Blackboard bold

	Email		Print
	Bookmark		Link

Blackboard bold

Blackboard bold is a typeface

Typeface

In typography, a typeface is a set of one or more fonts, in one or more sizes, designed with stylistic unity, each comprising a coordinated set of glyphs....
style often used for certain symbols in mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
and physics

Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
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

Typewriter

A typewriter is a Machine or electromechanical device with a set of "keys" that, when pressed, cause Typeface to be printed on a medium, usually paper....
. In some texts these symbols are simply shown in bold; blackboard bold in fact originated from the attempt to write bold letters on blackboard

Chalkboard

A chalkboard or blackboard is a reusable writing surface on which text or drawings are made with sticks of calcium sulfate, known, when used for this purpose, as chalk ....
s 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 mathematician

Mathematician

A mathematician is a person whose primary area of study and/or research is the field of mathematics....
s known as Nicolas Bourbaki

Nicolas Bourbaki

Nicolas Bourbaki is the collective pseudonym under which a group of 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935....
.

Discussion

Ask a question about 'Blackboard bold'

Start a new discussion about 'Blackboard bold'

Answer questions from other users

Full Discussion Forum

Encyclopedia

Blackboard bold is a typeface

Typeface

Mathematics

Physics

Typewriter

Chalkboard

Mathematician

A mathematician is a person whose primary area of study and/or research is the field of mathematics....
s known as Nicolas Bourbaki

Nicolas Bourbaki

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.
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

TeX is a typesetting system designed and mostly written by Donald Knuth. Together with the METAFONT language for font description and the Computer Modern typefaces, it was designed with two main goals in mind: to allow anybody to produce high-quality books using a reasonable amount of effort, and to provide a system that would give the exact...
, 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

American Mathematical Society

The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematics research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians....
provides this facility; a blackboard bold R is written as \mathbb.

In Unicode

Unicode

Unicode is a computing industry standard allowing computers to consistently represent and manipulate Character expressed in most of the world's writing systems....
, 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)

Mapping of Unicode characters

Unicode?s Universal Character Set has a potential capacity to support over 1 million characters. Each UCS character is mapped to a code point which is an integer between 0 and 1,114,111 used to represent each character within the internal logic of text processing software ....
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

LaTeX

LaTeX is a document markup language and Word processor for the TeX typesetting program. Within the typesetting system, its name is styled as ....
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.

