An example of blackboard bold letters.

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 describe number sets. Blackboard bold symbols are also referred to as double struck, although attempting to produce them by double striking on a typewriter is unlikely to give satisfactory results. The symbols are nearly universal in their interpretation, unlike their normally-typeset counterparts, which are constantly reused. Image File history File links Blackboard_bold. ... Image File history File links Blackboard_bold. ... For the origin and evolution of fonts, see History of western typography. ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Physics (from the Greek, (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space, and time. ... A number is an abstract entity that represents a count or measurement. ... Mechanical desktop typewriters, such as this Underwood Five, were long time standards of government agencies, newsrooms, and sales offices. ...

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. Bold Bold, see Bold (disambiguation). ... A blackboard, with multiple colors of chalk A blackboard or chalkboard is a reusable writing surface on which text or drawings are made with chalk or other erasable markers. ...

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. Leonhard Euler is considered by many to be one of the greatest mathematicians of all time A mathematician is the person whose primary area of study and research is the field of mathematics. ... Nicolas Bourbaki is the collective allonym under which a group of mainly French 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. ... Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ...

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 regular text and as mathbb{R} in math mode. TeX (IPA: as in Greek, often in English; sometimes written TEX in imitation of the logo) is a typesetting system created by Donald Knuth. ... The American Mathematical Society (AMS) is dedicated to the interests of mathematical research and education, which it does with various publications and conferences as well as annual monetary awards to mathematicians. ...

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. Because of technical limitations, some web browsers may not display some special characters in this article. ...

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 usage in mathematical texts. The LaTeX logo, typeset with LaTeX LATEX, written as LaTeX in plain text, is a document markup language and document preparation system for the TeX typesetting program. ...

LaTeX	Unicode	Symbol	Mathematics usage
	U+1D538	𝔸	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).
	U+1D539	𝔹	Represents a ball. Sometimes represents a boolean domain.
	U+2102	ℂ	Represents the complex numbers.
	U+1D53B	𝔻	Represents the unit disk in the complex plane, or the decimal fractions (see number).
	U+1D53C	𝔼	Represents the expected value of a random variable.
	U+1D53D	𝔽	Represents a field. Often used for finite fields, with a subscript to indicate the order. Also represents a Hirzebruch surface.
	U+1D53E	𝔾	Represents a Grassmannian.
	U+210D	ℍ	Represents the quaternions (the H stands for Hamilton), or the upper half-plane, or the hyperbolic space.
	U+1D541	𝕁	Sometimes represents the irrational numbers, RQ.
	U+1D542	𝕂	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.
	U+2115	ℕ	Represents the natural numbers. May or may not include zero.
	U+1D546	𝕆	Represents the octonions.
	U+2119	ℙ	Represents projective space, the probability of an event, the prime numbers, a power set, or the positive reals.
	U+211A	ℚ	Represents the rational numbers. (The Q stands for quotient.)
	U+211D	ℝ	Represents the real numbers.
	U+1D54A	𝕊	Represents the sedenions, or a sphere.
	U+1D54B	𝕋	Represents a torus, or the circle group.
	U+2124	ℤ	Represents the integers. (The Z is for Zahlen, which is German for "numbers".)

In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ... ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ... The solid interior of a sphere or circle; in mathematics, latter terms refer specifically to the (n-1)-dimensional surface of an n-dimensional solid ball. ... A boolean domain B is a generic 2-element set, say, B = {0, 1}, whose elements are interpreted as logical values, typically 0 = false and 1 = true. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... A number is an abstract entity that represents a count or measurement. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... A random variable is a mathematical function that maps outcomes of random experiments to numbers. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ... In algebraic geometry, a branch of mathematics, a rational surface is a surface birationally equivialent to the projective plane. ... In mathematics, a Grassmannian is the space of all k-dimensional subspaces of an n-dimensional vector space V, often denoted Gk(V) or simply Gk,n. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... William Rowan Hamilton Sir William Rowan Hamilton (August 4, 1805 – September 2, 1865) was an Irish mathematician, physicist, and astronomer who made important contributions to the development of optics, dynamics, and algebra. ... In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ... In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. ... In mathematics, an irrational number is any real number that is not a rational number, i. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... 0 (zero) is both a number and a numerical digit used to represent that number in numerals. ... In mathematics, the octonions are a nonassociative extension of the quaternions. ... In mathematics, a projective space is a fundamental construction from any vector space. ... Probability is the extent to which something is likely to happen or be the case[1]. Probability theory is used extensively in areas such as statistics, mathematics, science, philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors. ... In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, a quotient is the end result of a division problem. ... In mathematics, the real numbers may be described informally in several different ways. ... The sedenions form a 16-dimensional algebra over the reals obtained by applying the Cayley-Dickson construction to the octonions. ... A sphere is a perfectly symmetrical geometrical object. ... In geometry, a torus (pl. ... In mathematics, the circle group, denoted by T (or in blackboard bold by ), is the multiplicative group of all complex numbers with absolute value 1. ... The integers are commonly denoted by the above symbol. ...

See also

• Mathematical alphanumeric symbols

Mathematical alphanumeric symbols are modifications of Latin and Greek letters and decimal digits that enable mathematicians to denote different notions with different letter styles (one example is blackboard bold, or double-struck (in Unicode terminology)). Unicode now includes many such symbols (in the range U+1D400 . ...

External links

• http://www.w3.org/TR/MathML2/double-struck.html shows blackboard bold symbols together with their Unicode encodings. Encodings in the BMP are highlighted in yellow.