In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition (while being unlikely to introduce errors or cause confusion). Abusing notation should be contrasted with "misusing" notation which should be avoided. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Mathematical notation is used in mathematics, and throughout the physical sciences, engineering, and economics. ...

Common examples occur when speaking of compound mathematical objects. For example, a topological space consists of a set T and a topology , and two topological spaces and can be quite different if they have different topologies. Nevertheless, it is common to refer to such a space simply as T when there is no danger of confusion or when it is implicitly clear what topology is being considered. Similarly, one often refers to a group as simply G when the group operation is clear from context. Another example is in the Leibniz notation for the derivative . Although the derivative is not strictly a fraction, abusing this notation leads to the correct chain rule . Often good notation is judged by whether or not its abuses will lead to correct interpretations. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... This picture illustrates how the hours on a clock form a group under modular addition. ... Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... In calculus, the Leibniz notation, named in honor of the 17th century German philosopher and mathematician Gottfried Wilhelm Leibniz (pronounced LIBE nits) was originally the use of dx and dy and so forth to represent infinitely small increments of quantities x and y, just as Δx and Δy represent finite... For a non-technical overview of the subject, see Calculus. ...

The new use may achieve clarity in the new area in an unexpected way, but it may borrow arguments from the old area that do not carry over, creating a false analogy. This article does not cite any references or sources. ...

Abuse of language is an almost synonymous expression that is usually used for non-notational abuses. For example, while the word representation properly designates a group homomorphism from a group G to GL(V) where V is a vector space, it is common to call V "a representation of G." Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...

Examples

John Harrison cites "the use of f(x) to represent both application of a function f to an argument x, and the image under f of a subset, x, of f's domain".

The computation of the vector product as the determinant of the matrix In mathematics, the cross product is a binary operation on vectors in three dimensions. ... For the square matrix section, see square matrix. ...

is a significant abuse of notation as are treated as scalars but are in fact vectors. In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ... A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ...

With Big O notation, we say that some function f "is" O(g(x)) (given some function g, where x is one of f's parameters). Intuitively this notation groups functions according to their growth respective to some parameter; as such, it would be appropriate to use the set membership notation and say that . However, the usual notation is f = O(g(x)), despite the fact that the implied relationship is not symmetric (which the symbol "=" would imply). One reason for this is that, as an extension of the abuse, it is useful to overload relation symbols such as < and ≤, such that, for example, f < O(g(x)) means that f's real growth is less than g(x). But this further abuse is not necessary if, following Knuth one distinguishes between O and the closely related o and Θ notations. For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ... Donald Ervin Knuth ( or Ka-NOOTH[1], Chinese: [2]) (b. ... For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ...

Another common abuse of notation is to blur the distinction between equality and isomorphism. In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...

A very common abuse of notation, which some may think is formal because of its frequent use,^{[citation needed]} is writing "infinite limits" as

When the limit is infinite then there exists no and, formally, can not be defined.

Quotation

"We will occasionally use this arrow notation unless there is no danger of confusion."

(Ronald L. Graham, Rudiments of Ramsey Theory) Ronald L. Graham (born October 31, 1935) is a mathematician credited by the American Mathematical Society with being one of the principal architects of the rapid development worldwide of discrete mathematics in recent years[1]. He has done important work in scheduling theory, computational geometry, Ramsey theory, and quasi-randomness. ... Ramsey theory, named for Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear. ...

See also

• Mathematical notation

Mathematical notation is used in mathematics, and throughout the physical sciences, engineering, and economics. ...

External links

• Section 2.2: Criticism and reconstruction from "Formalized Mathematics", by John Harrison
• "Strong Symbols", by Henning Thielemann (PDF Slides) Section 5: Common abuse of notation