Blackboard bold

An example of blackboard bold letters.

Blackboard bold is a typeface style often used for certain symbols in mathematics and physics texts, in which certain lines of the symbol (usually vertical, or near-vertical lines) are doubled. The symbols usually describe number sets. Blackboard bold symbols are also referred to as double struck, although attempting to produce them by double striking on a typewriter is unlikely to give satisfactory results. The symbols are nearly universal in their interpretation, unlike their normally-typeset counterparts, which are constantly reused.

In some texts, these symbols are simply shown in bold, and 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.

It is frequently claimed that the symbols were first introduced by the group of mathematicians known as Nicolas Bourbaki. There are several reasons to doubt this claim: (1) the symbols do not appear in Bourbaki publications (rather, ordinary bold is used) at or near the era when they began to be used elsewhere, for instance, in typewritten lecture notes from Princeton University (achieved in some cases by overstriking R or C with I), and (an apparent first) typeset in Gunning and Rossi's textbook on several complex variables; (2) Jean-Pierre Serre, a member of the Bourbaki group, has publicly inveighed against the use of "blackboard bold" anywhere other than on a blackboard.

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; a blackboard bold R is written as \mathbb{R}.

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, 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). Being outside the BMP, these are relatively new and not widely supported.

Examples

The following table shows some of the more common uses of blackboard bold.

The first column shows the letter as typically rendered by the ubiquitous LaTeX markup system. The second column shows the Unicode codepoint. The third column shows the symbol itself (which will only display correctly if your browser supports Unicode and has access to a suitable font). The fourth column describes typical usage in mathematical texts.

LaTeX	Unicode	Symbol	Mathematics usage
$\mathbb {A}$	U+1D538	𝔸	Represents affine space or the ring of adeles. Sometimes represents the algebraic numbers, the algebraic closure of Q (although a Q with an overline is often used instead).
$\mathbb {B}$	U+1D539	𝔹	Represents a ball. Sometimes represents a boolean domain.
$\mathbb {C}$	U+2102	ℂ	Represents the complex numbers.
$\mathbb {D}$	U+1D53B	𝔻	Represents the unit disk in the complex plane, or the decimal fractions (see number).
$\mathbb {E}$	U+1D53C	𝔼	Represents the expected value of a random variable, or Euclidean space.
$\mathbb {F}$	U+1D53D	𝔽	Represents a field. Often used for finite fields, with a subscript to indicate the order. Also represents a Hirzebruch surface.
$\mathbb {G}$	U+1D53E	𝔾	Represents a Grassmannian or a group, especially an algebraic group.
$\mathbb {H}$	U+210D	ℍ	Represents the quaternions (the H stands for Hamilton), or the upper half-plane, or hyperbolic space, or hyperhomology of a complex.
$\mathbb {J}$	U+1D541	𝕁	Sometimes represents the irrational numbers, R\Q.
$\mathbb {K}$	U+1D542	𝕂	Represents a 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.
$\mathbb {L}$	U+1D543	𝕃	Represents the Lefschetz motive. See motives.
$\mathbb {N}$	U+2115	ℕ	Represents the natural numbers. May or may not include zero.
$\mathbb {O}$	U+1D546	𝕆	Represents the octonions.
$\mathbb {P}$	U+2119	ℙ	Represents projective space, the probability of an event, the prime numbers, a power set, the positive reals, or a forcing poset.
$\mathbb {Q}$	U+211A	ℚ	Represents the rational numbers. (The Q stands for quotient.)
$\mathbb {R}$	U+211D	ℝ	Represents the real numbers.
$\mathbb {S}$	U+1D54A	𝕊	Represents the sedenions, or a sphere.
$\mathbb {T}$	U+1D54B	𝕋	Represents a torus, or the circle group or a Hecke algebra (Hecke denoted his operators as Tn.)
$\mathbb {Z}$	U+2124	ℤ	Represents the integers. (The Z is for Zahlen, which is German for "numbers".)

A blackboard bold Greek letter mu (not found in Unicode) is sometimes used by number theorists and algebraic geometers (with a subscript n) to designate the group (or more specifically group scheme) of n-th roots of unity. A blackboard bold numeral 1 is often used in set theory for the top element of a forcing poset.

See also

• Mathematical alphanumeric symbols

External links

• http://www.w3.org/TR/MathML2/double-struck.html shows blackboard bold symbols together with their Unicode encodings. Encodings in the BMP are highlighted in yellow.