# Blackboard bold

[itex]\mathbb{B}\mathbb{O}\mathbb{L}\mathbb{D}[itex]

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

Note that PNZQRCHOS.

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