Amathematical symbol is a figure or a combination of figures that is used to represent amathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the othersymbols that occur in aformula or amathematical expression. More formally, amathematical symbol is anygrapheme used in mathematical formulas and expressions. As formulas and expressions are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.
The most basic symbols are thedecimal digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and the letters of theLatin alphabet. The decimal digits are used for representing numbers through theHindu–Arabic numeral system. Historically, upper-case letters were used for representingpoints in geometry, and lower-case letters were used forvariables andconstants. Letters are used for representing many other types ofmathematical object. As the number of these types has increased, theGreek alphabet and someHebrew letters have also come to be used. For more symbols, other typefaces are also used, mainlyboldface,script typeface (the lower-case script face is rarely used because of the possible confusion with the standard face),German fraktur, andblackboard bold (the other letters are rarely used in this face, or their use is unconventional). It is commonplace to use alphabets, fonts and typefaces to group symbols by type (for example, boldface is often used forvectors and uppercase formatrices).
The use of specific Latin and Greek letters as symbols for denoting mathematical objects is not described in this article. For such uses, seeVariable § Conventional variable names andList of mathematical constants. However, some symbols that are described here have the same shape as the letter from which they are derived, such as and.
These letters alone are not sufficient for the needs of mathematicians, and many other symbols are used. Some take their origin inpunctuation marks anddiacritics traditionally used intypography; others by deformingletter forms, as in the cases of and. Others, such as+ and=, were specially designed for mathematics.
Normally, entries of aglossary 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 longsection 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 theirsyntax, 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 is used for representing the neighboring parts of a formula that contains the symbol. See§ Brackets for examples of use.
Most symbols have two printed versions. They can be displayed asUnicode characters, or inLaTeX format. With the Unicode version, usingsearch engines andcopy-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 ananchor, 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.
1. Denotesaddition and is read asplus; for example,3 + 2.
2. Denotes that a number ispositive and is read asplus. Redundant, but sometimes used for emphasizing that a number ispositive, specially when other numbers in the context are or may be negative; for example,+2.
3. Placeholder used for replacing an indeterminate element. For example, saying "theabsolute value is denoted by| · |" is perhaps clearer than saying that it is denoted as| |.
Widely used for denotingdivision in Anglophone countries, it is no longer in common use in mathematics and its use is "not recommended".[1] In some countries, it can indicate subtraction.
Denotessquare root and is read asthe square root of. Rarely used in modern mathematics without a horizontal bar delimiting the width of its argument (see the next item). For example,√2.
Any of these is sometimes used for naming amathematical object. Thus, and are each an abbreviation of the phrase "let", where is anexpression and is avariable.This is similar to the concept ofassignment in computer science, which is variously denoted (depending on theprogramming language used)
3. Between twogroups, may mean that the second one is aproper subgroup of the first one.
≤
1. Means "less than or equal to". That is, whateverA andB are,A ≤B is equivalent toA <B orA =B.
2. Between twogroups, may mean that the first one is asubgroup of the second one.
≥
1. Means "greater than or equal to". That is, whateverA andB are,A ≥B is equivalent toA >B orA =B.
2. Between twogroups, may mean that the second one is asubgroup of the first one.
1. Means "much less than" and "much greater than". Generally,much is not formally defined, but means that the lesser quantity can be neglected with respect to the other. This is generally the case when the lesser quantity is smaller than the other by one or severalorders of magnitude.
2. Inmeasure theory, means that the measure is absolutely continuous with respect to the measure.
A rarely used symbol, generally a synonym of≤.
Often used for denoting anorder or, more generally, apreorder, when it would be confusing or not convenient to use< and>.
Symmetric difference: that is, or is the set formed by the elements that belong to exactly one of the two setsA andB.
1. With a subscript, denotes aset complement: that is, if, then.
2. Without a subscript, denotes theabsolute complement; that is,, whereU is a set implicitly defined by the context, which contains all sets under consideration. This setU is sometimes called theuniverse of discourse.
3. Incategory theory, denotes thedirect product (often called simplyproduct) of two objects, which is a generalization of the preceding concepts of product.
Denotes thedisjoint union. That is, ifA andB are sets then is a set ofpairs whereiA andiB are distinct indices discriminating the members ofA andB in.
1. Used for thedisjoint union of a family of sets, such as in
Denoteslogical negation, and is read as "not". IfE is alogical predicate, is the predicate that evaluates totrue if and only ifE evaluates tofalse. For clarity, it is often replaced by the word "not". Inprogramming languages and some mathematical texts, it is sometimes replaced by "~" or "!", which are easier to type on some keyboards.
