Editing Blackboard bold

{{short description|Typeface style}}
[[File:Blackboard bold on a blackboard.jpg|thumb|Blackboard bold used on a blackboard|alt=A blackboard displaying the definition: The set of complex numbers consists of all quantities a + bi such that a and b are elements of the real numbers, and i squared equals negative one. Symbolically, {{not a typo|ℂ = {a + bi ∣ a, b ∈ ℝ, i² = −1}.}}]]
'''Blackboard bold''' is a style of writing [[Emphasis (typography)|bold]] symbols on a [[blackboard]] by doubling certain strokes, commonly used in mathematical [[lecture]]s, and the derived style of [[typeface]] used in printed mathematical texts. The style is most commonly used to represent the [[Set (mathematics)#Special sets of numbers in mathematics|number sets]] <math>\N</math> ([[natural number]]s), <math>\Z</math> ([[integer]]s), <math>\Q</math> ([[rational number]]s), <math>\R</math> ([[real number]]s), and <math>\C</math> ([[complex number]]s).

To imitate a bold typeface on a [[typewriter]], a character can be typed over itself (called ''double-striking'');{{r|gilreath}} symbols thus produced are called '''double-struck''', and this name is sometimes adopted for blackboard bold symbols,{{r|rosendorf}} for instance in [[Unicode]] [[glyph]] names.

In [[typography]], a typeface with characters that are not solid is called ''inline'', ''handtooled'', or ''open face''.{{r|typography}}

==History==
[[File:Blackboard_bold_in_typewritten_notes_from_Gunning_(1966).jpg|thumb|right|upright=1.25|Typewritten lecture notes by [[Robert C. Gunning|Gunning]] (1966), showing "blackboard bold" style '''R''' and '''C''' achieved by double-striking each letter with significant offset{{r|gunning}}]]
[[File:Blackboard_bold_in_typewritten_notes_from_Narasimhan_(1966).png|thumb|right|upright=1.25|Typewritten lecture notes by [[Raghavan Narasimhan|Narasimhan]] (1966), with "blackboard bold" style '''R''' and '''C''' achieved with an inline typewriter face{{r|narasimhan}}]]

Traditionally, various symbols were indicated by [[boldface]] in print but on blackboards and in [[manuscript (publishing)|manuscript]]s "by wavy underscoring, or enclosure in a circle, or even by wavy overscoring".{{r|manuscripts}}

Most typewriters have no dedicated bold characters at all. To produce a bold effect on a typewriter, a character can be ''double-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.{{r|karch}} 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.{{r|webb|waite-arca}}

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 the [[Princeton University]] mathematics department in the early 1960s.{{r|webb}}{{r|rudolph}} Mathematical authors began typing faux-bold letters by double-striking them with a significant offset or over-striking them with the letter ''I'', creating new symbols such as
<!-- Blackboard bold R, N, accomplished with over-struck I; bold C accomplished by 2 copies. Sorry for the mess of raw HTML: -->
<span style='font-size: 1.15em; font-family: Courier, "Courier New", "Lucida Typewriter", "Nimbus Mono L", "Liberation Mono", monospace; letter-spacing: -.25em; padding: 0 .28em 0 0;'>IR</span>,
<span style='font-size: 1.15em; font-family: Courier, "Courier New", "Lucida Typewriter", "Nimbus Mono L", "Liberation Mono", monospace; letter-spacing: -.22em; padding: 0 .2em 0 0;'>IN</span>,
<span style='font-size: 1.15em; font-family: Courier, "Courier New", "Lucida Typewriter", "Nimbus Mono L", "Liberation Mono", monospace; letter-spacing: -.4em; padding: 0 .4em 0 0;'>CC</span>,
or <span style='font-size: 1.15em; font-family: Courier, "Courier New", "Lucida Typewriter", "Nimbus Mono L", "Liberation Mono", monospace; letter-spacing: -.37em; padding: 0 .37em 0 0;'>ZZ</span>;
at the blackboard, lecturers began writing bold symbols with certain doubled strokes.{{r|webb}}{{r|rudolph}} The notation caught on: blackboard bold spread from classroom to classroom and is now used around the world.{{r|webb}}

[[File:Blackboard bold in print in Loomis and Sternberg (1968).jpg|thumb|upright=1.25|A page from [[Lynn Harold Loomis|Loomis]] & [[Shlomo Sternberg|Sternberg]] (1968), showing an early example of "blackboard bold" style '''R''' and '''C''' in a printed book{{r|loomis-sternberg}}]]
The style made its way into print starting in the mid 1960s. Early examples include [[Robert Gunning (mathematician)|Robert Gunning]] and [[Hugo Rossi]]'s ''Analytic Functions of Several Complex Variables'' (1965){{r|gunning-rossi}}{{r|rudolph}} and [[Lynn Harold Loomis|Lynn Loomis]] and [[Shlomo Sternberg]]'s ''Advanced Calculus'' (1968).{{r|loomis-sternberg}} Initial adoption was sporadic, however, and most publishers continued using boldface. In 1979, [[Wiley (publisher)|Wiley]] recommended its authors avoid "double-backed shadow or outline letters, sometimes called blackboard bold", because they could not always be printed;{{r|wiley}} in 1982, Wiley refused to include blackboard bold characters in mathematical books because the type was difficult and expensive to obtain.{{r|krantz}}

