Blackboard bold

From Academic Kids

<math>\mathbb{B}\mathbb{O}\mathbb{L}\mathbb{D}<math>

An example of blackboard bold letters.

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 mathematicians 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) Jean-Pierre 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
<math>\mathbb{A}<math> U+1D538 Template:Unicode 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).
<math>\mathbb{B}<math> U+1D539 Template:Unicode Represents a ball.
<math>\mathbb{C}<math> U+2102 Template:Unicode Represents the complex numbers.
<math>\mathbb{D}<math> U+1D53B Template:Unicode Represents the unit disk in the complex plane.
<math>\mathbb{E}<math> U+1D53C Template:Unicode Represents the Expected value of a random variable.
<math>\mathbb{F}<math> U+1D53D Template:Unicode Represents a field. Often used for finite fields, with a subscript to indicate the order. Also represents a Hirzebruch surface.
<math>\mathbb{G}<math> U+1D53E Template:Unicode Represents a Grassmannian.
<math>\mathbb{H}<math> U+210D Template:Unicode Represents the quaternions (the H stands for Hamilton), or the upper half plane.
<math>\mathbb{J}<math> U+1D541 Template:Unicode Sometimes represents the irrational numbers, R\Q.
<math>\mathbb{K}<math> U+1D542 Template:Unicode Represents a field. This is derived from the German word Körper, which is German mathematical jargon for field (actually 'body').
<math>\mathbb{N}<math> U+2115 Template:Unicode Represents the natural numbers. May or may not include zero.
<math>\mathbb{O}<math> U+1D546 Template:Unicode Represents the octonions.
<math>\mathbb{P}<math> U+2119 Template:Unicode Represents projective space, the probability of an event, the prime numbers, or a power set.
<math>\mathbb{Q}<math> U+211A Template:Unicode Represents the rational numbers. (The Q stands for quotient.)
<math>\mathbb{R}<math> U+211D Template:Unicode Represents the real numbers.
<math>\mathbb{S}<math> U+1D54A Template:Unicode Represents the sedenions, or a sphere.
<math>\mathbb{T}<math> U+1D54B Template:Unicode Represents a torus.
<math>\mathbb{Z}<math> U+2124 Template:Unicode Represents the integers. (The Z is for Zahlen, which is German for "numbers".)

Note that PNZQRCHOS.

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