Blackboard bold
![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, ℂ = {a + bi ∣ a, b ∈ ℝ, i² = −1}.](https://upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Blackboard_bold_on_a_blackboard.jpg/220px-Blackboard_bold_on_a_blackboard.jpg)
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(natural numbers),(integers),(rational numbers),(real numbers), and(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 inUnicodeglyphnames.
Intypography,a typeface with characters that are not solid is calledinline,handtooled,oropen face.[3]
History
[edit]![](https://upload.wikimedia.org/wikipedia/en/thumb/e/ef/Blackboard_bold_in_typewritten_notes_from_Gunning_%281966%29.jpg/280px-Blackboard_bold_in_typewritten_notes_from_Gunning_%281966%29.jpg)
![](https://upload.wikimedia.org/wikipedia/en/thumb/4/42/Blackboard_bold_in_typewritten_notes_from_Narasimhan_%281966%29.png/280px-Blackboard_bold_in_typewritten_notes_from_Narasimhan_%281966%29.png)
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".[6]
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.[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 Universitymathematics 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 as IR, 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]
![](https://upload.wikimedia.org/wikipedia/en/thumb/9/9f/Blackboard_bold_in_print_in_Loomis_and_Sternberg_%281968%29.jpg/280px-Blackboard_bold_in_print_in_Loomis_and_Sternberg_%281968%29.jpg)
The style made its way into print starting in the mid 1960s. Early examples includeRobert GunningandHugo Rossi'sAnalytic Functions of Several Complex Variables(1965)[12][10]andLynn LoomisandShlomo Sternberg'sAdvanced Calculus(1968).[11]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;[13]in 1982, Wiley refused to include blackboard bold characters in mathematical books because the type was difficult and expensive to obtain.[14]
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.[14]When Knuth's 1984The TeXbookneeded an example of blackboard bold for the index, he producedusing the lettersIandRwith a negative space between;[15]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.[16]
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\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]
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".[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 Serrehas 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 listsRas the symbol for the real numbers but notes "the symbolsIRandare also used ", and similarly forN,Z,Q,C,andP(prime numbers).[21]
Encoding
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Blackboard_bold_number_sets.svg/280px-Blackboard_bold_number_sets.svg.png)
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 theamssymb
package, which includes a blackboard bold typeface for uppercase Latin letters accessed using\mathbb
(e.g.\mathbb{R}
produces).[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.
Usage
[edit]The following table shows all available Unicode blackboard bold characters.[24]
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.[25]Some of the symbols (particularlyand) are nearly universal in their interpretation,[14]while others are more varied in use.
LaTeX | Unicode code point(hex) | Unicode symbol | Mathematics usage |
---|---|---|---|
U+1D538
|
𝔸 | Representsaffine space,,or thering of adeles.Occasionally represents thealgebraic numbers,[26]thealgebraic closureof(more commonly writtenorQ), or thealgebraic integers,an important subring of the algebraic numbers. | |
U+1D539
|
𝔹 | Sometimes represents aball,aboolean domain,or theBrauer groupof a field. | |
U+2102
|
ℂ | Represents thesetofcomplex numbers.[14] | |
U+1D53B
|
𝔻 | Represents theunit diskin thecomplex plane,for example as theconformal disk modelof thehyperbolic plane.By generalisationmay mean then-dimensionalball.Occasionallymay mean the decimal fractions (seenumber),split-complex numbers,ordomain of discourse. | |
U+1D53C
|
𝔼 | Represents theexpected valueof arandom variable,orEuclidean space,or afieldin atower of fields,or theEudoxus reals. | |
U+1D53D
|