[[Donald Knuth]] preferred boldface to blackboard bold and so did not include blackboard bold in the [[Computer Modern]] typeface that he created for the [[TeX]] mathematical typesetting system he first released in 1978.{{r|krantz}} When Knuth's 1984 ''[[The TeXbook]]'' needed an example of blackboard bold for the index, he produced <math>\mathrm{I\!R}</math> using the letters ''I'' and ''R'' with a negative space between;{{r|knuth}} in 1988 Robert Messer extended this to a full set of "poor man's blackboard bold" macros, overtyping each capital letter with carefully placed ''I'' characters or vertical lines.{{r|messer}}

Not all mathematical authors were satisfied with such workarounds. The [[American Mathematical Society]] created a simple chalk-style blackboard bold typeface in 1985 to go with the [[AMS-TeX]] package created by [[Michael Spivak]], accessed using the <code>\Bbb</code> command (for "blackboard bold"); in 1990, the AMS released an update with a new inline-style blackboard bold font intended to better match [[Times New Roman|Times]].{{r|amstex}} 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.{{r|vieth}}

[[Unicode]] included the most common blackboard bold letters among the "[[Letterlike Symbols]]" in version 1.0 (1991), inherited from the [[Xerox Character Code Standard]]. Later versions of Unicode extended this set to all uppercase and lowercase [[Latin alphabet|Latin]] letters and a variety of other symbols, among the "[[Mathematical Alphanumeric Symbols]]".{{r|unicode}}

In professionally typeset books, publishers and authors have gradually adopted blackboard bold, and its use is now commonplace,{{r|krantz}} 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.{{r|serre}} The ''[[Chicago 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 standard [[ISO 80000-2]]:2019 lists {{math|'''R'''}} as the symbol for the real numbers but notes "the symbols {{math|1=<span style='letter-spacing: -.12em; padding: 0 .12em 0 0;'>IR</span>}} and <math>\R</math> are also used", and similarly for {{math|'''N'''}}, {{math|'''Z'''}}, {{math|'''Q'''}}, {{math|'''C'''}}, and {{math|'''P'''}} ([[prime numbers]]).{{r|iso}}

== Encoding ==
[[File:Blackboard bold number sets.svg|thumb|upright=1.25|Blackboard bold variants; from top to bottom: "poor man's blackboard bold", AMSFonts mathbb based on [[Times New Roman|Times]], doublestroke package based on [[Computer Modern]],{{r|doublestroke}} [[STIX Fonts project|STIX Two]] inspired by [[Monotype]] Special Alphabets 4 ]]

[[TeX]], the standard typesetting system for mathematical texts, does not contain direct support for blackboard bold symbols, but the [[American Mathematical Society]] distributes the [[AMSFonts]] collection, loaded from the <code>amssymb</code> package, which includes a blackboard bold typeface for uppercase Latin letters accessed using <code>\mathbb</code> (e.g. <code>\mathbb{R}</code> produces {{nobr|<math>\mathbb{R}</math>).{{r|pakin}}}}

In [[Unicode]], a few of the more common blackboard bold characters ({{not a typo|ℂ, ℍ, ℕ, ℙ, ℚ, ℝ, and ℤ}}) are encoded in the [[Basic Multilingual Plane]] (BMP) in the ''[[Letterlike Symbols]]'' (2100–214F) area, named DOUBLE-STRUCK CAPITAL C etc. The rest, however, are encoded outside the BMP, in ''[[Mathematical 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 in [[Arabic Mathematical Alphabetic Symbols]] (1EE00–1EEFF), specifically 1EEA1–1EEBB.

== Usage ==
{{redirects here|𝕏|the social network|Twitter|the display server|X Window System}}
<!--'Blackboard bold letters used in mathematics' redirects here-->
The following table shows all available Unicode blackboard bold characters.{{r|w3c}}

The first column shows the letter as typically rendered by the [[LaTeX]] 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.{{r|weisstein}} Some of the symbols (particularly <math>\mathbb{C}, \mathbb{Q}, \mathbb{R}</math> and <math>\mathbb{Z}</math>) are nearly universal in their interpretation,{{r|krantz}} while others are more varied in use. 

