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Содержание
Введение
Стиль шрифта Blackboard bold — это стиль выделения математических символов на доске путем утолщения определенных штрихов, часто используемый на математических лекциях, и производный стиль шрифта, применяемый в печатных математических текстах. Этот стиль чаще всего используется для обозначения множеств чисел: натуральных чисел (ℕ), целых чисел (ℤ), рациональных чисел (ℚ), действительных чисел (ℝ) и комплексных чисел (ℂ). Для имитации жирного шрифта на пишущей машинке символ можно напечатать дважды (это называется двойным ударом); символы, полученные таким образом, называются двойными, и это название иногда применяется к символам Blackboard bold, например, в названиях глифов Unicode. В типографии шрифт с незаполненными символами называется инлайном, handtooled или open face.
Typeface style
Blackboard bold is a style of writing bold symbols on a blackboard by doubling certain strokes, commonly used in mathematical lectures, and the derived style of typeface used in printed mathematical texts. The style is most commonly used to represent the number sets (natural numbers), (integers), (rational numbers), (real numbers), and (complex numbers). To imitate a bold typeface on a typewriter, a character can be typed over itself (called double striking); symbols thus produced are called double struck, and this name is sometimes adopted for blackboard bold symbols, for instance in Unicode glyph names. In typography, a typeface with characters that are not solid is called inline, handtooled, or open face.
Кодирование
TeX, стандартная система набора текста для математических текстов, не имеет встроенной поддержки символов «доски» (blackboard bold), но Американское математическое общество распространяет популярный пакет AMSFonts, загружаемый из пакета amssymb, который включает шрифт «доски» для заглавных латинских букв, доступный с помощью \mathbb (например, \mathbb{R} даёт
TeX, the standard typesetting system for mathematical texts, does not contain direct support for blackboard bold symbols, but the American Mathematical Society distributes the popular AMSFonts collection, loaded from the amssymb package, which includes a blackboard bold typeface for uppercase Latin letters accessed using \mathbb (e. g. \mathbb{R} produces
В Unicode несколько наиболее распространённых символов «доски» (ℂ, ℍ, ℕ, ℙ, ℚ, ℝ и ℤ) закодированы в базовой многоязыковой плоскости (BMP) в области буквоподобных символов (2100–214F), обозначенной как DOUBLE STRUCK CAPITAL C и т.д. Остальные же символы закодированы вне BMP, в области математических буквенно-цифровых символов (1D400–1D7FF), конкретно от U+1D538 до U+1D550 (заглавные буквы, исключая закодированные в BMP), от U+1D552 до U+1D56B (строчные буквы) и от U+1D7D8 до U+1D7E1 (цифры).
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, 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).
Использование
В следующей таблице показаны все доступные символы Unicode с жирным шрифтом, используемые для записи на доске. Первый столбец показывает символ, как он обычно отображается в системе LaTeX. Второй столбец показывает кодовую точку Unicode. Третий столбец показывает сам символ Unicode (который будет отображаться правильно только в браузерах, поддерживающих Unicode, и имеющих доступ к подходящему шрифту). Четвертый столбец описывает типичное использование в математических текстах. Некоторые символы (в частности, 𝔸 и ℂ) имеют почти универсальную интерпретацию, в то время как другие используются более разнообразно.
