A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.
The most basic symbols are the decimal digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and the letters of the Latin alphabet. The decimal digits are used for representing numbers through the Hindu–Arabic numeral system. Historically, upper-case letters were used for representing points in geometry, and lower-case letters were used for variables and constants. Letters are used for representing many other sorts of mathematical objects. As the number of these sorts has remarkably increased in modern mathematics, the Greek alphabet and some Hebrew letters are also used. In mathematical formulas, the standard typeface is italic type for Latin letters and lower-case Greek letters, and upright type for upper case Greek letters. For having more symbols, other typefaces are also used, mainly boldface, script typeface (the lower-case script face is rarely used because of the possible confusion with the standard face), German fraktur, and blackboard bold (the other letters are rarely used in this face, or their use is unconventional).
These letters alone are not sufficient for the needs of mathematicians, and many other symbols are used. Some take their origin in punctuation marks and diacritics traditionally used in typography; others by deforming letter forms, as in the cases of and . Others, such as + and =, were specially designed for mathematics.
Layout of this article
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 is related the symbol that they are looking for, 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 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.
1. Denotes addition and is read as plus; for example, 3 + 2.
2. Denotes that a number is positive and is read as plus. Redundant, but sometimes used for emphasizing that a number is positive, specially when other numbers in the context are or may be negative; for example, +2.
2. Used for naming a mathematical object in a sentence like "let ", where E is an expression. On a blackboard and in some mathematical texts, this may be abbreviated as or This is related to the concept of assignment in computer science, which is variously denoted (depending on the programming language used)
1. Means "less than or equal to". That is, whatever A and B are, A ≤ B is equivalent to A < B or A = B.
2. Between two groups, may mean that the first one is a subgroup of the second one.
1. Means "greater than or equal to". That is, whatever A and B are, A ≥ B is equivalent to A > B or A = B.
2. Between two groups, may mean that the second one is a subgroup 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 several orders of magnitude.
2. In measure theory, means that the measure is absolutely continuous with respect to the measure .
1. A rarely used symbol, generally used as a synonym of ≤.
≺ , ≻
1. Often used for denoting an order or, more generally, a preorder, when it would be confusing or not convenient to use < and >.
2. Without a subscript, denotes the absolute complement; that is, , where U is a set implicitly defined by the context, which contains all sets under consideration. This set U is sometimes called the universe of discourse.
Denotes logical negation, and is read as "not". If E is a logical predicate, is the predicate that evaluates to true if and only if E evaluates to false. For clarity, it is often replaced by the word "not". In programming languages and some mathematical texts, it is sometimes replaced by "~" or "!", which are easier to type on some keyboards.
The blackboard boldtypeface 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 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).
Denotes the set of natural numbers, or sometimes . It is often denoted also by . When the distinction is important and readers might assume either definition, and are used, respectively, to denote one of them unambiguously.
Denotes the set of integers. It is often denoted also by .
Newton's notation, most commonly used for the derivative with respect to time: If x is a variable depending on time, then is its derivative with respect to time. In particular, if x represents a moving point, then is its velocity.
4. Over a variable name, means that the variable represents a vector, in a context where ordinary variables represent scalars; for example, . Boldface () or a circumflex () are often used for the same purpose.
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 or ; 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.
Denotes the tensor product. If E and F are abelian groups, vector spaces, or modules over a commutative ring, then the tensor product of E and F, denoted is an abelian group, a vector space or a module (respectively), equipped with a bilinear map from to , such that the bilinear maps from to any abelian group, vector space or module G can be identified with the linear maps from to G. If E and F are vector spaces over a fieldR, or modules over a ring R, the tensor product is often denoted to avoid ambiguity.
1. Transpose: if A is a matrix, denotes the transpose of A, that is, the matrix obtained by exchanging rows and columns of A. Notation is also used. The symbol is often replaced by the letter T or t.
1. Inner semidirect product: if N and H are subgroups of a groupG, such that N is a normal subgroup of G, then and mean that G is the semidirect product of N and H, that is, that every element of G can be uniquely decomposed as the product of an element of N and an element of H. (Unlike for the direct product of groups, the element of H may change if the order of the factors is changed.)
1. The symbol is read as infinity. As an upper bound of a summation, an infinite product, an integral, etc., means that the computation is unlimited. Similarly, in a lower bound means that the computation is not limited toward negative values.
Many sorts of brackets 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 an expression for specifying that the sub-expression between the parentheses has to be considered as a single entity; typically used for specifying the order of operations.
□(□) □(□, □) □(□, ..., □)
1. Functional notation: if the first is the name (symbol) of a function, denotes the value of the function applied to the expression between the parentheses; for example, , . In the case of a multivariate 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 denote variables.
2. Number of elements: If S is a set, may denote its cardinality, that is, its number of elements. is also often used, see #.
3. Length of a line segment: If P and Q are two points in a Euclidean space, then often denotes the length of the line segment that they define, which is the distance from P to Q, and is often denoted .
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.
Used for marking the end of a proof and separating it from the current text. The initialismQ.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 "because" or "since". Placed between two assertions, it means that the first one is implied by the second one. For example: "11 is prime ∵ 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.
^The LaTeX equivalent to both Unicode 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 an interpoint, 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.