{| class=wikitable style="max-width:50em;"
! scope="col" style="width: 4em;" | LaTeX
! scope="col" style="width: 6.5em;" | Unicode code point {{nobold|1=(hex)}}
! scope="col" style="width: 4em;" | Unicode symbol
! Mathematics usage
|-
! colspan="4" | Uppercase Latin
|-
|<math>\mathbb{A}</math>
|<code>U+1D538</code>
|style="font-size: 105%"|{{not a typo|𝔸}}
|Represents [[affine space]], <math>\mathbb{A}^n</math>, or the [[adele ring|ring of adeles]]. Occasionally represents the [[algebraic number]]s,{{r|sevryuk}} the [[algebraic closure]] of <math>\mathbb{Q}</math> (more commonly written <math>\overline{\mathbb{Q}}</math> or <span style="text-decoration: overline">'''Q'''</span>), or the [[algebraic integer]]s, an important subring of the algebraic numbers.
|-
|<math>\mathbb{B}</math>
|<code>U+1D539</code>
|style="font-size: 105%"|{{not a typo|𝔹}}
|Sometimes represents a [[ball (mathematics)|ball]], a [[boolean domain]], or the [[Brauer group]] of a field.
|-
|<math>\mathbb{C}</math>
|<code>U+2102</code>
|style="font-size: 105%"|{{not a typo|ℂ}}
|Represents the [[set (mathematics)|set]] of [[complex number]]s.{{r|krantz}}
|-
|<math>\mathbb{D}</math>
|<code>U+1D53B</code>
|style="font-size: 105%"|{{not a typo|𝔻}}
|Represents the [[unit disk]] in the [[complex plane]], for example as the [[Poincaré disk model|conformal disk model]] of the [[hyperbolic plane]]. By generalisation <math>\mathbb{D}^n</math> may mean the ''n''-dimensional [[ball (mathematics)|ball]]. Occasionally <math>\mathbb{D}</math> may mean the decimal fractions (see [[number]]), [[split-complex number]]s, or [[domain of discourse]].
|-
|<math>\mathbb{E}</math>
|<code>U+1D53C</code>
|style="font-size: 105%"|{{not a typo|𝔼}}
|Represents the [[expected value]] of a [[random variable]], or [[Euclidean space]], or a [[Field (mathematics)|field]] in a [[tower of fields]], or the [[Eudoxus reals]]. 
|-
|<math>\mathbb{F}</math>
|<code>U+1D53D</code>
|style="font-size: 105%"|{{not a typo|𝔽}}
|Represents a [[Field (mathematics)|field]].{{r|sevryuk}}  Often used for [[finite field]]s, with a subscript to indicate the order.{{r|sevryuk}}  Also represents a [[Hirzebruch surface]] or a [[free group]], with a subscript to indicate the number of generators (or generating set, if infinite).
|-
|<math>\mathbb{G}</math>
|<code>U+1D53E</code>
|style="font-size: 105%"|{{not a typo|𝔾}}
|Represents a [[Grassmannian]] or a [[group (mathematics)|group]], especially an [[algebraic group]].
|-
|<math>\mathbb{H}</math>
|<code>U+210D</code>
|style="font-size: 105%"|{{not a typo|ℍ}}
|Represents the [[quaternions]] (the H stands for [[William Rowan Hamilton|Hamilton]]),{{r|sevryuk}} or the [[upper half-plane]], or [[hyperbolic space]],{{r|sevryuk}} or [[hyperhomology]] of a complex.
|-
|<math>\mathbb{I}</math>
|<code>U+1D540</code>
|style="font-size: 105%"|{{not a typo|𝕀}}
|The closed [[unit interval]] or the [[Ideal (set theory)|ideal]] of [[polynomial]]s vanishing on a [[subset]]. Occasionally the [[identity function|identity mapping]] on an [[algebraic structure]], or an [[indicator function]]. The set of purely [[imaginary numbers]] (i.e., the set of all real multiples of the [[imaginary unit]]).
|-
|<math>\mathbb{J}</math>
|<code>U+1D541</code>
|style="font-size: 105%"|{{not a typo|𝕁}}
|Sometimes represents the [[irrational numbers]], <math>\mathbb{R}\smallsetminus\mathbb{Q}</math>.
|-
|<math>\mathbb{K}</math>
|<code>U+1D542</code>
|style="font-size: 105%"|{{not a typo|𝕂}}
|Represents a [[Field (mathematics)|field]].{{r|sevryuk}} This is derived from the [[German language|German]] word {{lang|de|Körper}}, which is German for field (literally, 'body'; in French the term is {{lang|fr|corps}}).  May also be used to denote a [[compact space]].
|-
|<math>\mathbb{L}</math>
|<code>U+1D543</code>
|style="font-size: 105%"|{{not a typo|𝕃}}
|Represents the Lefschetz motive. See [[Motive (algebraic geometry)]].
|-
|<math>\mathbb{M}</math>
|<code>U+1D544</code>
|style="font-size: 105%"|{{not a typo|𝕄}}
|Sometimes represents the [[monster group]]. The set of all {{nowrap|''m''-by-''n''}} [[Matrix (mathematics)|matrices]] is sometimes denoted  <math>\mathbb{M}(m, n)</math>. In [[geometric algebra]], represents the motor group of rigid motions. In [[functional programming]] and [[formal semantics (linguistics)|formal semantics]], denotes the type constructor for a [[monad (functional programming)|monad]].
|-
|<math>\mathbb{N}</math>
|<code>U+2115</code>
|style="font-size: 105%"|{{not a typo|ℕ}}
|Represents the set of [[natural number]]s.{{r|iso}} May or may not include [[0 (number)|zero]].
