical Society distributes theAMSFonts collection, loaded from theamssymb package, which includes a blackboard bold typeface for uppercase Latin letters accessed using\mathbb (e.g.\mathbb{R} produces${\displaystyle \mathbb{R}}$).^[23]

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.

The following table shows all available Unicode blackboard bold characters.^[24]

The first column shows the letter as typically rendered by theLaTeX markup 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.^[25] Some of the symbols (particularly${\displaystyle \mathbb{C},\mathbb{Q},\mathbb{R}}$ and${\displaystyle \mathbb{Z}}$) are nearly universal in their interpretation,^[14] while others are more varied in use.

LaTeX	Unicode code point(hex)	Unicode symbol	Mathematics usage
Uppercase Latin
${\displaystyle \mathbb{A}}$	U+1D538	𝔸	Representsaffine space,${\displaystyle {\mathbb{A}}^n}$, or thering of adeles. Occasionally represents thealgebraic numbers,[26] thealgebraic closure of${\displaystyle \mathbb{Q}}$ (more commonly written${\displaystyle \overline{\mathbb{Q}}}$ orQ), or thealgebraic integers, an important subring of the algebraic numbers.
${\displaystyle \mathbb{B}}$	U+1D539	𝔹	Sometimes represents aball, aboolean domain, or theBrauer group of a field.
${\displaystyle \mathbb{C}}$	U+2102	ℂ	Represents theset ofcomplex numbers.[14]
${\displaystyle \mathbb{D}}$	U+1D53B	𝔻	Represents theunit disk in thecomplex plane, for example as theconformal disk model of thehyperbolic plane. By generalisation${\displaystyle {\mathbb{D}}^n}$ may mean then-dimensionalball. Occasionally${\displaystyle \mathbb{D}}$ may mean the decimal fractions (seenumber),split-complex numbers, ordomain of discourse.
${\displaystyle \mathbb{E}}$	U+1D53C	𝔼	Represents theexpected value of arandom variable, orEuclidean space, or afield in atower of fields, or theEudoxus reals.
${\displaystyle \mathbb{F}}$	U+1D53D	𝔽	Represents afield.[26] Often used forfinite fields, with a subscript to indicate the order.[26] Also represents aHirzebruch surface or afree group, with a subscript to indicate the number of generators (or generating set, if infinite).
${\displaystyle \mathbb{G}}$	U+1D53E	𝔾	Represents aGrassmannian or agroup, especially analgebraic group.
${\displaystyle \mathbb{H}}$	U+210D	ℍ	Represents thequaternions (the H stands forHamilton),[26] or theupper half-plane, orhyperbolic space,[26] orhyperhomology of a complex.
${\displaystyle \mathbb{I}}$	U+1D540	𝕀	The closedunit interval or theideal ofpolynomials vanishing on asubset. Occasionally theidentity mapping on analgebraic structure, or anindicator function. The set of purelyimaginary numbers (i.e., the set of all real multiples of theimaginary unit).
${\displaystyle \mathbb{J}}$	U+1D541	𝕁	Sometimes represents theirrational numbers,${\displaystyle \mathbb{R}\setminus \mathbb{Q}}$.
${\displaystyle \mathbb{K}}$	U+1D542	𝕂	Represents afield.[26] This is derived from theGerman wordKörper, which is German for field (literally, 'body'; in French the term iscorps). May also be used to denote acompact space.
${\displaystyle \mathbb{L}}$	U+1D543	𝕃	Represents the Lefschetz motive. SeeMotive (algebraic geometry).
${\displaystyle \mathbb{M}}$	U+1D544	𝕄	Sometimes represents themonster group. The set of allm-by-nmatrices is sometimes denoted${\displaystyle \mathbb{M}(m,n)}$. Ingeometric algebra, represents the motor group of rigid motions. Infunctional programming andformal semantics, denotes the type constructor for amonad.
${\displaystyle \mathbb{N}}$	U+2115	ℕ	Represents the set ofnatural numbers.[21] May or may not includezero.
${\displaystyle \mathbb{O}}$	U+1D546	𝕆	Represents theoctonions.[26]
${\displaystyle \mathbb{P}}$	U+2119	ℙ	Representsprojective space, theprobability of an event,[26] theprime numbers,[21] apower set, the positive reals, theirrational numbers, or aforcingposet.
${\displaystyle \mathbb{Q}}$	U+211A	ℚ	Represents the set ofrational numbers.[14] (The Q stands forquotient.) With a prime number in the subscript, represents thep-adic numbers.
${\displaystyle \mathbb{R}}$	U+211D	ℝ	Represents the set ofreal numbers.[14]
${\displaystyle \mathbb{S}}$	U+1D54A	𝕊	Represents asphere, or thesphere spectrum, or occasionally thesedenions.
${\displaystyle \mathbb{T}}$	U+1D54B	𝕋	Represents thecircle group, particularly theunit circle in the complex plane (and${\displaystyle {\mathbb{T}}^n}$ then-dimensionaltorus), occasionally thetrigintaduonions, or aHecke algebra (Hecke denoted his operators asTn or${\displaystyle {\mathbb{T}}_n}$), or thetropical semiring, ortwistor space.
${\displaystyle \mathbb{U}}$	U+1D54C	𝕌
${\displaystyle \mathbb{V}}$	U+1D54D	𝕍	Represents avector space or anaffine variety generated by a set of polynomials, or in probability theory and statistics thevariance.
${\displaystyle \mathbb{W}}$	U+1D54E	𝕎	Represents thewhole numbers (here in the sense of non-negative integers), which also are represented by${\displaystyle {\mathbb{N}}_0}$.
${\displaystyle \mathbb{X}}$	U+1D54F	𝕏	Occasionally used to denote an arbitrarymetric space.
${\displaystyle \mathbb{Y}}$	U+1D550	𝕐
${\displaystyle \mathbb{Z}}$	U+2124	ℤ	Represents the set ofintegers.[14] (The Z is forZahlen, German for 'numbers', andzählen, German for 'to count'.) When it has a positive integer subscript, it can mean thefinite cyclic group of that size, or thep-adic integers if the subscript is prime.
Lowercase Latin
	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	Sometimes used to represent theimaginary unit.[27]
	U+1D55B	𝕛
${\displaystyle \mathbb{k}}$	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
Italic Latin
	U+2145	ⅅ
	U+2146	ⅆ
	U+2147	ⅇ
	U+2148	ⅈ
	U+2149	ⅉ
Greek
	U+213E	ℾ
	U+213D	ℽ
	U+213F	ℿ
	U+213C	ℼ
	U+2140	⅀
Digits
	U+1D7D8	𝟘	In algebra of logical propositions, it represents a contradiction or falsity.
	U+1D7D9	𝟙	Inset theory, thetop element of aforcingposet, or occasionally the identity matrix in amatrix ring. Also used for theindicator function and theunit step function, and for theidentity operator oridentity 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
Arabic
	U+1EEA1	𞺡	Arabic Mathematical Double-StruckBeh (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 oramsmath LaTeX) is sometimes used bynumber theorists andalgebraic geometers to designate thegroup scheme ofn-throots of unity.^[28]

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

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

• Mathematical notation
• Mathematical typefaces

• Articles with short description
• Short description is different from Wikidata
• Articles containing German-language text
• Articles containing French-language text