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 nearvertical lines) are doubled. The symbols usually denote number sets. One way of producing blackboard bold is to doublestrike a character with a small offset on a typewriter. Thus, they are also referred to as double struck.^{[citation needed]}
In typography, such a font with characters that are not solid is called an "inline", "shaded", or "tooled" font.^{[citation needed]}
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 nonbold letters (by using the edge rather than the point of a chalk). It then made its way back into print form as a separate style from ordinary bold,^{[citation needed]} possibly starting with the original 1965 edition of Gunning and Rossi's textbook on complex analysis.^{[1]}^{[2]}
In the 1960s and 1970s, blackboard bold spread quickly in classrooms and is now widely used in the English and Frenchspeaking worlds. In textbooks, however, the situation is not so clear cut. Many mathematicians adopted blackboard bold, but many others still prefer to use bold.^{[citation needed]}
Wellknown books where the blackboard bold style is used include Lindsay Childs' A Concrete Introduction to Higher Algebra,^{[3]} which is widely used as a text for undergraduate courses in the US, John Stillwell's Elements of Number Theory,^{[4]} and Edward Barbeau's "University of Toronto Mathematics Competition (2001–2015)",^{[5]} which is often used to prepare for mathematics competitions.^{[citation needed]}^{[nonprimary source needed]}
JeanPierre Serre used doublestruck letters when he wrote bold on the blackboard,^{[6]}^{[nonprimary source needed]} whereas his published works (like his wellknown "Cohomologie galoisienne"^{[7]}) have consistently used ordinary bold for the same symbols.^{[citation needed]}
Donald Knuth also preferred boldface to blackboard bold and so did not include blackboard bold in the Computer Modern fonts that he created for the TeX mathematical typesetting system.^{[8]}
Serge Lang also used boldface instead of blackboard bold in his highly influential^{[9]} Algebra.^{[10]}^{[nonprimary source needed]}
The Chicago Manual of Style evolved over this issue. In 1993, for the 14th edition, it advised that "blackboard bold should be confined to the classroom" (13.14). In 2003, for the 15th edition, it stated that "openfaced (blackboard) symbols are reserved for familiar systems of numbers" (14.12).
TeX, the standard typesetting system for mathematical texts, does not contain direct support for blackboard bold symbols, but the addon AMS Fonts package (amsfonts
) by the American Mathematical Society provides this facility for uppercase letters (e.g., is written as \mathbb{R}
). The amssymb
package loads amsfonts
.^{[11]}
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 DOUBLESTRUCK CAPITAL C etc. The rest, however, are encoded outside the BMP, in Mathematical Alphanumeric Symbols (1D400–1D7FF), specifically 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).
The following table shows all available Unicode blackboard bold characters.^{[12]}
The first column shows the letter as typically rendered by the LaTeX markup system. The second column shows the Unicode code point. The third column shows the Unicode symbol itself (which will only display correctly on browsers that support Unicode and have access to a suitable font). The fourth column describes some typical usage in mathematical texts. Some of the symbols (particularly and ) are nearly universal in their interpretation,^{[8]} while others are more varied in use.
Unicode Code Point (hex)  Unicode Symbol  Mathematics usage  

U+1D538

𝔸  Represents affine space, , or the ring of adeles. Occasionally represents the algebraic numbers, the algebraic closure of (more commonly written or Q ), or the algebraic integers, an important subring of the algebraic numbers.  
U+1D552

𝕒  
U+1D539

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

𝕓  
U+2102

ℂ  Represents the set of complex numbers.  
U+1D554

𝕔  
U+1D53B

𝔻  Represents the unit (open) disk in the complex plane (and by generalisation may mean the ndimensional ball) — for example as a model of the hyperbolic plane and the domain of discourse. Occasionally may mean the decimal fractions (see number) or splitcomplex numbers.  
U+1D555

𝕕  
U+2145

ⅅ  
U+2146

ⅆ  May represent the differential symbol.  
U+1D53C

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

𝕖  
U+2147

ⅇ  Occasionally used for the mathematical constant e.  
U+1D53D

𝔽  Represents 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 subscript to indicate the number of generators (or generating set, if infinite).  
U+1D557