The following table shows all available Unicode blackboard bold characters. 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 typeface). The fourth column describes some typical usage in mathematical texts. Some of the symbols (particularly and ) are nearly universal in their interpretation, while others are more varied in use. LaTeX Unicode code point 1=(hex) Unicode symbol Mathematics usageU+1D538𝔸Represents 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+1D552𝕒U+1D539𝔹Sometimes represents a ball, a boolean domain, or the Brauer group of a field. U+1D553𝕓U+2102ℂRepresents the set of complex numbers. U+1D554𝕔U+1D53B𝔻Represents the unit disk in the complex plane, for example as the conformal disk model of the hyperbolic plane. By generalisation may mean the n dimensional ball. Occasionally may mean the decimal fractions (see number), split complex numbers, or domain of discourse. U+1D555𝕕U+2145ⅅU+2146ⅆMay represent the differential symbol. U+1D53C𝔼Represents the expected value of a random variable, or Euclidean space, or a field in a tower of fields, or the Eudoxus reals. U+1D556𝕖U+2147ⅇOccasionally used for the mathematical constant [[e (mathematical constant). U+1D53D𝔽Represents 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+1D557𝕗U+1D53E𝔾Represents a Grassmannian or a group, especially an algebraic group. U+1D558𝕘U+210DℍRepresents the quaternions (the H stands for Hamilton), or the upper half plane, or hyperbolic space, or hyperhomology of a complex. U+1D559𝕙U+1D540𝕀The 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+1D55A𝕚U+2148ⅈOccasionally used for the imaginary unit. U+1D541𝕁Sometimes represents the irrational numbers, U+1D55B𝕛U+2149ⅉU+1D542𝕂Represents 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+1D55C𝕜Represents a field. U+1D543𝕃Represents the Lefschetz motive. See Motive (algebraic geometry). U+1D55D𝕝U+1D544𝕄Sometimes represents the monster group. The set of all m by n 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+1D55E𝕞U+2115ℕRepresents the set of natural numbers. May or may not include zero. U+1D55F𝕟U+1D546𝕆Represents the octonions. U+1D560𝕠U+2119ℙRepresents projective space, the probability of an event, the prime numbers, a power set, the positive reals, the irrational numbers, or a forcing poset. U+1D561𝕡U+211AℚRepresents the set of rational numbers. (The Q stands for quotient. )U+1D562𝕢U+211DℝRepresents the set of real numbers. U+1D563𝕣U+1D54A𝕊Represents a sphere, or the sphere spectrum, or occasionally the sedenions. U+1D564𝕤U+1D54B𝕋Represents the circle group, particularly the unit circle in the complex plane (and the n dimensional torus), or a Hecke algebra (Hecke denoted his operators as Tn or ), or the tropical semiring, or twistor space. U+1D565𝕥U+1D54C𝕌U+1D566𝕦U+1D54D𝕍Represents a vector space or an affine variety generated by a set of polynomials, or in probability theory and statistics the variance. U+1D567𝕧U+1D54E𝕎Represents the whole numbers (here in the sense of non negative integers), which also are represented by U+1D568𝕨U+1D54F𝕏Occasionally used to denote an arbitrary metric space. U+1D569𝕩U+1D550𝕐U+1D56A𝕪U+2124ℤRepresents the set of integers. (The Z is for Zahlen, German for 'numbers', and zählen, German for 'to count'.) When it has a positive integer subscript, it can mean the finite cyclic group of that size. U+1D56B𝕫U+213EℾU+213DℽU+213FℿU+213CℼU+2140⅀U+1D7D8𝟘In algebra of logical propositions, it represents a contradiction or falsity. U+1D7D9𝟙In 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+1D7DA𝟚In category theory, the interval category. U+1D7DB𝟛U+1D7DC𝟜U+1D7DD𝟝U+1D7DE𝟞U+1D7DF𝟟U+1D7E0𝟠U+1D7E1𝟡
In addition, a blackboard bold μn (not found in Unicode or LaTeX) is sometimes used by number theorists and algebraic geometers to designate the group scheme of n th roots of unity. Note: Only uppercase Roman letters are given LaTeX renderings because Wikipedia's implementation uses the AMSFonts blackboard bold typeface, which does not support other characters.
LaTeX | Unicode code point (hex) | Unicode symbol | Использование в математике
------- | -------- | -------- | --------
U+1D538 | A | 𝔸 | Представляет аффинное пространство, 𝔸ⁿ, или кольцо аделей. Иногда представляет алгебраические числа, алгебраическое замыкание (обычно записывается как ℚ̄ или ℂ), или алгебраические целые, важный подкольцо алгебраических чисел.
U+1D552 | a | 𝕒 |
U+1D539 | B | 𝔹 | Иногда представляет шар, булеву область или группу Брауэра поля.
U+1D553 | b | 𝕓 |
U+2102 | C | ℂ | Представляет множество комплексных чисел.
U+1D554 | c | 𝕔 |
U+1D53B | D | 𝔻 | Представляет единичный диск в комплексной плоскости, например, как модель диска Пуанкаре гиперболической плоскости. Обобщая, 𝔻ⁿ может означать n-мерный шар. Иногда 𝔻 может означать десятичные дроби (см. число), расщепленные комплексные числа или область определения.
U+1D555 | d | 𝕕 |
U+2145 | D | ⅅ |
U+2146 | d | ⅆ | Может представлять символ дифференциала.
U+1D53C | E | 𝔼 | Представляет математическое ожидание случайной величины, евклидово пространство, поле в башне полей или действительные числа Эвдокса.
U+1D556 | e | 𝕖 |
U+2147 | e | ⅇ | Иногда используется для математической константы *e*.
