Glossary of mathematical symbols

• Normally, entries of a glossary are structured by topics and sorted alphabetically. This is not possible here, as there is no natural order on symbols, and many symbols are used in different parts of mathematics with different meanings, often completely unrelated. Therefore, some arbitrary choices had to be made, which are summarized below. • The article is split into sections that are sorted by an increasing level of technicality. That is, the first sections contain the symbols that are encountered in most mathematical texts, and that are supposed to be known even by beginners. On the other hand, the last sections contain symbols that are specific to some area of mathematics and are ignored outside these areas. However, the long section on brackets has been placed near to the end, although most of its entries are elementary: this makes it easier to search for a symbol entry by scrolling. • Most symbols have multiple meanings that are generally distinguished either by the area of mathematics where they are used or by their syntax, that is, by their position inside a formula and the nature of the other parts of the formula that are close to them. • As readers may not be aware of the area of mathematics to which the symbol that they are looking for is related, the different meanings of a symbol are grouped in the section corresponding to their most common meaning. • When the meaning depends on the syntax, a symbol may have different entries depending on the syntax. For summarizing the syntax in the entry name, the symbol \Box is used for representing the neighboring parts of a formula that contains the symbol. See for examples of use. • Most symbols have two printed versions. They can be displayed as Unicode characters, or in LaTeX format. With the Unicode version, using search engines and copy-pasting are easier. On the other hand, the LaTeX rendering is often much better (more aesthetic), and is generally considered a standard in mathematics. Therefore, in this article, the Unicode version of the symbols is used (when possible) for labelling their entry, and the LaTeX version is used in their description. So, for finding how to type a symbol in LaTeX, it suffices to look at the source of the article. • For most symbols, the entry name is the corresponding Unicode symbol. So, for searching the entry of a symbol, it suffices to type or copy the Unicode symbol into the search textbox. Similarly, when possible, the entry name of a symbol is also an anchor, which allows linking easily from another Wikipedia article. When an entry name contains special characters such as , , and , there is also an anchor, but one has to look at the article source to know it. • Finally, when there is an article on the symbol itself (not its mathematical meaning), it is linked to in the entry name. == Arithmetic operators ==

{{defn|no=1|1=Denotes division and is read as divided by or over. Often replaced by a horizontal bar. For example, or {{tmath| \tfrac{3}{2} }}.}} {{defn|no=1|1=Denotes square root and is read as the square root of. For example, {{tmath| \sqrt{3+2} }}.}} {{defn|no=2|1=With an integer greater than 2 as a left superscript, denotes an th root. For example, \sqrt[7]{3} denotes the 7th root of 3.}} == Equality, equivalence and similarity ==

{{defn|Any of these is sometimes used for naming a mathematical object. Thus, , x\mathrel{\stackrel{\scriptscriptstyle \mathrm{def}}{=}}E, x \mathrel{:=} E and E \mathrel{=:} x are each an abbreviation of the phrase "let ", where is an expression and is a variable.}} == Comparison ==

{{term|much-greater-or-less|content= \ll \text{ and }\gg}} {{term|less-equal sign|content= \leqq \text{ and } \geqq}} {{term|pred-succ|content= \prec \text{ and } \succ}} == Set theory ==

{{defn|Denotes the empty set, and is more often written . Using set-builder notation, it may also be denoted {{tmath| \{\} }}.}} {{defn|term=sharp|no=1 {{defn|term=sharp|no=2 {{defn|term=sharp|no=3 {{defn|term=⊂|no=1 {{defn|term=⊂|no=2 {{defn|Denotes set-theoretic union, that is, A\cup B is the set formed by the elements of and together. That is, A\cup B=\{x\mid (x\in A) \lor (x\in B)\}.}} {{defn|Denotes set-theoretic intersection, that is, A\cap B is the set formed by the elements of both and . That is, A\cap B=\{x\mid (x\in A) \land (x\in B)\}.}} {{defn|Symmetric difference: that is, A\ominus B or \ A \operatorname{\triangle}B\ is the set formed by the elements that belong to exactly one of the two sets and .}} {{defn|Denotes the disjoint union. That is, if and are sets then A\sqcup B=\left(A\times\{i_A\}\right)\cup\left(B\times\{i_B\}\right) is a set of pairs where and are distinct indices discriminating the members of and in .}} {{term|∐|content=\bigsqcup \text{ or } \coprod }} {{defn|no=1|Used for the disjoint union of a family of sets, such as in \bigsqcup_{i\in I}A_i.}} == Basic logic ==

Several logical symbols are widely used in all mathematics, and are listed here. For symbols that are used only in mathematical logic, or are rarely used, see List of logic symbols. == Blackboard bold ==

