Blackboard bold

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 near-vertical lines) are doubled. The symbols usually denote number sets. One way of producing blackboard bold is to double-strike a character with a small offset on a typewriter. Thus they are also referred to as double struck.

An example of blackboard bold letters

In typography, such a font with characters that are not solid is called an "inline", "shaded", or "tooled" font.

History

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 non-bold letters (i.e., by using the edge rather than the point of a chalk). It then made its way back into print form as a separate style from ordinary bold, possibly starting with the original 1965 edition of Gunning and Rossi's textbook on complex analysis.[1][2]

Use in textbooks

In the 1960s and 1970s, blackboard bold spread quickly in classrooms; it is now widely used in the English- and French-speaking worlds. In textbooks, however, the situation is not so clear cut: many mathematicians adopted blackboard bold, while many others still prefer to use bold.

Well-known books that have used blackboard bold style include Lindsay Childs's "A Concrete Introduction to Higher Algebra",[3] which is widely used as a text for undergraduate courses in the U.S., John Stillwell's "Elements of Number Theory",[4] and Edward Barbeau's "University of Toronto Mathematics Competition (2001-2015)",[5] which is often used to prepare for mathematics competitions.

Jean-Pierre Serre used double-struck letters when writing bold on the blackboard,[6] whereas his published works (e.g., his well-known "Cohomologie galoisienne"[7]) have consistently used ordinary bold for the same symbols.

Donald Knuth also preferred boldface to blackboard bold, and consequently did not include blackboard bold in the Computer Modern fonts that he created for the TeX mathematical typesetting system.[8]

Serge Lang used blackboard bold in his famous "Algebra",[9] which was widely used as a text for graduate courses in the U.S. for at least two decades.

The Chicago Manual of Style has evolved over this issue. In 1993, for the 14th edition, it advised: "blackboard bold should be confined to the classroom" (13.14). Later, in 2003, for the 15th edition, it stated that "open-faced (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 add-on AMS Fonts package (amsfonts) by the American Mathematical Society provides this facility (e.g., $\mathbb {R}$ 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 (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 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).

Usage

The following table shows all available Unicode blackboard bold characters.[10]

The symbols are nearly universal in their interpretation, unlike their normally-typeset 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 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 font). The fourth column describes known typical (but not universal) usage in mathematical texts.

	Unicode Code Point (Hex)	Unicode Symbol	Mathematics usage
$\mathbb {A}$	`U+1D538`	𝔸	Represents affine space or the ring of adeles. Occasionally represents the algebraic numbers, the algebraic closure of ℚ (more commonly written ℚ or Q), or the algebraic integers, an important subring of the algebraic numbers.
	`U+1D552`	𝕒
$\mathbb {B}$	`U+1D539`	𝔹	Sometimes represents a ball, a boolean domain, or the Brauer group of a field.
	`U+1D553`	𝕓
$\mathbb {C}$	`U+2102`	ℂ	Represents the set of complex numbers.
	`U+1D554`	𝕔
$\mathbb {D}$	`U+1D53B`	𝔻	Represents the unit (open) disk in the complex plane (and by generalisation 𝔻ⁿ may mean the n-dimensional ball) — for example as a model of the Hyperbolic plane and the domain of discourse. Occasionally 𝔻 may mean the decimal fractions (see number) or split-complex numbers.
	`U+1D555`	𝕕
$D\!\!\!\!D$	`U+2145`	ⅅ
$\,d\!\!\!\!d$	`U+2146`	ⅆ	May represent the differential symbol.
$\mathbb {E}$	`U+1D53C`	𝔼	Represents the expected value of a random variable, or Euclidean space, or a field in a tower of fields, or the Eudoxus reals.
	`U+1D556`	𝕖
$e\!\!e$	`U+2147`	ⅇ	Occasionally used for the mathematical constant $e$ .
$\mathbb {F}$	`U+1D53D`	𝔽	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`	𝕗
$\mathbb {G}$	`U+1D53E`	𝔾	Represents a Grassmannian or a group, especially an algebraic group.
	`U+1D558`	𝕘
$\mathbb {H}$	`U+210D`	ℍ	Represents the quaternions (the H stands for Hamilton), or the upper half-plane, or hyperbolic space, or hyperhomology of a complex.
	`U+1D559`	𝕙
$\mathbb {I}$	`U+1D540`	𝕀	The closed unit interval or the ideal of polynomials vanishing on a subset. Occasionally the identity mapping on an algebraic structure, or an indicator function, or the set of imaginary numbers (i.e., the set of all real multiples of the imaginary unit, more commonly indicated iℝ)
	`U+1D55A`	𝕚
$i\!i$	`U+2148`	ⅈ	Occasionally used for the imaginary unit.
$\mathbb {J}$	`U+1D541`	𝕁	Occasionally represents the set of irrational numbers, R\Q (ℝ\ℚ).
	`U+1D55B`	𝕛
$j\!\!j$	`U+2149`	ⅉ
$\mathbb {K}$	`U+1D542`	𝕂	Represents a field, typically a scalar field. This is derived from the German word Körper, which is German for field (literally, "body"; cf. the French term corps). May also be used to denote a compact space.
$\mathbb {k}$	`U+1D55C`	𝕜
$\mathbb {L}$	`U+1D543`	𝕃	Represents the Lefschetz motive. See Motive (algebraic geometry).
	`U+1D55D`	𝕝
$\mathbb {M}$	`U+1D544`	𝕄	Sometimes represents the monster group. The set of all m-by-n matrices is sometimes denoted 𝕄(m, n).
	`U+1D55E`	𝕞
$\mathbb {N}$	`U+2115`	ℕ	Represents the set of natural numbers. May or may not include zero.
	`U+1D55F`	𝕟
$\mathbb {O}$	`U+1D546`	𝕆	Represents the octonions.
	`U+1D560`	𝕠
$\mathbb {P}$	`U+2119`	ℙ	Represents projective space, the probability of an event, the prime numbers, a power set, the irrational numbers, or a forcing poset.
	`U+1D561`	𝕡
$\mathbb {Q}$	`U+211A`	ℚ	Represents the set of rational numbers. (The Q stands for quotient.)
	`U+1D562`	𝕢
$\mathbb {R}$	`U+211D`	ℝ	Represents the set of real numbers. $\mathbb {R} _{>0}$ represents the positive reals, while $\mathbb {R} _{\geq 0}$ represents the non-negative real numbers.
	`U+1D563`	𝕣
$\mathbb {S}$	`U+1D54A`	𝕊	Represents a sphere, or the sphere spectrum, or occasionally the sedenions.
	`U+1D564`	𝕤
$\mathbb {T}$	`U+1D54B`	𝕋	Represents the circle group, particularly the unit circle in the complex plane (and 𝕋ⁿ the n-dimensional torus), or a Hecke algebra (Hecke denoted his operators as T_n or 𝕋_𝕟), or the tropical semi-ring, or twistor space.
	`U+1D565`	𝕥
$\mathbb {U}$	`U+1D54C`	𝕌
	`U+1D566`	𝕦
$\mathbb {V}$	`U+1D54D`	𝕍	Represents a vector space or an affine variety generated by a set of polynomials.
	`U+1D567`	𝕧
$\mathbb {W}$	`U+1D54E`	𝕎	Occasionally represents the set of whole numbers (here in the sense of non-negative integers), which also are represented by ℕ₀.
	`U+1D568`	𝕨
$\mathbb {X}$	`U+1D54F`	𝕏	Occasionally used to denote an arbitrary metric space.
	`U+1D569`	𝕩
$\mathbb {Y}$	`U+1D550`	𝕐
	`U+1D56A`	𝕪
$\mathbb {Z}$	`U+2124`	ℤ	Represents the set of integers. (The Z is for Zahlen, German for "numbers", and zählen, German for "to count".)
	`U+1D56B`	𝕫