U+1D53D | F | 𝔽 | Представляет поле. Часто используется для конечных полей, с индексом, указывающим порядок. Также представляет поверхность Хирцебруха или свободную группу, с индексом, указывающим количество образующих (или образующего множества, если бесконечно).
U+1D557 | f | 𝕗 |
U+1D53E | G | 𝔾 | Представляет грассманиан или группу, особенно алгебраическую группу.
U+1D558 | g | 𝕘 |
U+210D | H | ℍ | Представляет кватернионы (H обозначает Гамильтона), верхнюю полуплоскость, гиперболическое пространство или гипергомологию комплекса.
U+1D559 | h | 𝕙 |
U+1D540 | I | 𝕀 | Замкнутый единичный интервал или идеал полиномов, обращающихся в нуль на подмножестве. Иногда отображение тождества на алгебраической структуре или индикаторная функция. Множество мнимых чисел (т.е. множество всех действительных кратных мнимой единицы).
U+1D55A | i | 𝕚 |
U+2148 | i | ⅈ | Иногда используется для мнимой единицы.
U+1D541 | J | 𝕁 | Иногда представляет иррациональные числа.
U+1D55B | j | 𝕛 |
U+2149 | j | ⅉ |
U+1D542 | K | 𝕂 | Представляет поле. Происходит от немецкого слова Körper, что означает "поле" (буквально "тело"; во французском языке термин - corps). Может также использоваться для обозначения компактного пространства.
U+1D55C | k | 𝕜 | Представляет поле.
U+1D543 | L | 𝕃 | Представляет мотив Лефшеца. См. Мотив (алгебраическая геометрия).
U+1D55D | l | 𝕝 |
U+1D544 | M | 𝕄 | Иногда представляет группу монстров. Множество всех матриц m × n иногда обозначается 𝕄<sub>m,n</sub>. В геометрической алгебре, 𝕄 представляет моторную группу жестких движений. В функциональном программировании и формальной семантике обозначает конструктор типа для монады.
U+1D55E | m | 𝕞 |
U+2115 | N | ℕ | Представляет множество натуральных чисел. Может включать или не включать ноль.
U+1D55F | n | 𝕟 |
U+1D546 | O | 𝕆 | Представляет октонионы.
U+1D560 | o | 𝕠 |
U+2119 | P | ℙ | Представляет проективное пространство, вероятность события, простые числа, булево множество, положительные действительные числа, иррациональные числа или принудительное частично упорядоченное множество.
U+1D561 | p | 𝕡 |
U+211A | Q | ℚ | Представляет множество рациональных чисел. (Q обозначает "частное").
U+1D562 | q | 𝕢 |
U+211D | R | ℝ | Представляет множество действительных чисел.
U+1D563 | r | 𝕣 |
U+1D54A | S | 𝕊 | Представляет сферу, спектр сферы или иногда седенионы.
U+1D564 | s | 𝕤 |
U+1D54B | T | 𝕋 | Представляет окружность (особенно единичную окружность в комплексной плоскости и n-мерный тор), или алгебру Хекке (Хекке обозначал свои операторы как T<sub>n</sub>), или тропическое полукольцо, или пространство твисторов.
U+1D565 | t | 𝕥 |
U+1D54C | U | 𝕌 |
U+1D566 | u | 𝕦 |
U+1D54D | V | 𝕍 | Представляет векторное пространство или аффинное многообразие, порожденное набором полиномов, или в теории вероятностей и статистике - дисперсию.
U+1D567 | v | 𝕧 |
U+1D54E | W | 𝕎 | Представляет целые числа (здесь в смысле неотрицательных целых чисел), которые также представлены символом ℕ.
U+1D568 | w | 𝕨 |
U+1D54F | X | 𝕏 | Иногда используется для обозначения произвольного метрического пространства.
U+1D569 | x | 𝕩 |
U+1D550 | Y | 𝕐 |
U+1D56A | y | 𝕪 |
U+2124 | Z | ℤ | Представляет множество целых чисел. (Z - это Zahlen, немецкое слово для "числа", и zählen, немецкое слово для "считать"). Если у него есть положительный индекс целого числа, он может означать конечную циклическую группу этого размера.
