Home > Blackboard bold
Blackboard bold is a style of typeface 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 describe sets of numbers and are also referred to as double struck, although attempting to produce them by double striking on a typewriter is unlikely to give satisfactory results. It is frequently claimed that the symbols were first introduced by the group of mathematicans known as Nicolas Bourbaki. There are several reasons to doubt this claim: (1) the symbols do not 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) Serre,
a member of the Bourbaki group, has publically inveighed against the use
of "blackboard bold" anywhere other than ... on a blackboard.
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. Wikipedia, too, uses ordinary bold in place of blackboard bold, as browser support for the latter is far from universal.
TeX, the standard typesetting system for mathematical texts, does not contain direct support for blackboard bold symbols, but the add-on AMS Fonts package by the American Mathematical Society provides this facility; a blackboard bold R is written as \Bbb{R} in regular text and as \mathbb{R} in math mode.
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).
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 very new and not widely supported.
The following table shows some of the more common uses of blackboard bold.
The first column shows the letter as rendered by Wikipedia's 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
| Usage
|
|
| U+1D504
| 𝔄
| Represents affine space. Sometimes represents the algebraic numbers, the algebraic closure of Q (although a Q with an overline is often used instead).
|
|
| U+1D505
| 𝔅
| Represents a ballGeometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. Examples With the ordinary (Euclidean) metric, if the space is the line, the ball is an interval, and if the space is the plane, the ball is t.
|
|
| U+2102
| ℂ
| Represents the complex numberThe complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. The complex numbers contain a number , the imaginary unit with , i. is a square root of. Every complex number can be represented in the form , whers.
|
|
| U+1D507
| 𝔇
| Represents the unit disk in the complex plane.
|
|
| U+1D509
| 𝔉
| Represents a fieldIn abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are famil. Often used for finite fieldIn abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theos, with a subscript to indicate the order.
|
|
| U+210D
| ℍ
| Represents the quaternions (the H stands for HamiltonSir William Rowan Hamilton ( August 4, 1805 September 2, 1865) was an Irish mathematician, physicist, and astronomer. Hamilton's discovery of quaternions is his best known investigation. Hamilton also contributed to the development of optics, dynamics, an), or the upper half planeIn mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. It is the domain of many functions of interest in complex analysis, especially modular forms. It is also a model of the hyperbolic plane. There is a group action.
|
|
| U+1D50D
| 𝔍
| Sometimes represents the irrational numbers, R\Q.
|
|
| U+1D50E
| 𝔎
| Represents a field. This is derived from the German word Körper, which is German mathematical jargon for field (actually 'body').
|
|
| U+2115
| ℕ
| Represents the natural numbers. May or may not include 0.
|
|
| U+1D512
| 𝔒
| Represents the octonions.
|
|
| U+2119
| ℙ
| Represents projective space, the probability of an event, or the prime numbers.
|
|
| U+211A
| ℚ
| Represents the rational numbers. (The Q stands for quotient.)
|
|
| U+211D
| ℝ
| Represents the real numbers.
|
|
| U+1D516
| 𝔖
| Represents the sedenions, or a sphere.
|
|
| U+1D517
| 𝔗
| Represents a torus.
|
|
| U+2124
| ℤ
| Represents the integers. (The Z is for Zahlen, which is German for "numbers".)
|
Note that P ⊆ N ⊆ Z ⊆ Q ⊆ (A ∩ R) ⊆ R ⊆ C ⊆ H ⊆ O ⊆ S, and Q ⊆ A ⊆ C.