𝕗  
U+1D53E

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

𝕘  
U+210D

ℍ  Represents the quaternions (the H stands for Hamilton), or the upper halfplane, or hyperbolic space, or hyperhomology of a complex.  
U+1D559

𝕙  
U+1D540

𝕀  The closed unit interval or the ideal of polynomials vanishing on a subset. Occasionally the identity mapping on an algebraic structure, or an indicator function. The set of imaginary numbers (i.e., the set of all real multiples of the imaginary unit).  
U+1D55A

𝕚  
U+2148

ⅈ  Occasionally used for the imaginary unit.  
U+1D541

𝕁  Sometimes represents the irrational numbers, .  
U+1D55B

𝕛  
U+2149

ⅉ  
U+1D542

𝕂  Represents a field. This is derived from the German word Körper, which is German for field (literally, "body"; in French the term is corps). May also be used to denote a compact space.  
U+1D55C

𝕜  Represents a field.  
U+1D543

𝕃  Represents the Lefschetz motive. See Motive (algebraic geometry).  
U+1D55D

𝕝  
U+1D544

𝕄  Sometimes represents the monster group. The set of all mbyn matrices is sometimes denoted . In geometric algebra, represents the motor group of rigid motions. In functional programming and formal semantics, denotes the type constructor for a monad.  
U+1D55E

𝕞  
U+2115

ℕ  Represents the set of natural numbers. May or may not include zero.  
U+1D55F

𝕟  
U+1D546

𝕆  Represents the octonions.  
U+1D560

𝕠  
U+2119

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

𝕡  
U+211A

ℚ  Represents the set of rational numbers. (The Q stands for quotient.)  
U+1D562

𝕢  
U+211D

ℝ  Represents the set of real numbers.  
U+1D563

𝕣  
U+1D54A

𝕊  Represents a sphere, or the sphere spectrum, or occasionally the sedenions.  
U+1D564

𝕤  
U+1D54B

𝕋  Represents the circle group, particularly the unit circle in the complex plane (and the ndimensional torus), or a Hecke algebra (Hecke denoted his operators as T_{n} or ), or the tropical semiring, or twistor space.  
U+1D565

𝕥  
U+1D54C

𝕌  
U+1D566

𝕦  
U+1D54D

𝕍  Represents a vector space or an affine variety generated by a set of polynomials.  
U+1D567

𝕧  
U+1D54E

𝕎  Represents the whole numbers (here in the sense of nonnegative integers), which also are represented by .  
U+1D568

𝕨  
U+1D54F

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

𝕩  
U+1D550

𝕐  
U+1D56A

𝕪  
U+2124

ℤ  Represents the set of integers. (The Z is for Zahlen, German for "numbers", and zählen, German for "to count".)  
U+1D56B

𝕫  
U+213E

ℾ  
U+213D

ℽ  
U+213F

ℿ  
U+213C

ℼ  
U+2140

⅀  
U+1D7D8

𝟘  In algebra of logical propositions, it represents a contradiction or falsity.  
U+1D7D9

𝟙  In set theory, the top element of a forcing poset, or occasionally the identity matrix in a matrix ring. Also used for the indicator function and the unit step function, and for the identity operator or identity matrix. In geometric algebra, represents the unit antiscalar, the identity element under the geometric antiproduct. In algebra of logical propositions, it represents a tautology.  
U+1D7DA

𝟚  In category theory, the interval category.  
U+1D7DB

𝟛  
U+1D7DC

𝟜  
U+1D7DD

𝟝  
U+1D7DE

𝟞  
U+1D7DF

𝟟  
U+1D7E0

𝟠  
U+1D7E1

𝟡 
In addition, a blackboardbold μ_{n} (not found in Unicode) is sometimes used by number theorists and algebraic geometers to designate the group scheme of nth roots of unity.^{[13]}
LaTeX notes:
amsmath
is used here.