|-
|<math>\mathbb{O}</math>
|<code>U+1D546</code>
|style="font-size: 105%"|{{not a typo|𝕆}}
|Represents the [[octonion]]s.{{r|sevryuk}}
|-
|<math>\mathbb{P}</math>
|<code>U+2119</code>
|style="font-size: 105%"|{{not a typo|ℙ}}
|Represents [[projective space]], the [[probability]] of an event,{{r|sevryuk}} the [[prime number]]s,{{r|iso}} a [[power set]], the positive reals, the [[irrational numbers]], or a [[forcing (mathematics)|forcing]] [[partially ordered set|poset]].
|-
|<math>\mathbb{Q}</math>
|<code>U+211A</code>
|style="font-size: 105%"|{{not a typo|ℚ}}
|Represents the set of [[rational number]]s.{{r|krantz}} (The Q stands for [[quotient]].) With a prime number in the subscript, represents the [[p-adic numbers|''p''-adic numbers]].
|-
|<math>\mathbb{R}</math>
|<code>U+211D</code>
|style="font-size: 105%"|{{not a typo|ℝ}}
|Represents the set of [[real number]]s.{{r|krantz}}
|-
|<math>\mathbb{S}</math>
|<code>U+1D54A</code>
|style="font-size: 105%"|{{not a typo|𝕊}}
|Represents a [[sphere]], or the [[sphere spectrum]], or occasionally the [[sedenion]]s.
|-
|<math>\mathbb{T}</math>
|<code>U+1D54B</code>
|style="font-size: 105%"|{{not a typo|𝕋}}
|Represents  the [[circle group]], particularly the [[unit circle]] in the complex plane (and <math>\mathbb{T}^n</math> the ''n''-dimensional [[torus]]), occasionally the [[trigintaduonion]]s, or a [[classical Hecke algebra|Hecke algebra]] (Hecke denoted his operators as ''T''<sub>''n''</sub> or <math>\mathbb{T}_n</math>), or the [[tropical semiring]], or [[twistor space]].
|-
|<math>\mathbb{U}</math>
|<code>U+1D54C</code>
|style="font-size: 105%"|{{not a typo|𝕌}}
|
|-
|<math>\mathbb{V}</math>
|<code>U+1D54D</code>
|style="font-size: 105%"|{{not a typo|𝕍}}
|Represents a [[vector space]] or an [[affine variety]] generated by a set of polynomials, or in probability theory and statistics the [[variance]].
|-
|<math>\mathbb{W}</math>
|<code>U+1D54E</code>
|style="font-size: 105%"|{{not a typo|𝕎}}
|Represents the [[natural number|whole numbers]] (here in the sense of non-negative integers), which also are represented by <math>\mathbb{N}_0</math>.
|-
|<math>\mathbb{X}</math>
|<code>U+1D54F</code>
|style="font-size: 105%"|{{not a typo|𝕏}}
|Occasionally used to denote an arbitrary [[metric space]].
|-
|<math>\mathbb{Y}</math>
|<code>U+1D550</code>
|style="font-size: 105%"|{{not a typo|𝕐}}
|
|-
|<math>\mathbb{Z}</math>
|<code>U+2124</code>
|style="font-size: 105%"|{{not a typo|ℤ}}
|Represents the set of [[integer]]s.{{r|krantz}} (The Z is for {{lang|de|Zahlen}}, German for 'numbers', and {{lang|de|zählen}}, German for 'to count'.)  When it has a positive integer subscript, it can mean the [[finite cyclic group]] of that size, or the [[p-adic integers|''p''-adic integers]] if the subscript is prime.
|-
! colspan="4" |Lowercase Latin
|-
|
|<code>U+1D552</code>
|style="font-size: 105%"|{{not a typo|𝕒}}
|
|-
|
|<code>U+1D553</code>
|style="font-size: 105%"|{{not a typo|𝕓}}
|
|-
|
|<code>U+1D554</code>
|style="font-size: 105%"|{{not a typo|𝕔}}
|
|-
|
|<code>U+1D555</code>
|style="font-size: 105%"|{{not a typo|𝕕}}
|
|-
|
|<code>U+1D556</code>
|style="font-size: 105%"|{{not a typo|𝕖}}
|
|-
|
|<code>U+1D557</code>
|style="font-size: 105%"|{{not a typo|𝕗}}
|
|-
|
|<code>U+1D558</code>
|style="font-size: 105%"|{{not a typo|𝕘}}
|
|-
|
|<code>U+1D559</code>
|style="font-size: 105%"|{{not a typo|𝕙}}
|
|-
|
|<code>U+1D55A</code>
|style="font-size: 105%"|{{not a typo|𝕚}}
|Sometimes used to represent the [[imaginary unit]].<ref>{{cite conference|first=Stephen|last=Wolfram|author-link=Stephen Wolfram|year=2000|url=https://www.stephenwolfram.com/publications/mathematical-notation-past-future/|contribution=Mathematical Notation: Past and Future|type=Transcript of keynote address|title=MathML and Math on the Web: MathML International Conference 2000 |access-date=2025-04-08}}</ref>
|-
|
|<code>U+1D55B</code>
|style="font-size: 105%"|{{not a typo|𝕛}}
|
|-
|<math>\mathbb{k}</math>
|<code>U+1D55C</code>
|style="font-size: 105%"|{{not a typo|𝕜}}
|Represents a [[field (mathematics)|field]].
|-
|
|<code>U+1D55D</code>
|style="font-size: 105%"|{{not a typo|𝕝}}
|
|-
|
|<code>U+1D55E</code>
|style="font-size: 105%"|{{not a typo|𝕞}}
|
|-
|
|<code>U+1D55F</code>
|style="font-size: 105%"|{{not a typo|𝕟}}
|
|-
|
|<code>U+1D560</code>
|style="font-size: 105%"|{{not a typo|𝕠}}
|
|-
|
|<code>U+1D561</code>
|style="font-size: 105%"|{{not a typo|𝕡}}
|
|-
|
|<code>U+1D562</code>
|style="font-size: 105%"|{{not a typo|𝕢}}
|
|-
|
|<code>U+1D563</code>
|style="font-size: 105%"|{{not a typo|𝕣}}