	`U+213E`	ℾ
	`U+213D`	ℽ
	`U+213F`	ℿ
	`U+213C`	ℼ
	`U+2140`	⅀

	`U+1D7D8`	𝟘
	`U+1D7D9`	𝟙	Often represents, in set theory, the top element of a forcing 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 operator or identity matrix.
	`U+1D7DA`	𝟚	Often represents, in category 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

In addition, a blackboard-bold μ_n (not found in Unicode) is sometimes used by number theorists and algebraic geometers to designate the group scheme of nth roots of unity.[11]

References

Gunning, Robert C.; Rossi, Hugo (1965). Analytic functions of several complex variables. Prentice-Hall.
Rudolph, Lee (Oct 5, 2003). "Re: History of blackboard bold?".
Childs, Lindsay N. (2009). A Concrete Introduction to Higher Algebra (3rd ed.). Springer.
Stillwell, John (2003). Elements of Number Theory. Springer.
Barbeau, Edward J. (2016). University of Toronto Mathematics Competition (2001-2015). Springer.
"Writing Mathematics Badly" video talk (part 3/3), starting at 7′08″
Serre, Jean-Pierre (1994). Cohomologie galoisienne. Springer.
Krantz, S. (2001). Handbook of Typography for the Mathematical Sciences. Chapman & Hall/CRC. p. 35.
Lang, Serge (2002). Algebra (revised 3rd ed.). Springer.
"Double Struck (Open Face, Blackboard Bold), which shows blackboard bold symbols together with their Unicode encodings. Encodings in the Basic Multilingual Plane are highlighted in yellow". www.w3.org. Retrieved 2016-01-01.
Milne, James S. (1980). Étale cohomology. Princeton University Press. pp. xiii, 66.

Bibliography

Weisstein, Eric W. "Doublestruck". MathWorld.

External links

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[1] Gunning, Robert C.; Rossi, Hugo (1965). Analytic functions of several complex variables. Prentice-Hall.

[2] Rudolph, Lee (Oct 5, 2003). "Re: History of blackboard bold?".

[3] Childs, Lindsay N. (2009). A Concrete Introduction to Higher Algebra (3rd ed.). Springer.

[4] Stillwell, John (2003). Elements of Number Theory. Springer.

[5] Barbeau, Edward J. (2016). University of Toronto Mathematics Competition (2001-2015). Springer.

[6] "Writing Mathematics Badly" video talk (part 3/3), starting at 7′08″

[7] Serre, Jean-Pierre (1994). Cohomologie galoisienne. Springer.

[8] Krantz, S. (2001). Handbook of Typography for the Mathematical Sciences. Chapman & Hall/CRC. p. 35.

[9] Lang, Serge (2002). Algebra (revised 3rd ed.). Springer.

[10] "Double Struck (Open Face, Blackboard Bold), which shows blackboard bold symbols together with their Unicode encodings. Encodings in the Basic Multilingual Plane are highlighted in yellow". www.w3.org. Retrieved 2016-01-01.

[11] Milne, James S. (1980). Étale cohomology. Princeton University Press. pp. xiii, 66.