𝔽 | Represents afield.[26]Often used forfinite fields,with a subscript to indicate the order.[26]Also represents aHirzebruch surfaceor afree group,with a subscript to indicate the number of generators (or generating set, if infinite). | |
U+1D53E
|
𝔾 | Represents aGrassmannianor agroup,especially analgebraic group. | |
U+210D
|
ℍ | Represents thequaternions(the H stands forHamilton),[26]or theupper half-plane,orhyperbolic space,[26]orhyperhomologyof a complex. | |
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). | |
U+1D541
|
𝕁 | Sometimes represents theirrational numbers,. | |
U+1D542
|
𝕂 | Represents afield.[26]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. | |
U+1D543
|
𝕃 | Represents the Lefschetz motive. SeeMotive (algebraic geometry). | |
U+1D544
|
𝕄 | Sometimes represents themonster group.The set of allm-by-nmatricesis sometimes denoted.Ingeometric algebra,represents the motor group of rigid motions. Infunctional programmingandformal semantics,denotes the type constructor for amonad. | |
U+2115
|
ℕ | Represents the set ofnatural numbers.[21]May or may not includezero. | |
U+1D546
|
𝕆 | Represents theoctonions.[26] | |
U+2119
|
ℙ | Representsprojective space,theprobabilityof an event,[26]theprime numbers,[21]apower set,the positive reals, theirrational numbers,or aforcingposet. | |
U+211A
|
ℚ | Represents the set ofrational numbers.[14](The Q stands forquotient.) | |
U+211D
|
ℝ | Represents the set ofreal numbers.[14] | |
U+1D54A
|
𝕊 | Represents asphere,or thesphere spectrum,or occasionally thesedenions. | |
U+1D54B
|
𝕋 | Represents thecircle group,particularly theunit circlein the complex plane (andthen-dimensionaltorus), or aHecke algebra(Hecke denoted his operators asTnor), or thetropical semiring,ortwistor space. | |
U+1D54C
|
𝕌 | ||
U+1D54D
|
𝕍 | Represents avector spaceor anaffine varietygenerated by a set of polynomials, or in probability theory and statistics thevariance. | |
U+1D54E
|
𝕎 | Represents thewhole numbers(here in the sense of non-negative integers), which also are represented by. | |
U+1D54F
|
𝕏 | Occasionally used to denote an arbitrarymetric space. | |
U+1D550
|
𝕐 | ||
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 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 | ||
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-boldμn(not found in Unicode oramsmath
LaTeX) is sometimes used bynumber theoristsandalgebraic geometersto designate thegroup schemeofn-throots of unity.[27]
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
[edit]- Latin letters used in mathematics, science, and engineering
- Mathematical Alpha numeric symbols
- Set notation
References
[edit]- ^Gilreath, Charles T. (1993)."Graphic cueing of text: The typographic and diagraphic dimensions".Visible Language.27(3): 336–361.
- ^Rosendorf, Theodore (2009).The Typographic Desk Reference.Oak Knoll Press. pp. 89–90.
- ^Bringhurst, Robert (1992)."Glossary of Typographic Terms".Elements of Typographic Style.Hartley & Marks. p. 234.ISBN0-88179-033-8.
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.
Hutchings, R.S. (1965)."Inlines and Outlines".A Manual of Decorated Typefaces.Hastings House. pp. 10–11.Consuegra, David (2004).American Type: Design & Designers.Allworth Press."Handtooled typefaces",p. 280;"Inline typefaces",p. 282;"Open face typefaces",p. 286–287.
- ^Gunning, Robert C.(1966).Lectures on Riemann Surfaces.Mathematical Notes. Princeton University Press. p. 1.
- ^Narasimhan, Raghavan(1966).Introduction to the Theory of Analytic Spaces.Lecture Notes in Mathematics. Vol. 25. Springer. p. 9.doi:10.1007/bfb0077071.ISBN978-3-540-03608-1.
- ^Hodgman, Charles D.; Selby, Samuel M.; Weast, Robert C., eds. (1959).C.R.C. Standard Mathematical Tables(12th ed.). Chemical Rubber Publishing Company. p. 494.Chaundy, Theodore W.; Barrett, P.R.; Batey, Charles (1954).The Printing of Mathematics.Oxford University Press. p. 52.
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.
- ^Karch, R. Randolph (1970).Graphic Arts Procedures.American Technical Society. p. 199.
- ^abcdWebb, Stephen (2018)."Set of Natural Numbers ℕ".Clash Of Symbols: A Ride Through The Riches Of Glyphs.Springer. pp. 198–199, 233.