1. Denotes thelogical or, and is read as "or". IfE andF arelogical predicates, is true if eitherE,F, or both are true. It is often replaced by the word "or".
1. Denotes thelogical and, and is read as "and". IfE andF arelogical predicates, is true ifE andF are both true. It is often replaced by the word "and" or the symbol "&".
2. Often used in plain text as an abbreviation of "for all" or "for every".
∃
1. Denotesexistential quantification and is read "there exists ... such that". IfE is alogical predicate, means that there exists at least one value ofx for whichE is true.
2. Often used in plain text as an abbreviation of "there exists".
∃!
Denotesuniqueness quantification, that is, means "there exists exactly onex such thatP (is true)". In other words, is an abbreviation of.
⇒
1. Denotesmaterial conditional, and is read as "implies". IfP andQ arelogical predicates, means that ifP is true, thenQ is also true. Thus, is logically equivalent with.
2. Often used in plain text as an abbreviation of "implies".
Theblackboard boldtypeface is widely used for denoting the basicnumber 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 incombinatorics, one should immediately know that this denotes thereal numbers, although combinatorics does not study the real numbers (but it uses them for many proofs).
Denotes the set ofnatural numbers or sometimes When the distinction is important and readers might assume either definition, and are used, respectively, to denote one of them unambiguously. Notation is also commonly used.
Denotes the set ofintegers It is often denoted also by
Newton's notation, most commonly used for thederivative with respect to time. Ifx is a variable depending on time, then read as "x dot", is its derivative with respect to time. In particular, ifx represents a moving point, then is itsvelocity.
1. Ify is a variable thatdepends onx, then, read as "d y over d x" (commonly shortened to "d y d x"), is the derivative ofy with respect tox.
2. Iff is afunction of a single variablex, then is the derivative off, and is the value of the derivative ata.
3. Total derivative: If is afunction of several variables thatdepend onx, then is the derivative off considered as a function ofx. That is,.
∂ □/∂ □
Partial derivative: If is afunction of several variables, is the derivative with respect to theith variable considered as anindependent variable, the other variables being considered as constants.
4. Mean value: Ifx is avariable that takes its values in some sequence of numbersS, then may denote the mean of the elements ofS.
5. Negation: Sometimes used to denote negation of the entire expression under the bar, particularly when dealing withBoolean algebra. For example, one ofDe Morgan's laws says that .
→
1. denotes afunction withdomainA andcodomainB. For naming such a function, one writes, which is read as "f fromA toB".
3. May denote alogical implication. For thematerial implication that is widely used in mathematics reasoning, it is nowadays generally replaced by⇒. Inmathematical logic, it remains used for denoting implication, but its exact meaning depends on the specific theory that is studied.
4. Over avariable name, means that the variable represents avector, in a context where ordinary variables representscalars; for example,. Boldface () or acircumflex () are often used for the same purpose.
(Note: the notation is not recommended for the four-gradient since both and are used to denote thed'Alembertian; see below.)
Quad, the 4-vector gradient operator orfour-gradient,.
or
(here an actual box, not a placeholder)
Denotes thed'Alembertian or squaredfour-gradient, which is a generalization of theLaplacian to four-dimensional spacetime. In flat spacetime with Euclidean coordinates, this may mean either or; the sign convention must be specified. In curved spacetime (or flat spacetime with non-Euclidean coordinates), the definition is more complicated. Also calledbox orquabla.
1. Denotes thesum of a finite number of terms, which are determined by subscripts and superscripts (which can also be placed below and above), such as in or.
1. Denotes theproduct of a finite number of terms, which are determined by subscripts and superscripts (which can also be placed below and above), such as in or.
1. Internaldirect sum: ifE andF are abelian subgroups of anabelian groupV, notation means thatV is the direct sum ofE andF; that is, every element ofV can be written in a unique way as the sum of an element ofE and an element ofF. This applies also whenE andF arelinear subspaces orsubmodules of thevector space ormoduleV.
2. Direct sum: ifE andF are twoabelian groups,vector spaces, ormodules, then their direct sum, denoted is an abelian group, vector space, or module (respectively) equipped with twomonomorphisms and such that is the internal direct sum of and. This definition makes sense because this direct sum is unique up to a uniqueisomorphism.
1. Transpose: ifA is a matrix, denotes thetranspose ofA, that is, the matrix obtained by exchanging rows and columns ofA. Notation is also used. The symbol is often replaced by the letterT ort.