|
|-
|
|<code>U+1D564</code>
|style="font-size: 105%"|{{not a typo|𝕤}}
|
|-
|
|<code>U+1D565</code>
|style="font-size: 105%"|{{not a typo|𝕥}}
|
|-
|
|<code>U+1D566</code>
|style="font-size: 105%"|{{not a typo|𝕦}}
|
|-
|
|<code>U+1D567</code>
|style="font-size: 105%"|{{not a typo|𝕧}}
|
|-
|
|<code>U+1D568</code>
|style="font-size: 105%"|{{not a typo|𝕨}}
|
|-
|
|<code>U+1D569</code>
|style="font-size: 105%"|{{not a typo|𝕩}}
|
|-
|
|<code>U+1D56A</code>
|style="font-size: 105%"|{{not a typo|𝕪}}
|
|-
|
|<code>U+1D56B</code>
|style="font-size: 105%"|{{not a typo|𝕫}}
|
|-
! colspan="4" |Italic Latin
|-
|
|<code>U+2145</code>
|style="font-size: 105%"|{{not a typo|ⅅ}}
|
|-
|
|<code>U+2146</code>
|style="font-size: 105%"|{{not a typo|ⅆ}}
|
|-
|
|<code>U+2147</code>
|style="font-size: 105%"|{{not a typo|ⅇ}}
|
|-
|
|<code>U+2148</code>
|style="font-size: 105%"|{{not a typo|ⅈ}}
|
|-
|
|<code>U+2149</code>
|style="font-size: 105%"|{{not a typo|ⅉ}}
|
|-
! colspan="4" |Greek
|-
|
|<code>U+213E</code>
|style="font-size: 105%"|{{not a typo|ℾ}}
|
|-
|
|<code>U+213D</code>
|style="font-size: 105%"|{{not a typo|ℽ}}
|
|-
|
|<code>U+213F</code>
|style="font-size: 105%"|{{not a typo|ℿ}}
|
|-
|
|<code>U+213C</code>
|style="font-size: 105%"|{{not a typo|ℼ}}
|
|-
|
|<code>U+2140</code>
|style="font-size: 105%"|{{not a typo|⅀}}
|
|-
! colspan="4" |Digits
|-
|
|<code>U+1D7D8</code>
|style="font-size: 105%"|{{not a typo|𝟘}}
|In algebra of logical propositions, it represents a contradiction or falsity.
|-
|
|<code>U+1D7D9</code>
|style="font-size: 105%"|{{not a typo|𝟙}}
|In [[set theory]], the [[Greatest element|top element]] of a [[forcing (mathematics)|forcing]] [[partially ordered set|poset]], or occasionally the identity matrix in a [[matrix ring]]. Also used for the [[indicator function]] and the [[unit step function]], and for the [[identity function|identity operator]] or [[identity matrix]]. In [[geometric algebra]], represents the unit antiscalar, the identity element under the geometric antiproduct. In algebra of logical propositions, it represents a tautology.
|-
|
|<code>U+1D7DA</code>
|style="font-size: 105%"|{{not a typo|𝟚}}
|In [[category theory]], the interval category.
|-
|
|<code>U+1D7DB</code>
|style="font-size: 105%"|{{not a typo|𝟛}}
|
|-
|
|<code>U+1D7DC</code>
|style="font-size: 105%"|{{not a typo|𝟜}}
|
|-
|
|<code>U+1D7DD</code>
|style="font-size: 105%"|{{not a typo|𝟝}}
|
|-
|
|<code>U+1D7DE</code>
|style="font-size: 105%"|{{not a typo|𝟞}}
|
|-
|
|<code>U+1D7DF</code>
|style="font-size: 105%"|{{not a typo|𝟟}}
|
|-
|
|<code>U+1D7E0</code>
|style="font-size: 105%"|{{not a typo|𝟠}}
|
|-
|
|<code>U+1D7E1</code>
|style="font-size: 105%"|{{not a typo|𝟡}}
|
|-
! colspan="4" |Arabic
|-
|
|<code>U+1EEA1</code>
|style="font-size: 105%"|{{not a typo|𞺡}}
|Arabic Mathematical Double-Struck [[Beh (letter)|Beh]] (based on ب)
|-
|
|<code>U+1EEA2</code>
|style="font-size: 105%"|{{not a typo|𞺢}}
|
|-
|
|<code>U+1EEA3</code>
|style="font-size: 105%"|{{not a typo|𞺣}}
|
|-
|
|<code>U+1EEA5</code>
|style="font-size: 105%"|{{not a typo|𞺥}}
|
|-
|
|<code>U+1EEA6</code>
|style="font-size: 105%"|{{not a typo|𞺦}}
|
|-
|
|<code>U+1EEA7</code>
|style="font-size: 105%"|{{not a typo|𞺧}}
|
|-
|
|<code>U+1EEA8</code>
|style="font-size: 105%"|{{not a typo|𞺨}}
|
|-
|
|<code>U+1EEA9</code>
|style="font-size: 105%"|{{not a typo|𞺩}}
|
|-
|
|<code>U+1EEAB</code>
|style="font-size: 105%"|{{not a typo|𞺫}}
|
|-
|
|<code>U+1EEAC</code>
|style="font-size: 105%"|{{not a typo|𞺬}}
|
|-
|
|<code>U+1EEAD</code>
|style="font-size: 105%"|{{not a typo|𞺭}}
|
|-
|
|<code>U+1EEAE</code>
|style="font-size: 105%"|{{not a typo|𞺮}}
|
|-
|
|<code>U+1EEAF</code>
|style="font-size: 105%"|{{not a typo|𞺯}}
|
|-
|
|<code>U+1EEB0</code>
|style="font-size: 105%"|{{not a typo|𞺰}}
|
|-
|
|<code>U+1EEB1</code>
|style="font-size: 105%"|{{not a typo|𞺱}}
|
|-
|
|<code>U+1EEB2</code>
|style="font-size: 105%"|{{not a typo|𞺲}}
|
|-
|
|<code>U+1EEB3</code>
|style="font-size: 105%"|{{not a typo|𞺳}}
|
|-
|
|<code>U+1EEB4</code>
|style="font-size: 105%"|{{not a typo|𞺴}}
|
|-
|
|<code>U+1EEB5</code>
|style="font-size: 105%"|{{not a typo|𞺵}}
|
|-
|
|<code>U+1EEB6</code>
|style="font-size: 105%"|{{not a typo|𞺶}}
|
|-
|
|<code>U+1EEB7</code>
|style="font-size: 105%"|{{not a typo|𞺷}}
|
|-
|
|<code>U+1EEB8</code>
|style="font-size: 105%"|{{not a typo|𞺸}}
|
|-
|
|<code>U+1EEB9</code>
|style="font-size: 105%"|{{not a typo|𞺹}}
|
|-
|
|<code>U+1EEBA</code>
|style="font-size: 105%"|{{not a typo|𞺺}}
|
|-
|
|<code>U+1EEBB</code>
|style="font-size: 105%"|{{not a typo|𞺻}}
|
|}