The blackboard bold typeface is widely used for denoting the basic number systems. These systems are often also denoted by the corresponding uppercase bold letter. A clear advantage of blackboard bold is that these symbols cannot be confused with anything else. This allows using them in any area of mathematics, without having to recall their definition. For example, if one encounters \mathbb R in combinatorics, one should immediately know that this denotes the real numbers, although combinatorics does not study the real numbers (but it uses them for many proofs). {{defn|Denotes the set of natural numbers \{1, 2,\ldots \}, or sometimes \{0, 1, 2, \ldots \}. When the distinction is important and readers might assume either definition, \mathbb{N}_1 and \mathbb{N}_0 are used, respectively, to denote one of them unambiguously. Notation \mathbf N is also commonly used.}} {{defn|Denotes the set of integers \{\ldots, -2, -1, 0, 1, 2,\ldots \}. It is often denoted also by \mathbf Z.}} {{term||content=\mathbb{Z}_p}} {{term||content=\mathbb{Q}_p}} {{term||content=\mathbb{F}_q}} == Calculus ==

{{defn|no=1|If is a variable that depends on , then \textstyle \frac{\mathrm{d}y}{\mathrm{d}x}, read as "d over d " (commonly shortened to "d d "), is the derivative of with respect to .}} {{defn|no=2|If is a function of a single variable , then \textstyle \frac{ \mathrm{d} f }{ \mathrm{d}x } is the derivative of , and \textstyle \frac{ \mathrm{d} f }{ \mathrm{d}x }(a) is the value of the derivative at . Sometimes \textstyle \frac{ \mathrm{d} }{ \mathrm{d}x } is used as a symbol for the derivative operator, which can be prepended to a function: {{nobr|\textstyle \frac{ \mathrm{d} }{ \mathrm{d}x } f(x).}}}} {{defn|no=3|Total derivative: If f(x_1, \ldots, x_n) is a function of several variables that depend on , then \textstyle \frac{\mathrm{d}f}{\mathrm{d}x} is the derivative of considered as a function of . That is, \textstyle\ \frac{\mathrm{d}f}{\mathrm{d}x} = \sum_{i=1}^n \frac{ \partial f }{ \partial x_i }\ \frac{\mathrm{d}x_i}{\mathrm{d}x} ~.}} {{defn|Partial derivative: If f(x_1, \ldots, x_n) is a function of several variables, \textstyle\frac{\partial f}{\partial x_i} is the derivative with respect to the th variable considered as an independent variable, the other variables being considered as constants.}} {{defn|Functional derivative: If f(y_1, \ldots, y_n) is a functional of several functions, \textstyle \frac{\delta f}{\delta y_{i}} is the functional derivative with respect to the th function considered as an independent variable, the other functions being considered constant.}} {{defn|no=1|Complex conjugate: If is a complex number, then \overline{z} is its complex conjugate. For example, \overline{a+bi} = a - b\ i ~.}} {{defn|no=2|Topological closure: If is a subset of a topological space , then \overline{S} is its topological closure, that is, the smallest closed subset of that contains .}} {{defn|no=3|Algebraic closure: If is a field, then \overline{F} is its algebraic closure, that is, the smallest algebraically closed field that contains . For example, \overline\mathbb Q is the field of all algebraic numbers.}} {{defn|no=4|Mean value: If is a variable that takes its values in some sequence of numbers , then \overline{x} may denote the mean of the elements of .}} {{defn|no=5|Negation: Sometimes used to denote negation of the entire expression under the bar, particularly when dealing with Boolean algebra. For example, one of De Morgan's laws says that {{tmath|1= \overline{A \land B} = \overline{A} \lor \overline{B} }}.}} {{defn|no=5|In Euclidean geometry and more generally in affine geometry, \overrightarrow{PQ} denotes the vector defined by the two points and , which can be identified with the translation that maps to . The same vector can be denoted also ; see Affine space.}} {{defn|no=2|Hadamard product of matrices: If and are two matrices of the same size, then A\circ B is the matrix such that (A\circ B)_{i,j} = (A)_{i,j}(B)_{i,j}. Possibly, \circ is also used instead of #⊙| for the Hadamard product of power series.}} {{defn|no=1|1=Without a subscript, denotes an antiderivative. For example, \textstyle\int x^2 \operatorname{d} x = \frac{\; x^3}{ 3 } + C .}} {{defn|no=2|1=With a subscript and a superscript, or expressions placed below and above it, denotes a definite integral. For example, \textstyle \int_a^b x^2 \operatorname{d} x = \frac{\;b^3 - a^3}{ 3 } ~.}} {{defn|no=3|1=With a subscript that denotes a curve, denotes a line integral. For example, \textstyle\ \int_C f\ =\ \int_a^b\ f\!\bigl(\ \!r\!\left(t\right)\ \!\!\bigr)\ r'\!\left(t\right) \operatorname{d} t\ , if is a parametrization of the curve , from point to .}} {{term|∇|\boldsymbol{\nabla} or \vec{\nabla}}} {{defn|no=1|Nabla, the gradient, vector derivative operator {{tmath| \textstyle \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right) }}, also called del or grad,}} {{defn|Laplace operator or Laplacian: {{tmath| \textstyle \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} }}. The forms \nabla^2 and \boldsymbol\nabla \cdot \boldsymbol\nabla represents the dot product of the gradient (\boldsymbol{\nabla} or {{tmath| \vec{\nabla} }}) with itself. Also notated (next item).}} (Capital Greek letter delta—not to be confused with , which may denote a geometric triangle or, alternatively, the symmetric difference of two sets.) {{term|four-gradient|\boldsymbol{\partial} or \partial_\mu}} (Note: Although \ \vec\Box\ may be used unambiguously, the notation \boldsymbol\Box is not recommended for the four-gradient since both \Box and {\Box}^2 are used to denote the d'Alembertian; see below.) {{defn|Quad, the 4-vector gradient operator or four-gradient, \textstyle \left( \frac{\ \partial }{\ \partial t\ }, \frac{\ \partial }{\ \partial x\ }, \frac{\ \partial }{\ \partial y\ }, \frac{\ \partial }{\ \partial z\ }\right) ~.}} {{term|d'Alembertian|\Box or {\Box}^2}} (here the symbol is an actual box, not a placeholder) {{defn|Denotes the d'Alembertian or squared four-gradient, which is a generalization of the Laplacian to four-dimensional spacetime. In flat spacetime with Euclidean coordinates, this may mean either \textstyle - \frac {\partial^2}{\partial t^2} + \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} or {{tmath| \textstyle + \frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} - \frac{\partial^2}{\partial z^2} }}; the sign convention must be specified. In curved spacetime (or flat spacetime with non-Euclidean coordinates), the definition is more complicated. Also called box or quabla.}} == Linear and multilinear algebra ==