U+1D56B | z | 𝕫 |
U+213E | ℾ |
U+213D | ℽ |
U+213F | ℿ |
U+213C | ℼ |
U+2140 | ⅀ |
U+1D7D8 | 𝟘 |
U+1D7D9 | 𝟙 |
U+1D7DA | 𝟚 |
U+1D7DB | 𝟛 |
U+1D7DC | 𝟜 |
U+1D7DD | 𝟝 |
U+1D7DE | 𝟞 |
U+1D7DF | 𝟟 |
U+1D7E0 | 𝟠 |
U+1D7E1 | 𝟡 |
The following table shows all available Unicode blackboard bold characters. 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 typeface). The fourth column describes some typical usage in mathematical texts. Some of the symbols (particularly and ) are nearly universal in their interpretation, while others are more varied in use. LaTeX Unicode code point 1=(hex) Unicode symbol Mathematics usageU+1D538𝔸Represents 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+1D552𝕒U+1D539𝔹Sometimes represents a ball, a boolean domain, or the Brauer group of a field. U+1D553𝕓U+2102ℂRepresents the set of complex numbers. U+1D554𝕔U+1D53B𝔻Represents the unit disk in the complex plane, for example as the conformal disk model of the hyperbolic plane. By generalisation may mean the n dimensional ball. Occasionally may mean the decimal fractions (see number), split complex numbers, or domain of discourse. U+1D555𝕕U+2145ⅅU+2146ⅆMay represent the differential symbol. U+1D53C𝔼Represents the expected value of a random variable, or Euclidean space, or a field in a tower of fields, or the Eudoxus reals. U+1D556𝕖U+2147ⅇOccasionally used for the mathematical constant [[e (mathematical constant). U+1D53D𝔽Represents 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+1D557𝕗U+1D53E𝔾Represents a Grassmannian or a group, especially an algebraic group. U+1D558𝕘U+210DℍRepresents the quaternions (the H stands for Hamilton), or the upper half plane, or hyperbolic space, or hyperhomology of a complex. U+1D559𝕙U+1D540𝕀The 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+1D55A𝕚U+2148ⅈOccasionally used for the imaginary unit. U+1D541𝕁Sometimes represents the irrational numbers, U+1D55B𝕛U+2149ⅉU+1D542𝕂Represents 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+1D55C𝕜Represents a field. U+1D543𝕃Represents the Lefschetz motive. See Motive (algebraic geometry). U+1D55D𝕝U+1D544𝕄Sometimes represents the monster group. The set of all m by n 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+1D55E𝕞U+2115ℕRepresents the set of natural numbers. May or may not include zero. U+1D55F𝕟U+1D546𝕆Represents the octonions. U+1D560𝕠U+2119ℙRepresents projective space, the probability of an event, the prime numbers, a power set, the positive reals, the irrational numbers, or a forcing poset. U+1D561𝕡U+211AℚRepresents the set of rational numbers. (The Q stands for quotient. )U+1D562𝕢U+211DℝRepresents the set of real numbers. U+1D563𝕣U+1D54A𝕊Represents a sphere, or the sphere spectrum, or occasionally the sedenions. U+1D564𝕤U+1D54B𝕋Represents the circle group, particularly the unit circle in the complex plane (and the n dimensional torus), or a Hecke algebra (Hecke denoted his operators as Tn or ), or the tropical semiring, or twistor space. U+1D565𝕥U+1D54C𝕌U+1D566𝕦U+1D54D𝕍Represents a vector space or an affine variety generated by a set of polynomials, or in probability theory and statistics the variance. U+1D567𝕧U+1D54E𝕎Represents the whole numbers (here in the sense of non negative integers), which also are represented by U+1D568𝕨U+1D54F𝕏Occasionally used to denote an arbitrary metric space. U+1D569𝕩U+1D550𝕐U+1D56A𝕪U+2124ℤRepresents the set of integers. (The Z is for Zahlen, German for 'numbers', and zählen, German for 'to count'.) When it has a positive integer subscript, it can mean the finite cyclic group of that size. U+1D56B𝕫U+213EℾU+213DℽU+213FℿU+213CℼU+2140⅀U+1D7D8𝟘In algebra of logical propositions, it represents a contradiction or falsity. U+1D7D9𝟙In 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+1D7DA𝟚In category theory, the interval category. U+1D7DB𝟛U+1D7DC𝟜U+1D7DD𝟝U+1D7DE𝟞U+1D7DF𝟟U+1D7E0𝟠U+1D7E1𝟡
In addition, a blackboard bold μn (not found in Unicode or LaTeX) is sometimes used by number theorists and algebraic geometers to designate the group scheme of n th roots of unity. Note: Only uppercase Roman letters are given LaTeX renderings because Wikipedia's implementation uses the AMSFonts blackboard bold typeface, which does not support other characters.