Normal subgroup of and normal subgroup of including equality, respectively. IfN andG are groups such thatN is a normal subgroup of (including equality)G, this is written.
⋉ ⋊
1. Innersemidirect product: ifN andH are subgroups of agroupG, such thatN is anormal subgroup ofG, then and mean thatG is the semidirect product ofN andH, that is, that every element ofG can be uniquely decomposed as the product of an element ofN and an element ofH. (Unlike for thedirect product of groups, the element ofH may change if the order of the factors is changed.)
1. The symbol is read asinfinity. As an upper bound of asummation, aninfinite product, anintegral, etc., means that the computation is unlimited. Similarly, in a lower bound means that the computation is not limited toward negative values.
With anordinali as a subscript, denotes theithaleph number, that is theith infinitecardinal. For example, is the smallest infinite cardinal, that is, the cardinal of the natural numbers.
Many types ofbracket 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.
Used in anexpression for specifying that the sub-expression between the parentheses has to be considered as a single entity; typically used for specifying theorder of operations.
□(□) □(□, □) □(□, ..., □)
1. Functional notation: if the first is the name (symbol) of afunction, denotes the value of the function applied to the expression between the parentheses; for example,,. In the case of amultivariate function, the parentheses contain several expressions separated by commas, such as.
2. May also denote a product, such as in. When the confusion is possible, the context must distinguish which symbols denote functions, and which ones denotevariables.
Denotes abinomial coefficient: Given twononnegative integers, is read as "n choosek", and is defined as the integer (ifk = 0, its value is conventionally1). Using the left-hand-side expression, it denotes apolynomial inn, and is thus defined and used for anyreal orcomplex value ofn.
4. Iverson bracket: ifP is apredicate, may denote the Iverson bracket, that is thefunction that takes the value1 for the values of thefree variables inP for whichP is true, and takes the value0 otherwise. For example, is theKronecker delta function, which equals one if, and zero otherwise.
5. In combinatorics or computer science, sometimes with denotes the set of positive integers up ton, with.
2. Number of elements: IfS is aset, may denote itscardinality, that is, its number of elements. is also often used, see#.
3. Length of aline segment: IfP andQ are two points in aEuclidean space, then often denotes the length of the line segment that they define, which is thedistance fromP toQ, and is often denoted.
1. Generated object: ifS is a set of elements in an algebraic structure, denotes often the object generated byS. If, one writes (that is, braces are omitted). In particular, this may denote
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 inclassical 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 ablack board for indicating relationships between formulas.
Used for marking the end of a proof and separating it from the current text. TheinitialismQ.E.D. or QED (Latin:quod erat demonstrandum, "as was to be shown") is often used for the same purpose, either in its upper-case form or in lower case.
Bourbaki dangerous bend symbol: Sometimes used in the margin to forewarn readers against serious errors, where they risk falling, or to mark a passage that is tricky on a first reading because of an especially subtle argument.
Abbreviation of "therefore". Placed between two assertions, it means that the first one implies the second one. For example: "All humans are mortal, and Socrates is a human. ∴ Socrates is mortal."
Abbreviation of "because" or "since". Placed between two assertions, it means that the first one is implied by the second one. For example: "11 isprime ∵ it has no positive integer factors other than itself and one."
∋
1. Abbreviation of "such that". For example, is normally printed "x such that".
2. Sometimes used for reversing the operands of; that is, has the same meaning as. See∈ in§ Set theory.
1. Factorial: ifn is apositive integer,n! is the product of the firstn positive integers, and is read as "n factorial".
2. Double factorial: ifn is apositive integer,n!! is the product of all positive integers up ton with the same parity asn; that is, ifn is odd, the product of all odd integers from 1 up to and includingn, and ifn is even, the product of all even integers, up to and includingn. It is read as "the double factorial ofn".
3. Subfactorial: ifn is a positive integer,!n is the number ofderangements of a set ofn elements, and is read as "the subfactorial of n".
^TheLaTeX equivalent to bothUnicode symbols ∘ and ○ is \circ. The Unicode symbol that has the same size as \circ depends on the browser and its implementation. In some cases ∘ is so small that it can be confused with aninterpoint, and ○ looks similar as \circ. In other cases, ○ is too large for denoting a binary operation, and it is ∘ that looks like \circ. As LaTeX is commonly considered as the standard for mathematical typography, and it does not distinguish these two Unicode symbols, they are considered here as having the same mathematical meaning.
^Rutherford, D. E. (1965).Vector Methods. University Mathematical Texts. Oliver and Boyd Ltd., Edinburgh.