{{defn|no=1|Denotes the sum of a finite number of terms, which are determined by subscripts and superscripts (which can also be placed below and above), such as in \textstyle \sum_{i=1}^n i^2 or \textstyle \sum_{0.}} {{defn|no=2|Denotes a series and, if the series is convergent, the sum of the series. For example, \textstyle \sum_{i=0}^\infty \frac {x^i}{i!}=e^x.}} {{defn|no=1|Denotes the product of a finite number of terms, which are determined by subscripts and superscripts (which can also be placed below and above), such as in \textstyle \prod_{i=1}^n i^2 or \textstyle \prod_{0.}} {{defn|no=2|Denotes an infinite product. For example, the Euler product formula for the Riemann zeta function is \textstyle\zeta(z) = \prod_{n=1}^{\infty} \frac{1}{1 - p_n^{-z}}.}} {{defn|Transpose: if is a matrix, A^\mathsf{T} denotes the transpose of , that is, the matrix obtained by exchanging rows and columns of . Notation ^\mathsf{T}\!\! A is also used.}} == Advanced group theory ==

{{defn|In group theory, G\wr H denotes the wreath product of the groups and . It is also denoted as G\operatorname{wr} H or G\operatorname{Wr} H; see for several notation variants.}} == Infinite numbers ==