Кроме того, иногда теоретики чисел и алгебраические геометры используют жирный шрифт blackboard μ<sub>n</sub> (отсутствует в Unicode или LaTeX) для обозначения групповой схемы n-х корней из единицы.
The following table shows all available Unicode blackboard bold characters. 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 typeface). The fourth column describes some typical usage in mathematical texts. Some of the symbols (particularly and ) are nearly universal in their interpretation, while others are more varied in use. LaTeX Unicode code point 1=(hex) Unicode symbol Mathematics usageU+1D538𝔸Represents 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+1D552𝕒U+1D539𝔹Sometimes represents a ball, a boolean domain, or the Brauer group of a field. U+1D553𝕓U+2102ℂRepresents the set of complex numbers. U+1D554𝕔U+1D53B𝔻Represents the unit disk in the complex plane, for example as the conformal disk model of the hyperbolic plane. By generalisation may mean the n dimensional ball. Occasionally may mean the decimal fractions (see number), split complex numbers, or domain of discourse. U+1D555𝕕U+2145ⅅU+2146ⅆMay represent the differential symbol. U+1D53C𝔼Represents the expected value of a random variable, or Euclidean space, or a field in a tower of fields, or the Eudoxus reals. U+1D556𝕖U+2147ⅇOccasionally used for the mathematical constant [[e (mathematical constant). U+1D53D𝔽Represents 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+1D557𝕗U+1D53E𝔾Represents a Grassmannian or a group, especially an algebraic group. U+1D558𝕘U+210DℍRepresents the quaternions (the H stands for Hamilton), or the upper half plane, or hyperbolic space, or hyperhomology of a complex. U+1D559𝕙U+1D540𝕀The 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+1D55A𝕚U+2148ⅈOccasionally used for the imaginary unit. U+1D541𝕁Sometimes represents the irrational numbers, U+1D55B𝕛U+2149ⅉU+1D542𝕂Represents 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+1D55C𝕜Represents a field. U+1D543𝕃Represents the Lefschetz motive. See Motive (algebraic geometry). U+1D55D𝕝U+1D544𝕄Sometimes represents the monster group. The set of all m by n 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+1D55E𝕞U+2115ℕRepresents the set of natural numbers. May or may not include zero. U+1D55F𝕟U+1D546𝕆Represents the octonions. U+1D560𝕠U+2119ℙRepresents projective space, the probability of an event, the prime numbers, a power set, the positive reals, the irrational numbers, or a forcing poset. U+1D561𝕡U+211AℚRepresents the set of rational numbers. (The Q stands for quotient. )U+1D562𝕢U+211DℝRepresents the set of real numbers. U+1D563𝕣U+1D54A𝕊Represents a sphere, or the sphere spectrum, or occasionally the sedenions. U+1D564𝕤U+1D54B𝕋Represents the circle group, particularly the unit circle in the complex plane (and the n dimensional torus), or a Hecke algebra (Hecke denoted his operators as Tn or ), or the tropical semiring, or twistor space. U+1D565𝕥U+1D54C𝕌U+1D566𝕦U+1D54D𝕍Represents a vector space or an affine variety generated by a set of polynomials, or in probability theory and statistics the variance. U+1D567𝕧U+1D54E𝕎Represents the whole numbers (here in the sense of non negative integers), which also are represented by U+1D568𝕨U+1D54F𝕏Occasionally used to denote an arbitrary metric space. U+1D569𝕩U+1D550𝕐U+1D56A𝕪U+2124ℤRepresents the set of integers. (The Z is for Zahlen, German for 'numbers', and zählen, German for 'to count'.) When it has a positive integer subscript, it can mean the finite cyclic group of that size. U+1D56B𝕫U+213EℾU+213DℽU+213FℿU+213CℼU+2140⅀U+1D7D8𝟘In algebra of logical propositions, it represents a contradiction or falsity. U+1D7D9𝟙In 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+1D7DA𝟚In category theory, the interval category. U+1D7DB𝟛U+1D7DC𝟜U+1D7DD𝟝U+1D7DE𝟞U+1D7DF𝟟U+1D7E0𝟠U+1D7E1𝟡
In addition, a blackboard bold μn (not found in Unicode or LaTeX) is sometimes used by number theorists and algebraic geometers to designate the group scheme of n th roots of unity. Note: Only uppercase Roman letters are given LaTeX renderings because Wikipedia's implementation uses the AMSFonts blackboard bold typeface, which does not support other characters.