LaTeX	Unicode	Symbol	Mathematics usage
	`U+1D538`		Represents affine space Affine space In mathematics, an affine space is a geometric structure that generalizes the affine geometry properties of Euclidean space. It can be thought of informally as a vector space where one has forgotten which point is the origin.... or the ring of adeles Adele ring In algebraic number theory and topological algebra, the adele ring is a topological ring which is built on the Field of rational numbers . It involves all the completions of the field.... . Sometimes represents the algebraic number Algebraic number In mathematics, an algebraic number is a complex number that is a root of a non-zero polynomial in one variable with rational number coefficients.... s, the algebraic closure Algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed field.... of Q (although a Q with an overline is often used instead). It may also represent the algebraic integer Algebraic integer This article deals with the ring of complex numbers integral over Z. For the general notion of algebraic integer, see Integrality.In number theory, an algebraic integer is a complex number that is a root of some monic polynomial with integer coefficients.... s, an important subring of the algebraic numbers.
	`U+1D539`		Sometimes represents a ball Ball (mathematics) In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general.... , a boolean domain Boolean domain In mathematics and abstract algebra, a Boolean domain is a Set consisting of exactly two elements whose interpretations include false and true.... , or the Brauer group Brauer group In mathematics, the Brauer group arose out of an attempt to classify division algebras over a given field K. It is an abelian group with elements equivalence classes of Azumaya algebra .... of a field.
	`U+2102`		Represents the complex number Complex number In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:... s.
	`U+1D53B`		Represents the unit disk in the complex plane Complex plane In mathematics, the complex plane is a geometric representation of the complex numbersestablished by the real axis and the orthogonal imaginary axis.... , or the decimal fractions (see number Number A number is a mathematical object used in counting and measurement. A notational symbol which represents a number is called a Numeral system, but in common usage the word number is used for both the abstract object and the symbol, as well as for the numeral for the number.... ).
	`U+1D53C`		Represents the expected value Expected value In probability theory and statistics, the expected value of a random variable is the Lebesgue integral of the random variable with respect to its probability measure.... of a random variable Random variable In mathematics, random variables are used in the study of Randomness and probability. They were developed to assist in the analysis of Game of chance, stochastic events, and the results of experiment by capturing only the mathematical properties necessary to answer probability questions.... , or Euclidean space Euclidean space Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space.... .
	`U+1D53D`		Represents a field Field (mathematics) In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms.... . Often used for finite field Finite field In abstract algebra, a finite field or Galois field is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory.... s, with a subscript to indicate the order. Also represents a Hirzebruch surface.
	`U+1D53E`		Represents a Grassmannian Grassmannian In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr1 is the space of lines through the origin in V, so it is the same as the projective space PV.... or a group Group (mathematics) In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element.... , especially an algebraic group Algebraic group In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety.... .
	`U+210D`		Represents the quaternions (the H stands for Hamilton William Rowan Hamilton Sir William Rowan Hamilton was an Ireland physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra.... ), or the upper half-plane Upper half-plane In mathematics, the upper half-plane H is the set of complex numberswith positive imaginary part y.The term is associated with a common visualization of complex numbers with points in the plane endowed with Cartesian coordinates, with the Y-axis pointing upwards: the "upper half-plane" corresponds to the half-plane above the X... , or hyperbolic space Hyperbolic space In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1.... , or hyperhomology Hyperhomology In homological algebra, the hyperhomology or hypercohomology of a complexof objects of an abelian category is an extension of the usual homology of an object to complexes.... of a complex.
	`U+1D541`		Sometimes represents the irrational number Irrational number In mathematics, an irrational number is any real number that is not a rational number ? that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero.... s, R\Q.
	`U+1D542`		Represents a field Field (mathematics) In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms.... . This is derived from the German German language German is a West Germanic languages, thus related to and classified alongside English language and Dutch language. It is one of the world's world language and the most widely spoken mother tongue in the European Union.... word Körper, which is German for field (literally, "body"; cf. the French term corps). May also be used to denote a compact space Compact space In mathematics, a topological space is called compact if each of its open covers has a finite set subcover.Note: Some authors such as Nicolas Bourbaki use the term "quasi-compact" for this instead, and reserve the term "compact" for topological spaces that are both Hausdorff spaces and "quasi-compact".... .
	`U+1D543`		Represents the Lefschetz motive. See motives Motive (algebraic geometry) In algebraic geometry, a motive refers to'some essential part of an algebraic variety'. Mathematically, the theory of motives is then the conjectural "universal" cohomology theory for such objects.... .
	`U+2115`		Represents the natural number Natural number In mathematics, a natural number can mean either an element of the Set = *n = = ? = ? ... s. May or may not include zero 0 (number) 0 is both a number and the numerical digit used to represent that number in numeral system. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures.... .
	`U+1D546`		Represents the octonion Octonion In mathematics, the octonions are a associative extension of the quaternions. Their 8-dimensional normed division algebra over the real numbers is the widest possible that can be obtained from the Cayley-Dickson construction.... s.
	`U+2119`		Represents projective space Projective space In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively.... , the probability Probability Probability, or wikt:chance, is a way of expressing knowledge or belief that an Event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about t... of an event, the prime number Prime number In mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC.... s, a power set Power set In mathematics, given a Set S, the power set of S, written , P, ℘ or Power set#Representing subsets as functions, is the set of all subsets of S.... , the positive reals, or a forcing Forcing (mathematics) In the mathematical discipline of set theory, forcing is a technique invented by Paul Cohen for proving consistency and independence results. It was first used, in 1962, to prove the independence of the continuum hypothesis and the axiom of choice from Zermelo-Fraenkel set theory.... poset.
	`U+211A`		Represents the rational number Rational number In mathematics, a rational number is a number which can be expressed as a quotient of two integers. Non-integer rational numbers are usually written as the vulgar fraction , where b is not 0 .... s. (The Q stands for quotient Quotient In mathematics, a quotient is the result of a division . For example, when dividing 6 by 3, the quotient is 2, while 6 is called the division , and 3 the divisor.... .)
	`U+211D`		Represents the real number Real number In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co... s.
	`U+1D54A`		Represents the sedenion Sedenion In abstract algebra, sedenions form a 16-dimension of a vector space algebra over a field over the real number. The set of sedenions is denoted as .... s, or a sphere Sphere A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface.... .
	`U+1D54B`		Represents a torus Torus In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle.... , or the circle group Circle group In mathematics, the circle group, denoted by T , is the multiplicative group of all complex numbers with absolute value 1, i.e., the unit circle in the complex plane.... or a Hecke algebra Hecke algebra In mathematics, the term Hecke algebra is the common name for several related types of associative rings in abstract algebra and representation theory.... (Hecke denoted his operators as T_n.)
	`U+1D54E`		Represents the whole number Whole number The term whole number is used by various authors to mean either:the nonnegative integer the positive integer *all integer ... s, which also are represented by N₀.
	`U+2124`		Represents the integer Integer The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set .... s. (The Z is for Zahlen, which is German for "numbers".)

A blackboard bold Greek letter mu

Mu (letter)

Mu is the 12th letter of the Greek alphabet. In the system of Greek numerals it has a value of 40. Mu was derived from the Egyptian hieroglyphic symbol for water which had been simplified by the Phoenicians and named after their word for water, to become Mem ....
(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

Group scheme

In mathematics, a group scheme is a group object in the category of schemes. That is, it is a scheme G with the equivalent properties* there is a group law expressible as a multiplication ? and inversion map ? on G; or...
) of n-th roots of unity

Root of unity

In mathematics, the nth roots of unity, or Abraham de Moivre numbers, are all the complex numbers that yield 1 when exponentiation to a given power n....
. A blackboard bold numeral 1 is often used in set theory

Set theory

Set theory is the branch of mathematics that studies Set , which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics....
for the top element of a forcing poset, or occasionally for the identity matrix in a matrix ring

Matrix ring

In abstract algebra the matrix ring M is the set of all n×n matrix over an arbitrary ring R. This set is itself a ring under matrix addition and matrix multiplication....
.

External links

shows blackboard bold symbols together with their Unicode encodings. Encodings in the BMP are highlighted in yellow.

Blackboard bold

Examples

See also

External links