Many types of bracket are used in mathematics. Their meanings depend not only on their shapes, but also on the nature and the arrangement of what is delimited by them, and sometimes what appears between or before them. For this reason, in the entry titles, the symbol is used as a placeholder for schematizing the syntax that underlies the meaning. Parentheses {{term|pmatrix|content=\begin{pmatrix} \Box & \cdots & \Box \\ \vdots & \ddots & \vdots \\ \Box & \cdots & \Box \end{pmatrix}}} {{term|binomial|content=\binom{\Box}{\Box}}} {{defn|Denotes a binomial coefficient: Given two nonnegative integers, \binom{n}{k} is read as " choose ", and is defined as the integer \frac{n(n-1)\cdots(n-k+1)}{1\cdot 2\cdots k}=\frac{n!}{k!\,(n-k)!} (if , its value is conventionally ). Using the left-hand-side expression, it denotes a polynomial in , and is thus defined and used for any real or complex value of .}} {{term|legendre|content=\left(\frac{\Box}{\Box}\right)}} {{defn|Legendre symbol: If is an odd prime number and is an integer, the value of \left(\frac{a}{p}\right) is if is a quadratic residue modulo ; it is if is a quadratic non-residue modulo ; it is if divides . The same notation is used for the Jacobi symbol and Kronecker symbol, which are generalizations where is respectively any odd positive integer, or any integer.}} Square brackets {{defn|no=5|In combinatorics or computer science, sometimes [n] with n\in\mathbb{N} denotes the set \{1,2,3,\ldots,n\} of positive integers up to , with [0]=\empty.}} {{defn|no=2|Commutator (group theory): if and belong to a group, then [a,b]=a^{-1}b^{-1}ab.}} {{term|bmatrix|content=\begin{bmatrix} \Box & \cdots & \Box \\ \vdots & \ddots & \vdots \\ \Box & \cdots & \Box \end{bmatrix}}} Braces {{defn|no=2|Set-builder notation for a singleton set: \{x\} denotes the set that has as a single element.}} {{defn|Set-builder notation: if P(x) is a predicate depending on a variable , then both \{x : P(x)\} and \{x\mid P(x)\} denote the set formed by the values of for which P(x) is true.}} {{defn|no=1|Used for emphasizing that several equations have to be considered as simultaneous equations; for example, \textstyle \begin{cases}2x+y=1\\3x-y=1\end{cases}.}} {{defn|no=2|Piecewise definition; for example, \textstyle |x|=\begin{cases}x&\text{if }x\ge 0\\-x&\text{if }x.}} {{defn|no=3|Used for grouped annotation of elements in a formula; for example, \textstyle \underbrace{ (a,b,\ldots,z) }_{26}, \textstyle \overbrace{ 1+2+\cdots+100 }^{=5050}, \textstyle \left.\begin{bmatrix}A\\B\end{bmatrix}\right\} m+n\text{ rows}}} Other brackets {{term|determinant|content=\textstyle\begin{vmatrix} \Box & \cdots & \Box \\ \vdots & \ddots & \vdots \\ \Box & \cdots & \Box \end{vmatrix}}} {{defn|\begin{vmatrix} x_{1,1} & \cdots & x_{1,n} \\ \vdots & \ddots & \vdots \\ x_{n,1} & \cdots & x_{n,n} \end{vmatrix} denotes the determinant of the square matrix \begin{bmatrix} x_{1,1} & \cdots & x_{1,n} \\ \vdots & \ddots & \vdots \\ x_{n,1} & \cdots & x_{n,n} \end{bmatrix}.}} {{defn|no=1|Generated object: if is a set of elements in an algebraic structure, \langle S \rangle denotes often the object generated by . If S=\{s_1,\ldots, s_n\}, one writes \langle s_1,\ldots, s_n \rangle (that is, braces are omitted). In particular, this may denote • the linear span in a vector space (also often denoted ), • the generated subgroup in a group, • the generated ideal in a ring, • the generated submodule in a module.}} {{term|bra–ket|content=\langle\Box| \text{ and } |\Box\rangle}} == Symbols that do not belong to formulas ==

In this section, the symbols that are listed are used as some sorts of punctuation marks in mathematical reasoning, or as abbreviations of natural language phrases. They are generally not used inside a formula. Some were used in classical logic for indicating the logical dependence between sentences written in plain language. Except for the first two, they are normally not used in printed mathematical texts since, for readability, it is generally recommended to have at least one word between two formulas. However, they are still used on a black board for indicating relationships between formulas. == Miscellaneous ==

{{defn|no=2|Parallel – the harmonic sum – an arithmetical operation used in electrical engineering for summing two impedances wired in parallel (e.g. parallel resistors) or two admittances wired in series (e.g. series capacitors): \ x \parallel y = \frac{ 1 }{\ \frac{\ 1\ }{ x } +\frac{\ 1\ }{ y }\ } = \frac{x\ y}{\ x + y\ } ~.}} {{defn|no=4|Concatenation: Typically used in computer science, x\mathbin{\vert\vert}y is said to represent the value resulting from appending the digits of to the end of .}} {{defn|no=5|{\displaystyle D_{\text{KL}}(P\parallel Q)}, denotes a statistical distance or measure of how one probability distribution P is different from a second, reference probability distribution Q.}} {{defn|Hadamard product of power series: if \textstyle S=\sum_{i=0}^\infty s_ix^i and \textstyle T=\sum_{i=0}^\infty t_ix^i, then \textstyle S\odot T=\sum_{i=0}^\infty s_i t_i x^i. Possibly, \odot is also used instead of #∘| for the Hadamard product of matrices.}} == See also ==