Примечание: приведены только символы больших латинских букв, поскольку реализация Wikipedia использует шрифт AMSFonts blackboard bold, который не поддерживает другие символы.
The following table shows all available Unicode blackboard bold characters. 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 typeface). The fourth column describes some typical usage in mathematical texts. Some of the symbols (particularly and ) are nearly universal in their interpretation, while others are more varied in use. LaTeX Unicode code point 1=(hex) Unicode symbol Mathematics usageU+1D538𝔸Represents 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+1D552𝕒U+1D539𝔹Sometimes represents a ball, a boolean domain, or the Brauer group of a field. U+1D553𝕓U+2102ℂRepresents the set of complex numbers. U+1D554𝕔U+1D53B𝔻Represents the unit disk in the complex plane, for example as the conformal disk model of the hyperbolic plane. By generalisation may mean the n dimensional ball. Occasionally may mean the decimal fractions (see number), split complex numbers, or domain of discourse. U+1D555𝕕U+2145ⅅU+2146ⅆMay represent the differential symbol. U+1D53C𝔼Represents the expected value of a random variable, or Euclidean space, or a field in a tower of fields, or the Eudoxus reals. U+1D556𝕖U+2147ⅇOccasionally used for the mathematical constant [[e (mathematical constant). U+1D53D𝔽Represents 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+1D557𝕗U+1D53E𝔾Represents a Grassmannian or a group, especially an algebraic group. U+1D558𝕘U+210DℍRepresents the quaternions (the H stands for Hamilton), or the upper half plane, or hyperbolic space, or hyperhomology of a complex. U+1D559𝕙U+1D540𝕀The 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+1D55A𝕚U+2148ⅈOccasionally used for the imaginary unit. U+1D541𝕁Sometimes represents the irrational numbers, U+1D55B𝕛U+2149ⅉU+1D542𝕂Represents 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+1D55C𝕜Represents a field. U+1D543𝕃Represents the Lefschetz motive. See Motive (algebraic geometry). U+1D55D𝕝U+1D544𝕄Sometimes represents the monster group. The set of all m by n 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+1D55E𝕞U+2115ℕRepresents the set of natural numbers. May or may not include zero. U+1D55F𝕟U+1D546𝕆Represents the octonions. U+1D560𝕠U+2119ℙRepresents projective space, the probability of an event, the prime numbers, a power set, the positive reals, the irrational numbers, or a forcing poset. U+1D561𝕡U+211AℚRepresents the set of rational numbers. (The Q stands for quotient. )U+1D562𝕢U+211DℝRepresents the set of real numbers. U+1D563𝕣U+1D54A𝕊Represents a sphere, or the sphere spectrum, or occasionally the sedenions. U+1D564𝕤U+1D54B𝕋Represents the circle group, particularly the unit circle in the complex plane (and the n dimensional torus), or a Hecke algebra (Hecke denoted his operators as Tn or ), or the tropical semiring, or twistor space. U+1D565𝕥U+1D54C𝕌U+1D566𝕦U+1D54D𝕍Represents a vector space or an affine variety generated by a set of polynomials, or in probability theory and statistics the variance. U+1D567𝕧U+1D54E𝕎Represents the whole numbers (here in the sense of non negative integers), which also are represented by U+1D568𝕨U+1D54F𝕏Occasionally used to denote an arbitrary metric space. U+1D569𝕩U+1D550𝕐U+1D56A𝕪U+2124ℤRepresents the set of integers. (The Z is for Zahlen, German for 'numbers', and zählen, German for 'to count'.) When it has a positive integer subscript, it can mean the finite cyclic group of that size. U+1D56B𝕫U+213EℾU+213DℽU+213FℿU+213CℼU+2140⅀U+1D7D8𝟘In algebra of logical propositions, it represents a contradiction or falsity. U+1D7D9𝟙In 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+1D7DA𝟚In category theory, the interval category. U+1D7DB𝟛U+1D7DC𝟜U+1D7DD𝟝U+1D7DE𝟞U+1D7DF𝟟U+1D7E0𝟠U+1D7E1𝟡
In addition, a blackboard bold μn (not found in Unicode or LaTeX) is sometimes used by number theorists and algebraic geometers to designate the group scheme of n th roots of unity. Note: Only uppercase Roman letters are given LaTeX renderings because Wikipedia's implementation uses the AMSFonts blackboard bold typeface, which does not support other characters.