# Blackboard bold explained

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 denote number sets. Blackboard bold symbols are also referred to as double struck, although they cannot actually be produced by double striking on a typewriter. In some texts these symbols are simply shown in bold; 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.

The symbols are nearly universal in their interpretation, unlike their normally-typeset counterparts, which are used for many different purposes.

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 (but not universal) usage in mathematical texts.

LaTeXUnicodeSymbolMathematics usage

A

U+1D538>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). It may also represent the algebraic integers, an important subring of the algebraic numbers.

B

U+1D539>Sometimes represents a ball, a boolean domain, or the Brauer group of a field.

C

U+2102>Represents the complex numbers.

D

U+1D53B>Represents the unit disk in the complex plane, or the decimal fractions (see number).

E

U+1D53C>Represents the expected value of a random variable, or Euclidean space.

F

U+1D53D>Represents a field. Often used for finite fields, with a subscript to indicate the order. Also represents a Hirzebruch surface.

G

U+1D53E>Represents a Grassmannian or a group, especially an algebraic group.

H

U+210D>Represents the quaternions (the H stands for Hamilton), or the upper half-plane, or hyperbolic space, or hyperhomology of a complex.

J

U+1D541>Sometimes represents the irrational numbers, R\Q.

K

U+1D542>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.

L

U+1D543>Represents the Lefschetz motive. See motives.

N

U+2115>Represents the natural numbers. May or may not include zero.

O

U+1D546>Represents the octonions.

P

U+2119>Represents projective space, the probability of an event, the prime numbers, a power set, the positive reals, or a forcing poset.

Q

U+211A>Represents the rational numbers. (The Q stands for quotient.)

R

U+211D>Represents the real numbers.

S

U+1D54A>Represents the sedenions, or a sphere.

T

U+1D54B>Represents a torus, or the circle group or a Hecke algebra (Hecke denoted his operators as Tn.)

W

U+1D54E>Represents the whole numbers, which also are represented by N0.

Z

U+2124>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, or occasionally for the identity matrix in a matrix ring.