# Blackboard bold explained

Blackboard bold is a typeface style that is 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 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.

The Chicago Manual of Style in 1993 (14th edition) advises: "[b]lackboard bold should be confined to the classroom" (13.14) whereas in 2003 (15th edition) it states that "[o]pen-faced (blackboard) symbols are reserved for familiar systems of numbers" (14.12).

In some texts these symbols are simply shown in bold type: 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, and then made its way back in print form as a separate style from ordinary bold,[1] possibly starting with the original 1965 edition of Gunning and Rossi's textbook on complex analysis.[2] Some mathematicians, therefore, do not recognize blackboard bold as a separate style from bold: Jean-Pierre Serre, for example, has publicly inveighed against the use of "blackboard bold" anywhere other than on a blackboard, and uses double-struck letters when writing bold on the blackboard,[3] whereas his published works consistently use ordinary bold for the same symbols.[4] Donald Knuth also advises against the use of blackboard bold in print.[5]

It is sometimes erroneously claimed[1] that Bourbaki introduced the blackboard bold notation, but whereas individual members of the Bourbaki group may have popularized double-striking bold characters on the blackboard, their printed books use ordinary bold.[6]

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

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 all available Unicode blackboard bold characters.

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 known typical (but not universal) usage in mathematical texts.

Unicode (Hex)SymbolMathematics usage

A

U+1D538Represents affine space or the ring of adeles. Sometimes represents the algebraic numbers, the algebraic closure of Q (or , although Q is often used instead). It may also represent the algebraic integers, an important subring of the algebraic numbers.
U+1D552

B

U+1D539Sometimes represents a ball, a boolean domain, or the Brauer group of a field.
U+1D553

C

U+2102Represents the complex numbers.
U+1D554

D

U+1D53BRepresents the unit (open) disk in the complex plane (for example as a model of the Hyperbolic plane), or the decimal fractions (see number).
U+1D555

D\

\!\DU+2145

d\

\!\dU+2146May represent the differential symbol.

E

U+1D53CRepresents the expected value of a random variable, or Euclidean space, or a field in a tower of fields.
U+1D556

e\

eU+2147Sometimes used for the mathematical constant e.

F

U+1D53DRepresents a field. Often used for finite fields, with a subscript to indicate the order. Also represents a Hirzebruch surface or a free group, with a subset to indicate the number of generators (or generating set, if infinite).
U+1D557

G

U+1D53ERepresents a Grassmannian or a group, especially an algebraic group.
U+1D558

H

U+210DRepresents the quaternions (the H stands for Hamilton), or the upper half-plane, or hyperbolic space, or hyperhomology of a complex.
U+1D559

I

U+1D540Occasionally used to denote the identity mapping on an algebraic structure, or the set of imaginary numbers (i.e., the set of all real multiples of the imaginary unit).
U+1D55A

ii

U+2148Occasionally used for the imaginary unit.

J

U+1D541Sometimes represents the irrational numbers, R\Q .
U+1D541

j\

jU+2149

K

U+1D542Represents a field, typically a scalar 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.
U+1D55C

L

U+1D543Represents the Lefschetz motive. See motives.
U+1D55D

M

U+1D544Represents the monster group.
U+1D55E

N

U+2115Represents the natural numbers. May or may not include zero.
U+1D55F

O

U+1D546Represents the octonions.
U+1D560

P

U+2119Represents projective space, the probability of an event, the prime numbers, a power set, the positive reals, the irrational numbers, or a forcing partially ordered set (poset).
U+1D561

Q

U+211ARepresents the rational numbers. (The Q stands for quotient.)
U+1D562

R

U+211DRepresents the real numbers.
U+1D563

S

U+1D54ARepresents the sedenions, or a sphere.
U+1D564

T

U+1D54BRepresents a torus, or the circle group or a Hecke algebra (Hecke denoted his operators as Tn or ), or the Tropical semi-ring.
U+1D565

U

U+1D54C
U+1D566

V

U+1D54DRepresents a vector space.
U+1D567

W

U+1D54ERepresents the whole numbers (here in the sense of non-negative integers), which also are represented by N0.
U+1D568

X

U+1D54FOccasionally used to denote an arbitrary metric space.
U+1D569

Y

U+1D550
U+1D56A

Z

U+2124Represents the integers. (The Z is for Zahlen, which is German for "numbers".)
U+1D56B
U+213E
U+213D
U+213F
U+213C
U+2140
U+1D7D8
U+1D7D9Often represents, in set theory, the top element of a forcing partially ordered set (poset), or occasionally for the identity matrix in a matrix ring.
U+1D7DA
U+1D7DB
U+1D7DC
U+1D7DD
U+1D7DE
U+1D7DF
U+1D7E0
U+1D7E1

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.[7]

Another option is to use the amssymb package. This will output a

R

and

Z

 \usepackage