In addition, a blackboard-bold [[mu (letter)|μ]]<sub>''n''</sub> (not found in Unicode or {{code|amsmath}} LaTeX) is sometimes used by [[number theory|number theorists]] and [[algebraic geometry|algebraic geometers]] to designate the [[group scheme]] of {{nowrap|''n''-th}} [[roots of unity]].{{r|milne}}

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

==See also==
* [[Latin letters used in mathematics, science, and engineering]]
*[[Mathematical alphanumeric symbols]]
*[[Set notation]]

==References==
{{Reflist|30em|refs=

<ref name=amstex>{{cite journal |title=Mathematical Symbols and Cyrillic Fonts Ready for Distribution |last=Beeton |first=Barbara |year=1985 |journal=TUGboat |volume=6 |number=2 |pages=59–63 |url=https://tug.org/TUGboat/tb06-2/tb12beet.pdf }} {{pb}} {{cite book |last=Spivak |first=Michael |year=1986 |title=The Joy of TeX: A Gourmet Guide to Typesetting with the AMS-TeX Macro Package |publisher=American Mathematical Society |page=260 |url=https://archive.org/details/joyoftexgourmetg0000spiv/page/260/mode/1up |url-access=limited }} {{pb}} {{cite journal |title=Coming in January from the American Mathematical Society |year=1989 |journal=TUGboat |volume=10 |number=3 |pages=365–366 |url=https://tug.org/TUGboat/tb10-3/tb25ams.pdf }} {{pb}} {{cite web |last=Beeton |first=Barbara |date=2020-09-05 |title=Re: Who designed the mathematical blackboard bold letters of AMS, and when?  |url=https://tex.stackexchange.com/a/561430 |website=TeX–LaTeX Stack Exchange |access-date=2023-07-27 |quote=The [1985] blackboard bold letters [...] are blocky in appearance, somewhat similar to those in the Monotype blackboard bold, but of much lower quality. (It's no surprise that Knuth did not like them.) }}</ref>

<ref name=doublestroke>{{cite web |last=Kummer |first=Olaf |year=2006 |title=doublestroke – Typeset mathematical double stroke symbols |url=https://www.ctan.org/pkg/doublestroke |website=[[CTAN|Comprehensive TeX Archive Network]] |access-date=2023-07-27 }}</ref>

<ref name=gilreath>{{cite journal |last=Gilreath |first=Charles T. |title=Graphic cueing of text: The typographic and diagraphic dimensions |journal=Visible Language |volume=27 |number=3 |year=1993 |pages=336–361 |url=https://journals.uc.edu/index.php/vl/article/view/5585 }}</ref>

<ref name=gunning>{{cite book |last=Gunning |first=Robert C. |author-link=Robert C. Gunning |year=1966 |title= Lectures on Riemann Surfaces |series=Mathematical Notes |publisher= Princeton University Press |page=1 |url=https://archive.org/details/lecturesonrieman0000unse/page/1/mode/1up }}</ref>

<ref name=gunning-rossi>{{cite book |last1=Gunning |first1=Robert C. |author1-link=Robert Gunning (mathematician) |last2=Rossi |first2=Hugo |author2-link=Hugo Rossi |year=1965 |title=Analytic functions of several complex variables |publisher=Prentice-Hall }}</ref>

<ref name=iso>{{cite book |title=''ISO 80000-2'' Quantities and Units: Mathematics |edition=2nd |date=August 2019 |url=https://www.iso.org/standard/64973.html |publisher=[[International Organization for Standardization]] |chapter=7. Standard number sets and intervals |at=Table 3, No. 2-7.4 }}</ref>