- ^An example of double-struck type produced by animpact printerof the early 1980s can be found in:Waite, Mitchell; Arca, Julie (1982).Word Processing Primer.BYTE/McGraw-Hill. pp. 76–77.
- ^abcRudolph, Lee (2003-10-06)."Re: History of blackboard bold?".Newsgroup:compt.text.tex.Archivedfrom the original on 2021-09-23.Retrieved2023-07-25.This usenet post (asmirrored by The Math Forum) seems to have been one of the sources forWebb 2018;seep. 233
- ^abLoomis, Lynn Harold;Sternberg, Shlomo(1968).Advanced Calculus.Addison Wesley. p. 241.The later revised editionis available from Sternberg's website.
- ^Gunning, Robert C.;Rossi, Hugo(1965).Analytic functions of several complex variables.Prentice-Hall.
- ^A guide for Wiley-Interscience and Ronald Press Authors in the Preparation and Production of Manuscript and Illustrations(2nd ed.). John Wiley & Sons. 1979.
- ^abcdefghKrantz, S. (2001). "2.8 Technical Issues".Handbook of Typography for the Mathematical Sciences.Chapman & Hall/CRC. p. 35.ISBN9781584881490.
- ^Knuth, Donald(1984).The TeXbook.Addison-Wesley. p. 460.
- ^Messer, Robert (1988)."Blackboard Bold"(PDF).TUGboat.9(1): 19–20.
- ^Beeton, Barbara (1985)."Mathematical Symbols and Cyrillic Fonts Ready for Distribution"(PDF).TUGboat.6(2): 59–63.Spivak, Michael (1986).The Joy of TeX: A Gourmet Guide to Typesetting with the AMS-TeX Macro Package.American Mathematical Society. p. 260."Coming in January from the American Mathematical Society"(PDF).TUGboat.10(3): 365–366. 1989.Beeton, Barbara (2020-09-05)."Re: Who designed the mathematical blackboard bold letters of AMS, and when?".TeX–LaTeX Stack Exchange.Retrieved2023-07-27.
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.)
- ^Vieth, Ulrik (2012)."OpenType math font development: Progress and challenges"(PDF).TUGboat.33(3): 302–308.
Design choices of Blackboard Bold Alpha bets 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.
- ^Aliprand, Joan; Allen, Julie; et al., eds. (2003)."Math Alphanumeric Symbols: U+1D400–U+1D7FF".The Unicode Standard, Version 4.0.Addison-Wesley. pp. 354–357.
- ^Example Serre lecture:"Writing Mathematics Badly" video talk (part 3/3),starting at 7′08″Example Serre book:Serre, Jean-Pierre(1994).Cohomologie galoisienne.Springer.
- ^abc"7. Standard number sets and intervals".ISO 80000-2Quantities and Units: Mathematics(2nd ed.).International Organization for Standardization.August 2019. Table 3, No. 2-7.4.
- ^Kummer, Olaf (2006)."doublestroke – Typeset mathematical double stroke symbols".Comprehensive TeX Archive Network.Retrieved2023-07-27.
- ^Pakin, Scott (25 June 2020).The Comprehensive LATEX Symbol List(PDF).Archived(PDF)from the original on 2022-10-09.
- ^Carlisle, David; Ion, Patrick (2023)."Double Struck (Open Face, Blackboard Bold)".XML Entity Definitions for Characters(Technical report) (3rd ed.). World Wide Web Consortium.Retrieved2023-07-27.
Note: Characters highlighted [in yellow] are in the Plane 0 [Basic Multilingual Plane], not in the Mathematical Alphanumeric Symbols block in Plane 1.
- ^Weisstein, Eric W."Doublestruck".mathworld.wolfram.Retrieved2022-12-21.
- ^abcdefgh Sevryuk, Mikhail B. (1998-12-02)."Writing on a computer: some discouraging experiences".Arnold's Mathematical Seminar.
- ^Milne, James S. (1980).Étale cohomology.Princeton University Press. pp. xiii, 66.