<ref name=karch>{{cite book |last=Karch |first=R. Randolph |title=Graphic Arts Procedures |year=1970 |publisher=American Technical Society |page=199 |url=https://archive.org/details/graphicartsproce0000unse/page/199/mode/1up }}</ref>

<ref name=knuth>{{cite book |last=Knuth |first=Donald |author-link=Donald Knuth |year=1984 |publisher=Addison-Wesley |title=The TeXbook |url=https://archive.org/details/texbook00dona/page/460/mode/1up |page=460 |url-access=limited }}</ref>

<ref name=krantz>{{cite book |last=Krantz |first=S. |title=Handbook of Typography for the Mathematical Sciences |chapter=2.8 Technical Issues |publisher=Chapman & Hall/CRC |year=2001 |page=35 |isbn=9781584881490 }}</ref>

<ref name=loomis-sternberg>{{cite book |last1=Loomis |first1=Lynn Harold |author1-link=Lynn Harold Loomis |last2=Sternberg |first2=Shlomo |author2-link=Shlomo Sternberg |year=1968 |title=Advanced Calculus |publisher=Addison Wesley |page=241 }} The later revised edition [https://people.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf is available from Sternberg's website].</ref>

<ref name=manuscripts>{{cite book |editor1-last=Hodgman |editor1-first=Charles D. 
|editor2-last=Selby |editor2-first=Samuel M. |editor3-last=Weast |editor3-first=Robert C. |title=C.R.C. Standard Mathematical Tables |publisher=Chemical Rubber Publishing Company |year=1959 |edition=12th |url=https://archive.org/details/crcstandardmathe12edunse/page/494/mode/1up |page=494 |url-access=limited }} {{pb}} {{cite book |last1=Chaundy |first1=Theodore W. |last2=Barrett |first2=P.R. |last3=Batey |first3=Charles |year=1954 |title=The Printing of Mathematics |publisher=Oxford University Press |page=52 |quote=The sign for bold type is a wavy line beneath the words or symbols in question; for security the word 'bold' may be added in the margin. |url = https://archive.org/details/printingofmathem0000chau/page/52/ |url-access=limited }}</ref>

<ref name=messer>{{cite journal |title=Blackboard Bold |last=Messer |first=Robert |year=1988 |journal=TUGboat |volume=9 |number=1 |pages=19–20 |url=https://www.tug.org/TUGboat/tb09-1/tb20messer.pdf }}</ref>

<ref name=milne>{{cite book | last=Milne | first=James S. | title=Étale cohomology | publisher=Princeton University Press | year=1980 | pages=xiii, 66 }}</ref>

<ref name=narasimhan>{{cite book |last=Narasimhan |first=Raghavan |author-link=Raghavan Narasimhan |year=1966 |title=Introduction to the Theory of Analytic Spaces |series=Lecture Notes in Mathematics |volume=25 |publisher=Springer |page=9 |doi= 10.1007/bfb0077071 |isbn=978-3-540-03608-1 }}</ref>

<ref name=pakin>{{cite book |last1=Pakin |first1=Scott |title=The Comprehensive LATEX Symbol List |date=25 June 2020 |url=https://mirror-hk.koddos.net/CTAN/info/symbols/comprehensive/symbols-a4.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://mirror-hk.koddos.net/CTAN/info/symbols/comprehensive/symbols-a4.pdf |archive-date=2022-10-09 |url-status=live}}</ref>

<ref name=rosendorf>{{cite book |last=Rosendorf |first=Theodore |year=2009 |title=The Typographic Desk Reference |publisher=Oak Knoll Press |pages=89–90 |url=https://archive.org/details/typographicdeskr0000rose/page/89/mode/1up |url-access=limited }}</ref>

<ref name=rudolph>{{cite newsgroup |last=Rudolph |first=Lee |title=Re: History of blackboard bold? |date=2003-10-06 |newsgroup=compt.text.tex | url =https://groups.google.com/g/comp.text.tex/c/Vb0DuAgmPO4 |access-date=2023-07-25 |archive-url=https://web.archive.org/web/20210923005736/https://groups.google.com/g/comp.text.tex/c/Vb0DuAgmPO4 |archive-date=2021-09-23 |url-status=live }} {{pb}} This usenet post (as [https://web.archive.org/web/20190203030359/http://mathforum.org/kb/message.jspa?messageID=515850 mirrored by The Math Forum]) seems to have been one of the sources for {{harvnb|Webb|2018}}; see [https://archive.org/details/webb-stephen-clash-of-symbols-a-ride-through-the-riches-of-glyphs/page/233/mode/1up {{p.|233}}]</ref>

<ref name=serre>Example Serre lecture: [http://www.dailymotion.com/video/xf88g3_jean-pierre-serre-writing-mathemati_tech "Writing Mathematics Badly" video talk (part 3/3)], starting at 7&prime;08&Prime; {{pb}} Example Serre book: {{cite book |last=Serre |first=Jean-Pierre |author-link=Jean-Pierre Serre |title=Cohomologie galoisienne |publisher=Springer |year=1994}}</ref>

<ref name=sevryuk>
{{cite web
|last=Sevryuk |first=Mikhail B. 
|title=Writing on a computer: some discouraging experiences.
|date=1998-12-02
|url=https://www.pdmi.ras.ru/~arnsem/papers/
|website=Arnold's Mathematical Seminar
}}</ref>

<ref name=typography>{{cite book 
|last=Bringhurst |first=Robert |year=1992 |title=Elements of Typographic Style |chapter=Glossary of Typographic Terms |publisher=Hartley & Marks |isbn=0-88179-033-8 |quote=''Inline'': A letter in which the inner portions of the main strokes have been carved away, leaving the edges more or less intact. Inline faces lighten the color while preserving the shapes and proportions of the original face. |page=234 |chapter-url=https://archive.org/details/elementsoftypogr0000brin_z4a6/page/234/mode/1up |chapter-url-access=limited }} {{pb}}
{{cite book |last=Hutchings |first=R.S. |year=1965 |title=A Manual of Decorated Typefaces |publisher=Hastings House |chapter=Inlines and Outlines |pages=10–11 |chapter-url=https://archive.org/details/manualofdecorate0000rshu/page/10/ |chapter-url-access=limited }} {{pb}}
{{cite book |last=Consuegra |first=David |year=2004 |chapter=|title=American Type: Design & Designers |publisher=Allworth Press |url=https://archive.org/details/americantypedesi0000cons/ |url-access=limited |at=[https://archive.org/details/americantypedesi0000cons/page/280/mode/1up "Handtooled typefaces"], {{p.|280}}; [https://archive.org/details/americantypedesi0000cons/page/282/mode/1up "Inline typefaces"], {{p.|282}}; [https://archive.org/details/americantypedesi0000cons/page/286/mode/1up "Open face typefaces"], {{p.|286–287}} }}</ref>

<ref name=unicode>{{cite book |editor1-last=Aliprand |editor1-first=Joan |editor2-last=Allen |editor2-first=Julie |editor3-last=Becker |editor3-first=Joe |editor4-last=Davis |editor4-first=Mark |display-editors=2 |year=2003 |title=The Unicode Standard, Version 4.0 |pages=354–357 |chapter=Math Alphanumeric Symbols: U+1D400–U+1D7FF |publisher=Addison-Wesley |chapter-url=https://archive.org/details/unicodestandard0000unse_p4z3/page/354/ |chapter-url-access=limited }}</ref>

<ref name=vieth>{{cite journal |last=Vieth |first=Ulrik |year=2012 |title=OpenType math font development: Progress and challenges |journal=TUGboat |volume=33 |number=3 |pages=302–308 |url=https://www.tug.org/TUGboat/tb33-3/tb105vieth.pdf |quote=Design choices of Blackboard Bold alphabets again fall into multiple groups. One group favors a serif design which is derived from the main serif font: [...] Another group favor a sans-serif design which may be unrelated to the main sans-serif font: [...] Finally, the designs of individual letters can vary significantly among different math fonts, and are an additional consideration in font choice. For example, some users may have fairly strong preferences regarding such details as to whether the stem or the diagonal of the letter 'N' is double-struck. }}</ref>

<ref name=waite-arca>An example of double-struck type produced by an [[impact printing|impact printer]] of the early 1980s can be found in: {{pb}} {{cite book |last1=Waite |first1=Mitchell |last2=Arca |first2=Julie |year=1982 |title=Word Processing Primer |publisher=BYTE/McGraw-Hill |pages=76–77 |url=https://archive.org/details/wordprocessingpr0000wait/page/76/mode/1up |url-access=limited }}</ref>

<ref name=webb>{{cite book |last=Webb |first=Stephen |year=2018 |chapter=Set of Natural Numbers ℕ |title=Clash Of Symbols: A Ride Through The Riches Of Glyphs |publisher=Springer |pages=198–199, 233 |chapter-url=https://archive.org/details/webb-stephen-clash-of-symbols-a-ride-through-the-riches-of-glyphs/page/198/mode/1up |chapter-url-access=limited }}</ref>

<ref name=weisstein>{{Cite web |last=Weisstein |first=Eric W. |title=Doublestruck |url=https://mathworld.wolfram.com/Doublestruck.html |access-date=2022-12-21 |website=mathworld.wolfram.com |language=en}}</ref>

<ref name=wiley>{{cite book |year=1979 |title=A guide for Wiley-Interscience and Ronald Press Authors in the Preparation and Production of Manuscript and Illustrations |edition=2nd |publisher=John Wiley & Sons |url=https://archive.org/details/guideforwileyint0000john/page/67/mode/1up |url-access=limited }}</ref>

<ref name=w3c>{{Cite tech report
|last1=Carlisle |first1=David
|last2=Ion |first2=Patrick
|chapter-url=https://www.w3.org/TR/xml-entity-names/double-struck.html
|chapter=Double Struck (Open Face, Blackboard Bold)
|title=XML Entity Definitions for Characters |edition=3rd |year=2023
|publisher=World Wide Web Consortium
|access-date=2023-07-27
|quote=Note: Characters highlighted [in yellow] are in the Plane 0 [Basic Multilingual Plane], not in the Mathematical Alphanumeric Symbols block in Plane 1.
}}</ref>

<!-- end refs --> }}

{{Typography terms}}
{{MathematicalSymbolsNotationLanguage}}

[[Category:Mathematical notation]]
[[Category:Mathematical typefaces]]